Heterogeneous Community Surveillance–Driven Physics-Informed Reformulation of Fine-Scale Convection–Diffusion Air Pollution Distribution

1  Introduction

Air quality has become a critical factor in efforts to enhance living conditions in urban environments. Consequently, many municipalities have introduced policies aimed at reducing pollution originating from industrial activities and road traffic. Numerous studies have demonstrated that air quality has a direct impact on human health, being strongly associated with serious respiratory and cardiovascular diseases such as asthma, lung cancer, stroke, and lower respiratory infections [1]. The World Health Organization (WHO) estimates that around seven million premature deaths occur annually due to air pollution, with 91% of the global population living in areas that do not meet air quality guidelines [2]. Given these severe health impacts, fine-grained air quality monitoring has become a major global priority. Access to block-level pollution data enables urban residents to better plan outdoor activities and reduce exposure risks, while for municipal authorities, accurate detection of airborne pollutants supports effective air quality control and more efficient city management [3]. Air quality predictions assisted by mobility data have been shown to reduce health impacts from pollution [4].

Urban air pollution is currently monitored using both stationary sensors and mobile devices. Although official monitoring stations provide highly accurate data and are widely used for routine assessments in many countries, they are costly, bulky, and resource-intensive. Consequently, their deployment density is insufficient to capture fine-grained pollution patterns. Over the past decade, mobile sensing platforms have emerged as a complementary approach [5–7]. These systems employ low-cost sensors carried by humans or vehicles, enabling broader coverage and finer spatial resolution. While mobility-based monitoring improves the ability to capture localized variations, the collected samples cannot fully represent the entire spatiotemporal domain. Thus, inference algorithms remain essential for reconstructing fine-grained pollution fields and providing a comprehensive view of the urban environment.

Deterministic frameworks representing physical and chemical processes have long underpinned conventional air quality modeling [8,9]. In parallel, the widespread adoption of Smart City initiatives has driven a rapid expansion in the deployment of sensors and monitoring networks, with air quality monitoring emerging as one of the most prominent applications in these systems [10]. Recent studies increasingly exploit data collected in smart cities for air quality monitoring, particularly through the use of machine learning techniques to predict the spatiotemporal evolution of air pollution.

Traditional regression models, such as Linear Regression and Support Vector Machines, as well as classification approaches including Decision Trees and Random Forests, are progressively being superseded by deep learning methods based on neural networks (NNs). Among these, physics-informed NNs (PINNs) have gained considerable attention due to their ability to integrate observational data with governing physical laws [11,12]. Unlike convolutional NNs (CNNs), which are designed for regularly structured data, PINNs incorporate physics-based constraints directly into the learning process, enabling physically consistent modeling of complex spatial and temporal dynamics. This property is particularly advantageous for urban air quality monitoring, where pollution processes are governed by atmospheric transport mechanisms and heterogeneous emission sources. By embedding physical laws into the loss function, PINN-based models have demonstrated improved predictive accuracy and enhanced physical consistency in air pollution forecasting [13,14].

However, physics-based simulations and PINN frameworks can be computationally expensive and often require substantial prior knowledge, such as accurate initial conditions and emission inventories. To alleviate these limitations, low-cost air quality sensors mounted on mobile carriers, including vehicles and pedestrians, are increasingly deployed to extend monitoring coverage and improve spatial resolution at a manageable cost [15,16]. Concurrently, data-driven deep learning (DL) approaches have gained popularity due to the growing availability of empirical air quality data [17,18]. Despite their strong predictive capability, DL models are frequently criticized as “black boxes” because of their limited interpretability and weak linkage to physical mechanisms. Moreover, the continuous movement of mobile sensors results in uneven spatiotemporal data distributions, making it difficult to capture complete neighborhood observations. As a result, accurately modeling spatiotemporal dependencies and generating reliable pollution visualizations remains a significant challenge [19].

In the present problem, two major challenges are evident:

(i) Despite regulatory efforts, PM2.5 and PM10 levels in Pakistan still frequently exceed WHO health standards, especially in urban and industrial regions.

(ii) Evidence suggests rising O3 concentrations in major cities due to vehicular emissions and industrial activities.

(iii) Accurate spatiotemporal mapping of O3, PM2.5 and PM10 is essential for targeted pollution control policies.

(iv) Modeling challenges include:

• Irregular data from mobile sensing: Unevenly distributed air quality measurements complicate capturing spatiotemporal dependencies.

• Non-interpretability of DL techniques: Traditional DL models without physical grounding often fail to extrapolate reliably.

(v) Phy-APMR innovations are as follows:

• Categorizes inputs based on atmospheric knowledge.

• Uses tanh-based self-attention to model differential input impact.

• Deep interaction module captures complex physicochemical pollutant interactions.

• Hierarchical physics-constrained loss improves model robustness.

(vi) Adaptive short-time update sampling (ASUS) allows analysis of predictor contributions and identification of key pollution drivers.

(vii) Spatiotemporal distributions of O3, PM2.5 and PM10 can be effectively mapped across Pakistan, supporting reliable policy-making and environmental management.

Fine-grained Phy-APMR therefore remains essential, as many regions are still unmonitored despite the broader coverage enabled by mobile sensors. As illustrated in Fig. 1, wireless sensors mounted on automobiles collect atmospheric contaminant measurements, which are then used to develop a reconstruction framework that estimates pollutant concentrations in unobserved areas. In recent years, the design of highly efficient reconstruction algorithms has become a central focus of research.

Figure 1: A demonstration of the fine-grained Phy-APMR procedure. The reconstruction model predicts pollution levels and generates the corresponding visualization, while mobile sensors enhance sensing coverage.

Reconstruction algorithms are generally developed using two principal methodologies: data-driven strategies and physics-inspired approaches [20]. Physics-based methods typically rely on theoretical diffusion models [21] to estimate air quality; however, they often perform poorly because they cannot fully capture the complex dynamics of real-world pollution forecasting. In contrast, evidence-based techniques, such as probabilistic regression with Gaussian processes [22] and NNs [23,24], aim to learn diffusion patterns directly from observed data. When sufficient observations are available, data-driven methods particularly those based on DL have demonstrated clear advantages over physics-informed approaches. Nonetheless, data-driven methods still face two major limitations:

(a1): In certain temporal and spatial domains, mobile sensing data are sparse. Data collection is often uneven, leaving gaps across both time and space due to the uncontrollable mobility of portable sensors (see Fig. 2). The performance of analytical methods depends strongly on the quantity and representativeness of the collected samples, that is, whether a sufficiently diverse range of conditions is covered. When sampling is inadequate, models fail to generalize effectively to sparsely observed regions.

Figure 2: A representation of the limited availability of mobile sensing data across spatiotemporal domains. The green backdrop indicates that, during daytime, urban sensing vehicles tend to concentrate in busy areas of the city (such as central and commercial zones). The light-pink backdrop highlights this tendency, which leads to reduced sensing coverage in other areas and times (such as residential neighborhoods and nighttime).

(a2): Relearning and rebuilding processes, which are often time-consuming, present an additional challenge. Continuous integration of new information is essential to adapt mobility sensor–driven reconstruction frameworks and to reflect the evolving characteristics of urban air contaminants. However, the large volume of training data required, combined with the complexity of advanced intelligence-driven architectures particularly DL models imposes significant computational demands. This problem is further compounded by the limited processing capacity of portable devices.

To address the challenges of accurately estimating air pollutant concentrations in Pakistan during 2023–2024, this study proposes a hybrid framework that integrates DL with physics-informed modeling. The framework consists of the Phy-APMR reconstruction model, supported by the ASUS module for efficient training and updates. In Phy-APMR, model inputs are categorized based on prior atmospheric knowledge. A tanh-based self-attention mechanism dynamically adjusts the contribution of each input to different pollutants, while a deep interaction module captures complex physicochemical interactions. A physics-constrained loss function ensures predictions are consistent with physical laws.

Phy-APMR is designed for reliable reconstruction of air quality in regions with sparse or missing data. In areas with sensors, the model aligns predictions with observed data, while in unmonitored regions, it enforces physical principles governing pollutant dispersion. Its architecture integrates a NN for inferential computation and a PINN to encode domain knowledge, typically expressed through partial differential equations (PDEs), ensuring robust reconstruction even in data-scarce locations.

To improve training efficiency, the ASUS module selects representative collocation points through efficiency-driven sampling, coverage-oriented sampling, and periodic temporal updates. This strategy accelerates convergence, reduces computational costs, and maintains high inference accuracy. The proposed framework was validated using two datasets. First, a large-scale deployment dataset collected between 2023 and 2024 from 78 fixed detectors across Lahore, Faisalabad, and Karachi, comprising over 15 million observations, was used for model development and large-scale performance analysis. Second, a controlled 10-day three-city evaluation campaign involving approximately 70 active detectors and a substantially smaller sample subset was conducted to assess short-term generalization and convergence behavior. Across both settings, Phy-APMR outperforms state-of-the-art benchmarks by 16% in predictive accuracy. In addition, ASUS reduces training completion time by 84% relative to baseline sampling strategies, without compromising reliability.

The core contributions of this framework are summarized below:

(i) Phy-APMR as the first hybrid PINN-based technique for reconstructing ambient air quality across spatially and temporally sparse regions in Pakistan.

(ii) An enhanced Phy-APMR version incorporating wind speed and additional ecological variables, improving reconstruction accuracy.

(iii) The ASUS methodology, which accelerates training while preserving robustness.

(iv) Comprehensive evaluation across three major metropolitan areas in Pakistan, demonstrating significant improvements in both accuracy and efficiency.

Together, the Phy-APMR + ASUS framework provides a robust, interpretable, and efficient solution for simultaneous estimation and reconstruction of multiple air pollutants in Pakistan, offering actionable insights for coordinated air pollution control and environmental management during 2023–2024.

2  Data Source

The datasets employed in this study cover in-situ measurements, outputs from remote sensing devices, and reanalysis data across Pakistan. To harmonize these datasets, a standard grid covering the study region was established with a spatial resolution of 0.05∘×0.05∘.

2.1 In-Situ Measurements

Real-time ground-based measurements of O3, PM2.5, and PM10 from 2023 to 2024 were collected from 1194 monitoring sites across Pakistan (Fig. 3). These sites are operated by the Pakistan Environmental Protection Agency (Pak-EPA) and affiliated provincial monitoring networks. The mean daily maximum 8-h average O3 concentrations and daily mean PM2.5 and PM10 concentrations were computed after removing invalid or anomalous readings caused by instrument errors. For grid cells containing multiple monitoring sites, the average of all sites within the cell was used to represent the cell-level value.

Figure 3: Distribution of in-situ air quality monitoring sites across Pakistan.

2.2 Geospatial Remote Sensing Data

Satellite-derived data from the Tropospheric Monitoring Instrument (TROPOMI) and the Moderate Resolution Imaging Spectroradiometer (MODIS) spanning 2023–2024 were used to provide essential information for joint air-quality estimation across Pakistan. TROPOMI offers advanced atmospheric detection capabilities with near-global coverage at a spatial resolution of 3.5 km2 × 5.5 km2 (https://sentinels.copernicus.eu/). Daily Level 3 (L3) products including total column O3 (S5P_O3), formaldehyde (S5P_HCHO), nitrogen dioxide (S5P_NO2), carbon monoxide (S5P_CO), and the absorbing aerosol index (S5P_AAI) were retrieved via the Google Earth Engine (GEE) platform. These datasets provide information on O3, PM2.5, and PM10, as well as key precursor gases involved in their formation and transformation. The L3 products were pre-processed by GEE with quality control measures addressing cloud interference and sensor anomalies; thus, no additional quality filtering was applied, and non-random missing data were not specifically treated in this study. All products were resampled to a 0.05° spatial resolution using bilinear interpolation to align with the standard analysis grid.

Aerosol Optical Depth (AOD) is a key proxy for PM2.5 and PM10 concentrations. MODIS-derived daily 1 km AOD products, generated using the Multi-Angle Implementation of Atmospheric Correction (MAIAC) algorithm, were obtained from NASA (https://ladsweb.modaps.eosdis.nasa.gov/). MAIAC AOD values greater than 3 were excluded based on the reported valid range (Khan et al. [25]). Missing AOD values due to cloud contamination were reconstructed following the approach detailed in our previous work (Bilal et al. [26]), with full methodology provided in the Supporting Information. These AOD datasets were similarly resampled to 0.05° resolution to maintain consistency with the TROPOMI products and the standard grid.

2.3 Reconstructed Climate/Air-Quality Data

For 2023–2024, meteorological and radiation data relevant to O3 and particulate matter, with a spatial resolution of 0.25°, were obtained from the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA5, https://cds.climate.copernicus.eu/) to complement measurements collected by wireless sensor vehicles across Pakistan. The variables include hourly boundary layer height (BLH), relative humidity (RH), surface pressure (SP), surface net solar radiation (SSR), surface net thermal radiation (STR), downward UV radiation at the surface (UVB), 2-m temperature (T2M), 10-m U wind (U10), 10-m V wind (V10), and surface O3 mass mixing ratio (GO3).

In addition, atmospheric composition data from the Copernicus Atmosphere Monitoring Service (CAMS, https://ads.atmosphere.copernicus.eu/), with a spatial resolution of 0.75° and a temporal resolution of 3 h, were collected. These datasets include surface concentrations of CO (CAMS_CO), ethane (CAMS_C2H6), formaldehyde (CAMS_HCHO), hydrogen peroxide (CAMS_H2O2), hydroxyl radical (CAMS_OH), isoprene (CAMS_C5H8), nitrogen dioxide (CAMS_NO2), nitrogen monoxide (CAMS_NO), peroxyacetyl nitrate (CAMS_PAN), propane (CAMS_C3H8), sulfur dioxide (CAMS_SO2), aerosol optical depths for black carbon (CAMS_BCAOD), dust (CAMS_DUAOD), sulfate (CAMS_SUAOD), and sea salt (CAMS_SSAOD), and surface PM1, PM2.5, and PM10 concentrations.

All reanalysis and CAMS datasets were averaged to daily means and resampled to a 0.05° spatial resolution using bilinear interpolation to align with the standard grid used for the wireless sensor vehicles. This harmonization ensures consistency across datasets, enabling robust joint modeling and comprehensive spatial analysis of air quality.

3  Methods

3.1 Classified Factors

Based on known relationships among O3, PM2.5, and PM10, the collected datasets for 2023–2024 in Pakistan were classified into four factor categories: O3-independent factors, PM-independent factors, shared factors, and interacting factors. Radiation strongly influences O3 formation by driving photochemical reactions, while it has minimal direct impact on PM generation (see Liu et al. [27]). Accordingly, O3-independent factors included radiation variables (SSR, STR, UVB), reanalyzed CAMS_GO3, and satellite-derived S5P_O3.

PM-independent factors consisted of AOD-related data, which represent the atmospheric extinction from surface to space, including MODIS AOD, S5P_AAI, and reanalyzed aerosol components (BCAOD, DUAOD, SUAOD, SSAOD), as well as CAMS surface PM1, PM2.5, and PM10. Shared factors comprised meteorological variables (U10, V10, T2M, RH, BLH, SP) and spatiotemporal indicators (longitude, latitude, and day of the year), which are commonly used in models for both O3 and PM and significantly influence pollutant transport and formation (see Souza et al. [28]).

Interacting factors included various precursor gases (S5P_HCHO, S5P_NO2, S5P_CO, CAMS_CO, CAMS_C2H6, CAMS_HCHO, CAMS_C5H8, CAMS_NO2, CAMS_NO, CAMS_PAN, CAMS_C3H8, CAMS_SO2) and oxidants (CAMS_H2O2, CAMS_OH), reflecting the chemical interactions that affect the formation and transformation of O3 and PM in the atmosphere (see Ma et al. [29]).

3.2 Air Pollution Mapping Model

The fine-grained Phy-APMR approach for Pakistan was developed to jointly estimate O3, PM2.5, and PM10 concentrations (Fig. 4). The categorized factors were initially transformed into high-dimensional abstract representations through a feature encoder. To address the data sparsity issue, an ASUS strategy was integrated with a cascade architecture consisting of an Attention module and an Interaction module, enabling the model to capture complex interactions among meteorological conditions, pollutant precursors, and air quality levels. Leveraging prior physical knowledge, a hierarchical physics-constrained loss function was applied to guide the estimations. Finally, the model outputs were interpreted to provide insights into the contributions of each predictor, supporting transparent and explainable air quality mapping across Pakistan.

Figure 4: Schematic diagram of the proposed fine-grained Phy-APMR model.

3.3 Feature Encoder (FE)

The deep NN-based FE was developed to transform the categorized factors into high-dimensional abstract representations. A typical deep NN comprises an input layer, multiple hidden layers, and an output layer, with neurons transmitting information through dense connections and performing nonlinear transformations via activation functions. Due to their strong representation learning capabilities and adaptability, deep NNs have achieved considerable success across numerous domains, including wireless sensor vehicles and related mobile sensing applications [30].

Since the primary goal of the FE was to map inputs into high-dimensional feature spaces, the standard including wireless sensor vehicles and related mobile sensing applications output layer was removed. In the fine-grained Phy-APMR framework, we implemented four feature encoders with identical architectures: FE(PM), FE(O3), FE(Share), and FE(Int). The computations in each layer can be expressed as:

Yl={ℏ(θlX+bl),l=1 (input layer)ℏ(θlYl−1+bl),l=2,…,L (hidden layers)

where X denotes the input, Yl is the output of the l-th layer, θl and bl are the weights and biases of the l-th layer, L is the total number of layers, and ℏ(⋅) represents the leaky Rectified Linear Unit (leakyReLU) activation function. LeakyReLU is a variant of the standard ReLU that mitigates the problem of neurons becoming inactive and failing to update their weights [31].

Each FE used three consecutive hidden layers with 512 neurons per layer, a configuration selected via grid search (see Supporting Information (S3) for details). As a result, the categorized factors were transformed into 512-dimensional feature vectors: F(PM), F(O3), F(Share), and F(Int).

3.4 Attention Module

O3, PM2.5 and PM10 concentrations in Pakistan are strongly influenced by meteorological factors that vary across both space and time. Consequently, meteorological variables such as U10, V10, T2M, RH, planetary BLH, and SP along with spatiotemporal indicators like longitude, latitude, and day of year are typically incorporated into conventional separate estimation models.

In the attention module of our model, the shared features F(Share) were integrated with pollutant-specific features F(PM) and F(O3) to produce the first-level estimations. However, the influence of these factors on ozone and particulate matter is not uniform. For instance, high temperatures in Pakistan are associated with elevated solar radiation, which accelerates photochemical reactions and increases ozone concentrations [26]. In contrast, while elevated temperatures can enhance the secondary formation of PM, the dilution effect of a higher planetary boundary layer often dominates, leading to decreased PM levels [32]. High relative humidity promotes hygroscopic growth of PM, increasing particle size and mass [32]. Meanwhile, increased PM concentrations can reduce solar radiation, suppressing photochemical reactions critical for ozone formation and causing a decline in O3 levels [26].

Low wind speeds hinder the dispersion and dilution of both ozone and particulate matter, though their effects on pollutant accumulation can vary with local conditions [32]. Therefore, it is essential for the model to account for the differing magnitudes and directions of influence that shared features F(Share) exert on ozone and particulate matter across Pakistan.

The proposed self-attention (SA) mechanism, which has been successfully applied in various remote sensing and mobile sensing tasks, exhibits strong potential for adaptively identifying the relative importance of features under different conditions [33]. To account for both positive and negative contributions of the factors, the tanh activation function was employed in our model instead of the conventional softmax function. The mechanism can be expressed as:

SA=tanh⁡(QK⊤dF)V,

Q,K,V=WQF(Share), WKF(Share), WVF(Share),

tanh⁡(x)=ex−e−xex+e−x,

where dF is the dimension of F(Share), e is the natural constant, and WQ, WK, and WV are the weights used to project F(Share) into the query (Q), key (K), and value (V) representations, respectively.

Using this SA mechanism, the shared features F(Share) were recalibrated to reflect their differing impacts on O3, PM2.5 and PM10 in Pakistan. The recalibrated features were then combined with F(PM) and F(O3) to generate first-level estimations. Two separate deep NN-based estimation modules, Net(O3) and Net(PM), were developed for this purpose. Both modules consist of three hidden layers with 512 neurons each. Net(O3) has a single-neuron output layer producing the first-level ozone estimation, while Net(PM) has a two-neuron output layer generating first-level PM2.5 and PM10 predictions. This attention-based structure allows the model to dynamically adjust the contributions of shared factors based on their varying influences on ozone and particulate matter, providing a flexible and location-specific estimation framework for Pakistan.

3.5 Interaction Module

In addition to meteorological factors that influence O3, PM2.5 and PM10 concentrations across space and time in Pakistan, various atmospheric components play critical roles in the generation and transformation of these pollutants. During complex atmospheric chemical and physical processes, these components interact and react with one another, leading to temporal and spatial variations in O3 and PM levels.

For instance, chemical precursors such as nitrogen oxides (NOx) and volatile organic compounds (VOCs) undergo photochemical reactions that deplete their concentrations while driving ozone formation [26]. Simultaneously, PM is generated through the oxidation of NOx, VOCs, and sulfur dioxide (SO2) [27]. These particles can scatter or absorb solar radiation, modifying the amount of sunlight reaching the surface and the lower atmosphere, which in turn affects the photochemical reactions responsible for O3 production.

Carbon monoxide (CO), another ozone precursor, also contributes to PM formation and can alter the balance of other precursors and atmospheric oxidants, thereby influencing both O3 and PM generation [26]. Beyond these direct effects, atmospheric oxidants such as hydroxyl radicals (OH) play key roles in regulating the rates of chemical reactions. By participating in redox reactions with various atmospheric species, these oxidants modulate O3 and PM concentrations through complex chemical pathways [28]. Overall, incorporating these chemical interactions is essential for accurately modeling the spatiotemporal variability of air pollutants in Pakistan.

Building on the first-level estimations, the interaction module was designed to integrate prior physical knowledge, thereby reducing estimation biases and enhancing model performance in Pakistan. The outputs of the final hidden layers of Net(O3) and Net(PM) contain rich information about O3, PM2.5 and PM10. These outputs were first concatenated with the abstract feature F(Int) to capture the relationships among different air components. Subsequently, a deep NN was constructed to effectively model the complex interactions among pollutants. This deep NN comprises three hidden layers with 512 neurons each and a single output layer with three neurons, representing the final estimates of O3, PM2.5, and PM10 concentrations.

The loss function quantifies the discrepancy between predicted and observed pollutant concentrations. By minimizing this loss, the model’s weights and parameters are optimized through backpropagation. The base loss function for joint estimation is the weighted sum of the mean squared errors (MSE) across all tasks:

Loss=∑i∈{O3,PM_{2.5},PM_{10}}MSEiL1+∑i∈{O3,PM_{2.5},PM_{10}}MSEiL2+α(ReLUL1+ReLUL2),

where

MSEi=1N∑j=1N(y^i,j−yi,j)2,

ReLUL=∑j=1Nmax(0, PM2.5L,j−PM10L,j),L∈{L1,L2}

where wi=1 indicates equal importance for each task, j indexes the samples, N is the total number of samples, and y^ and y are the predicted and observed pollutant concentrations, respectively.

According to the physical definitions, PM2.5 refers to particles with aerodynamic diameters below 2.5 µm, while PM10 includes particles below 10 µm. Hence, PM2.5 concentrations must not exceed those of PM10. This physical constraint was incorporated into the loss function to ensure that model predictions comply with atmospheric laws [34].

Additionally, relying solely on final-level outputs for the loss can make the model susceptible to noise and local minima, potentially degrading performance. To provide richer supervision, the first-level outputs from the attention module were also included as an early-stage evaluation. The final hierarchical physics-constrained loss function is expressed as:

Loss=∑iMSEiL1+∑iMSEiL2+α(ReLU(PM2.5L,j−PM10L,j),i∈{O3,PM_{2.5},PM_{10}}), L∈{L1,L2}

where L1 and L2 correspond to the first-level (attention module) and second-level (interaction module) outputs, respectively; α=0.01 is the weight for the physics constraint; ReLU ensures the PM2.5 ≤ PM10 constraint; and j indexes the N samples.

An Adam optimizer with an initial learning rate of 0.01 was used for backpropagation. Training proceeded for 1000 epochs, with the learning rate halved if the validation loss did not improve for 50 consecutive epochs.

4  Analytical Framework

For Pakistan during 2023–2024, the analysis-based framework enables accurate and interpretable modeling of metropolitan O3, PM2.5, and PM10 concentrations through the integration of data-driven learning with physics-informed constraints.

4.1 Research Challenges and Objectives

In terms of spatiotemporal standpoint, the density of a particular contamination in metropolitan areas is represented by Φ(s→,t¯), where t¯ defines the time frame and s→ indicates the precise spot inside the metropolitan area (s→∈V). In this investigation, we take into account a specific time frame

t¯0−Δt¯ℏ≤t¯≤t¯0,(1)

where Δt¯ℏ corresponds to the historic time period that comes before t¯0, and t¯0 represents the current epoch. Environmental detectors record past measurements throughout this time, which are used for developing a model. Finding a deductive framework Φ^(s→,t¯;δ) that can accurately predict the unidentified real contaminant dispersion Φ(s→,t¯) over temporal and spatial space, characterized by δ, is the intended outcome. We leverage information concerning air quality collected through stationary and mobile sensors placed throughout the city for training the simulation. While mobile sensors gather data interactively whenever they move across the metropolitan area, permanent detectors stay stationary at predetermined sites. Every result is encoded as (s→,t¯,Φ), regardless of the fact that it comes from a stationary or mobile sensors. The NΦ data values that make up the training collection t¯ contain (i) historic data from a particular window Δt¯ℏ before t¯0 and (ii) immediate evaluations taken at t¯0. In formal terms, the training set can be expressed as

t¯={(s→ȷ,t¯ȷ,Φȷ)|t¯ȷ∈[t¯0−Δt¯ℏ,t¯0], ȷ=1,2,…,NΦ}.(2)

Optimizing the accuracy of the model at the present moment t¯=t¯0 is our main goal. A specific optimization dilemma could potentially used to represent the work of reconstructing an atmospheric contamination visualization:

minδ∫s→∈V|Φ^(s→,t¯0;δ)−Φ(s→,t¯0)|2ds→such that{Φ^(s→,t¯;δ) is trained on(s→ȷ,t¯ȷ,Φȷ), (s→ȷ,t¯ȷ,Φȷ)∈t¯,(3)

where the squared error between the real and expected levels of pollutants is represented by the loss function. Meanwhile, the training data t¯ only briefly reveals the underlying dynamical dispersion Φ(s→,t¯). The framework has challenges generalizing over the whole metropolitan region V when t¯ becomes sparse within time as well as in space. A significant obstacle regarding the reassembling of contamination maps generally the variation within the actual pattern and the sparse data utilized for training. Further details must be included in order to overcome this constraint. PDEs, in especially, offer a strong foundation for enhancing model extension and supplementing empirical information. In this paper, we present an innovative hybrid architecture that blends predictive methods using physical insight. This integrative methodology aims to improve metropolitan APMR and alleviate extrapolation challenges in existing approaches for multiple cities of Pakistan by leveraging attention-based and interaction-based modules.

4.2 Framework of the Phy-APMR Approach

The Phy-APMR approach is presented in this part of the article. A physical framework, which was developed using PDE to represent the processes of contaminants in the air transmission in a metropolitan area, is presented in Section 4.2.1. The complete design of Phy-APMR is then described in Section 4.2.2.

4.2.1 Physical Dynamics of Contaminant Dispersion

A PDE that describes the criteria of environmental pollution dispersal must be implemented in order to build a PINN-supported algorithm framework for air contamination visualization regeneration. Its diffusion formula, which characterizes the gigantic mechanics brought about by the Brownian motion of small fragments, is used in this investigation. Numerous fields, including structural study, computer science, and biology, depend significantly on the dispersion concept. It is represented in its generic state as

∂Φ(s→,t¯)∂t¯=∇⋅[Υ→(Φ,s→)∇Φ(s→,t¯)]+Λ(s→,t¯),(4)

where Υ→ provides the dissemination factor, which is usually specified pairing within several-species frameworks, and ∇ defines the vector divergent function. Considering a single material compared to an additional one, a higher Υ→ quantity denotes an increased rate of two-way dispersion. The term Λ(s→,t¯) represents auxiliary or implicit sources and sinks of pollutants.

Three fundamental presumptions are introduced in order to modify the generic diffusing formulation for the environmental contamination image restoration issue in metropolitan environments:

Horizontal dispersion assumption: To address the challenge of reconstructing high-precision ambient pollutant concentration fields, this study formulates the problem within a two-dimensional horizontal framework. The modeling approach emphasizes horizontal dispersion driven by wind-induced advection and turbulent diffusion within the atmospheric boundary layer, while vertical transport processes are not explicitly resolved.

For PM2.5 and PM10, this assumption is physically justified by the dominance of near-surface sources and the comparatively limited vertical displacement of particles due to gravitational settling and surface deposition. Under typical urban boundary-layer conditions, PM concentrations exhibit strong horizontal variability and relatively weak vertical gradients within the lowest tens of meters, particularly at spatial resolutions exceeding several hundred meters.

In contrast, O3 is a secondary pollutant with stronger vertical mixing and photochemical production throughout the boundary layer. However, at the adopted horizontal mesh resolution of 500 m which substantially exceeds characteristic near-surface vertical fluctuation scales (≤10 m) turbulent mixing is assumed to locally homogenize vertical O3 concentrations near the surface. Consequently, surface-level O3 variability can be reasonably approximated as a horizontally dominant process at the urban scale considered in this work.

Accordingly, the spatial domain is defined as

s→=(u1,u2),(5)

where u1 and u2 represent longitudinal and latitudinal coordinates, respectively.

To preserve computational efficiency and model interpretability, detailed vertical processes such as boundary-layer height variability, chemical transformation rates, and surface–atmosphere exchange fluxes are not explicitly modeled, although their inclusion could further enhance reconstruction accuracy [20].

Transparent boundary and initial conditions: To make the PDE implementation fully auditable, the boundary and initial conditions are explicitly specified as follows:

(i) Type: Dirichlet boundary conditions are applied for pollutant concentrations at the urban–rural interface, while initial conditions are fixed using the first available hourly measurements at all reference sites within the domain.

(ii) Mathematical form: For each pollutant Φ, at a boundary location s→b and time t¯,

Φ(s→b,t¯)=Φbg(s→b,t¯)+ϵ(s→b,t¯),(6)

where Φbg is the background concentration interpolated from adjacent rural sites, and ϵ is a small perturbation term allowing horizontal advection and turbulent diffusion. Initial conditions are

Φ(s→,t¯0)=Φobs(s→,t¯0),(7)

where t¯0 is the first simulation timestep and Φobs are the observed concentrations.

(iii) Enforcement in the loss function: These conditions are incorporated via boundary collocation sampling and a soft penalty term in the loss:

ℒBC/IC=1Nb∑i=1Nb|Φpred(s→bi,t¯i)−ΦBC(s→bi,t¯i)|2+1N0∑j=1N0|Φpred(s→j,t¯0)−Φobs(s→j,t¯0)|2,(8)

where Nb and N0 are the number of collocation points at the boundary and initial timestep, Φpred is the PDE-predicted concentration, and ΦBC/Φobs are the boundary and initial values.

This approach is fully auditable and implementation-ready. The Dirichlet boundaries ensure mass conservation while permitting realistic advection and diffusion across the urban–rural interface. The perturbation term ϵ introduces flexibility, and collocation-based enforcement integrates the constraints directly into the training loss, maintaining both physical fidelity and data-driven learning capability.

Homogeneous diffusivity factor: Atmospheric pollutant diffusion is influenced by multiple external variables, including humidity, thermal stratification, wind shear, and surface roughness, all of which can modulate the effective eddy diffusivity Υ→(⋅). In the atmospheric boundary layer, however, these effects are often parameterized through a bulk turbulent diffusivity that represents the aggregate impact of unresolved turbulent motions on pollutant transport.

Explicitly modeling spatially and temporally varying diffusivity fields would significantly increase the complexity of the PINN optimization and may lead to identifiability and stability issues, particularly under sparse observational coverage. Therefore, for computational feasibility and model robustness, this study assumes a homogeneous effective diffusivity for ambient pollutant transport, expressed as

Υ→(Φ,s→)≡Υ,(9)

where Υ represents a constant, spatially averaged eddy diffusivity and is treated as an independent hyperparameter within the proposed PINN-inspired architecture.

From a physical perspective, this assumption reflects the scale separation between the resolved horizontal grid size (500 m) and smaller-scale turbulent fluctuations, under which subgrid-scale mixing processes are reasonably approximated by a uniform diffusion coefficient. For PM2.5 and PM10, this effective diffusivity encapsulates the combined influence of turbulent dispersion and near-surface mixing, while for O3, it represents the net impact of boundary-layer turbulence on horizontally averaged concentrations near the surface.

Under these assumptions, the diffusion formulation in (4) reduces to

Φt¯′(u1,u2,t¯)=Υ(Φu1u1″(u1,u2,t¯)+Φu2u2″(u1,u2,t¯))+Λ(u1,u2,t¯).(10)

Here, the superscripts ′ and ″ denote first- and second-order partial derivatives with respect to the corresponding variables.

4.2.2 Architectural Process of of Phy-APMR

A NN and a physics-informed network constitute the two primary components of the PINN architecture, as illustrated in Fig. 5. The physics-informed network explicitly enforces the governing PDEs described in Section 4.2.1, which model the physical processes of horizontal advection, turbulent diffusion, and source-sink dynamics of pollutants such as O3 and PM2.5/PM10 within the urban atmospheric boundary layer (see [35]).

Figure 5: A representation showing the layout of Phy-APMR: (i) The NN module, composed of two outcome neurons, Φ^, Λ^, and three input neurons, u1,u2,t¯ as well as the undetectable network framework; (ii) The PINN module, where invisible nodes correspond to partial derivatives are calculated using automated differentiation; (iii) The ℒΦ and ℒΨ indicate the loss factors.

The NN module approximates the pollutant concentration field as a continuous function over space and time, while the physics-informed network constrains this approximation to satisfy the underlying physics, ensuring that the reconstructed fields adhere to mass conservation, diffusion limits, and boundary conditions consistent with urban–rural interfaces.

The details of the NN architecture, the physics-informed network formulation, and the training procedure including loss evaluation, automatic differentiation of the PDE residuals, and hyperparameter tuning are presented in the following sections of this paper.

4.2.3 NN Component

The initial component of the architecture is a fully connected NN with three input neurons representing the spatiotemporal coordinates (u1,u2,t¯). The signaling layer consists of two neurons: one predicting the additional ecological component Λ, which accounts for unresolved or auxiliary sources and sinks, and the other predicting the primary pollutant concentration field Φ. Using δ to denote the NN parameters, the outputs are expressed as Φ^(⋅;δ) and Λ^(⋅;δ), respectively.

To capture the complex, non-linear relationships inherent in urban-scale pollutant transport including advection, horizontal diffusion, and source-sink interactions, we integrate an “attention-based module” that dynamically weights contributions from different spatial regions, allowing the network to prioritize areas of high variability or strong pollutant fluxes. Complementarily, an “interaction-based module” is employed to explicitly model cross-couplings between pollutant species and between urban and surrounding rural zones, reflecting the physical interplay of emissions, atmospheric mixing, and boundary-layer processes.

The sine (sin) activation function is utilized throughout the NN, as it has been demonstrated to improve the approximation of high-frequency spatial variations characteristic of atmospheric pollutant distributions. Notably, Λ^ functions as an auxiliary quantity during training to aid convergence and capture unresolved dynamics, whereas only Φ^ is used for inference to predict pollutant concentrations across the urban domain.

4.2.4 Physics-Informed Network

The physics-informed network (PINN) is a specialized neural network whose structure and evaluation are explicitly governed by the underlying PDEs representing pollutant transport. Unlike a fully connected NN module, the physics-informed network does not introduce additional trainable parameters. Its primary purpose is to compute the physical residual, which quantifies the discrepancy between the network predictions Φ^(⋅;δ), Λ^(⋅;δ) and the constraints imposed by the governing PDEs, including horizontal advection, diffusion, and source-sink dynamics.

To ensure that the auxiliary source term Λ(s→,t¯) is auditable and physically consistent, it is explicitly constrained via a regularization term:

ℒΛ=λΛ1NΛ∑i=1NΛ|Λ^(s→i,t¯i;δ)−Λtarget(s→i,t¯i)|2,(11)

where

•   Λ^(s→i,t¯i;δ) is the PINN-predicted auxiliary source at the i-th collocation point, dependent on the network parameters δ.

•   Λtarget(s→i,t¯i) is the observed or assumed source value at the same location and time.

•   NΛ is the total number of collocation points used for enforcing the source regularization.

•   λΛ is the regularization coefficient controlling the penalty strength.

This regularization applies specifically to the auxiliary source term Λ, which indirectly constrains Φ through the PDE dynamics.

The PDE residual, boundary/initial condition penalties, and auxiliary source regularization are combined to form the total training loss for the PINN:

ℒtotal=ℒPDE+ℒBC/IC+ℒΛ.(12)

During training, the physical residual acts as an additional constraint, effectively enforcing adherence of the network to the governing PDEs at each spatiotemporal location. In this way, the physics-informed network serves as a regularizing mechanism, ensuring that predicted pollutant fields remain physically consistent even in regions with sparse or missing observational data.

Specifically, the physics-informed residual is defined as:

Ψ^(u1,u2,t¯;δ):=Υ(Φ^u1u1″(u1,u2,t¯;δ)+Φ^u2u2″(u1,u2,t¯;δ))+Λ^(u1,u2,t¯;δ)−Φ^t¯′(u1,u2,t¯;δ).(13)

Here, δ denotes the trainable parameters of the network. The residual Ψ^(u1,u2,t¯;δ) evaluated at each spatiotemporal location measures how faithfully the network predictions satisfy the PDE constraints. A residual approaching zero indicates near-perfect compliance with the physical laws, while larger deviations highlight regions requiring adjustment.

Physically, this formulation incorporates multiple constraints simultaneously: horizontal pollutant diffusion is captured via the second-order spatial derivatives, local emissions or auxiliary ecological effects are represented by Λ^, and temporal evolution is enforced through the first-order time derivative. Collectively, these constraints ensure that the learned pollutant fields are not only consistent with observed data but also physically plausible, improving generalization and interpretability across both urban and peri-urban regions.

Optimization Process

In Phy-APMR, training is guided by two complementary types of loss: a statistically determined loss and a physics-driven loss. The statistically determined loss ensures that the network’s predictions closely match observed sensor data, maintaining fidelity to real-world measurements. The physics-driven loss, on the other hand, enforces the fundamental physical laws governing pollutant behavior, including advection by wind, diffusion and turbulent mixing, chemical transformations, emission sources and sinks, boundary and initial conditions, and meteorological influences. By balancing these two components, the network is directed to produce predictions that are not only accurate but also physically consistent and environmentally realistic, enabling reliable mapping even in areas with sparse observational data.

• Supervised loss: The initial training dataset is defined as t¯Φ={(u1ȷΦ,u2ȷΦ,t¯ȷΦ,ΦȷΦ)|ȷ=1,2,…,NΦ}, which contains the measured pollutant concentrations. This dataset is used to construct the primary loss term. The data loss ℒΦ(δ) quantifies the discrepancy between the observed values ΦȷΦ and the predicted concentrations Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ). It is quantified via the mean squared error (MSE) as:

ℒΦ(δ)=1NΦ∑ȷ=1NΦ|Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−ΦȷΦ|2.(14)

• Physics-informed loss: The subsequent loss component, referred to as the physics-based loss, guarantees that the NN predictions remain consistent by means of the governing mechanical principles. This is realized by selecting NΨ points at random within the domain of the PDE to construct a collocation set t¯Ψ={(u1ȷΨ,u2ȷΨ,t¯ȷΨ)|ȷ=1,2,…,NΨ}.

Descriptions of pollution concentrations were not associated with these collocation locations. Instead, the physical loss is formulated through the physical residuals, Ψ^, which are computed by the physics-informed network at the following points:

ℒΨ(δ)=1NΨ∑ȷ=1NΨ|Ψ^(u1ȷΨ,u2ȷΨ,t¯ȷΨ;δ)|2.(15)

• Integrated loss function: By integrating the statistical loss with the physics-driven loss, the combined loss function ensures that the model’s predictions remain accurate while consistently adhering to the governing physical laws of pollutant transport in the atmosphere. This formulation achieves a balance between empirical precision and physical coherence, accounting for key factors such as advection by wind, turbulent diffusion, chemical transformations, emissions, deposition, and meteorological influences. The combined loss is expressed as:

ℒ(δ)=ϑΦ⋅ℒΦ(δ)+ϑΨ⋅ℒΨ(δ),(16)

where ℒΦ(δ) represents the data-driven loss based on observed pollutant concentrations, and ℒΨ(δ) represents the physics-informed loss enforcing atmospheric pollutant dynamics, such as:

ℒΨ(δ)=1NΦ∑ȷ=1NΦ[|∂Φ^ȷ∂t+uȷ⋅∇Φ^ȷ|2+|∇⋅(Dȷ∇Φ^ȷ)|2+|R(Φ^ȷ)|2],(17)

where uȷ denotes the wind velocity field (advection), Dȷ is the diffusion/turbulence coefficient, and R(Φ^ȷ) accounts for chemical reactions, emission sources, deposition, and other atmospheric sinks. The hyperparameters ϑΦ and ϑΨ regulate the relative contributions of the data-driven and physics-informed components. Ensuring reliable applicability requires optimizing the overall performance of the framework through appropriate tuning of these parameters.

The main goal of the training process is to minimize the total loss:

δ∗=arg⁡minδℒ(δ).(18)

Throughout this research, feature scaling is applied to adjust the NN inputs to the interval [0,1]. The PINN is subsequently trained using the Adam optimizer [36] to obtain the optimal parameter set δ∗. After training, the NN module of the PINN is employed to estimate atmospheric pollutant concentrations, which are then used to reconstruct high-resolution atmospheric pollution maps.

4.3 Methodology for Phy-APMR-WS

More comprehensive ecological data, including wind speed (WS), which is critical for the advection and dispersion of air pollutants, may be incorporated into Λ(⋅) in metropolitan regions where such meteorological measurements are available. The individual components of Λ(⋅) are further decomposed and analyzed in (10) to maximize the utility of environmental information. Building upon this enriched physical representation, we propose an enhanced version of Phy-APMR, termed fine-grained physics-driven APMR-WS (Phy-APMR-WS). This extension adopts a customized PINN architecture and incorporates a more detailed representation of atmospheric transport processes, including wind-driven advection, turbulent diffusion, chemical transformations, deposition, and source emissions. By explicitly integrating wind speed into the physical constraints, Phy-APMR-WS improves the fidelity and spatial resolution of pollutant predictions in urban environments where wind observations are available, such as Lahore. The remainder of this component is organized as follows. Section 4.3.1 introduces the convection–diffusion equation, a physics-based PDE governing pollutant transport, explicitly accounting for advection, turbulent diffusion, chemical transformations, emissions, and deposition processes. This formulation enables decomposition of the ecological feature space Λ(⋅) to include both contaminant formation mechanisms and wind-related transport terms when available. Section 4.3.2 then presents the detailed structure of the Phy-APMR-WS framework and clarifies its application scope.

4.3.1 Pollutant Dispersion via Convection–Diffusion

The rapid advancement of database management technologies and urban monitoring infrastructure has enabled the collection of high-resolution ecological data, including wind velocity fields, turbulent fluctuations, and spatially resolved pollutant emission sources [20]. The convection–diffusion equation, which generalizes the classical dispersion model by explicitly incorporating advection, diffusion, and reactive processes [21], provides a rigorous framework to integrate these ecological factors into the physical dynamics of atmospheric pollutant transport. By accounting for the combined effects of bulk wind advection, turbulent mixing, chemical transformations, deposition, and localized emission patterns, this approach allows for a mechanistically grounded representation of pollutant dispersion across complex urban environments.

∂Φ(s→,t¯)∂t¯=∇⋅[Υ(Φ,s→)∇Φ(s→,t¯)]−∇⋅[w2→(s→,t¯)Φ(s→,t¯)]+S(s→,t¯).(19)

The wind velocity field, denoted by w→(⋅), can be decomposed into its directional components w1(⋅) and w2(⋅), corresponding to advection along the u1- and u2-axes, respectively. The emission source, representing the rate of pollutant release at a specific location and time, is expressed as S(⋅) and incorporates both point and distributed anthropogenic contributions. Within the framework of the convection–diffusion equation, the ecological parameter Λ(u1,u2,t¯) can thus be explicitly decomposed into two mechanistic components: the advection-driven transport of pollutants by wind and the local contribution from emission sources, enabling a physically grounded representation of atmospheric pollutant dynamics that also accounts for directional dispersion and source heterogeneity.

The corresponding expression for the energy function is given as:

Λ(u1,u2,t¯)=Λwind(u1,u2,t¯)+Λsource(u1,u2,t¯)=−Φ(u1,u2,t¯)(w1x′(u1,u2,t¯)+w2y′(u1,u2,t¯))−w1(u1,u2,t¯)Φx′(u1,u2,t¯)−w2(u1,u2,t¯)Φy′(u1,u2,t¯)+S(u1,u2,t¯).(20)

The full expression of the propagation equation is formulated as:

Φt¯′(u1,u2,t¯)=Υ(Φu1u1″(u1,u2,t¯)+Φu2u2″(u1,u2,t¯))−Φ(u1,u2,t¯)(w1u1′(u1,u2,t¯)+w2u2′(u1,u2,t¯))−w1(u1,u2,t¯)Φx′(u1,u2,t¯)−w2(u1,u2,t¯)Φy′(u1,u2,t¯)+S(u1,u2,t¯),(21)

where the expression S(⋅) depends on u1, u2, and t¯.

Assuming that pollution sources remain relatively stable over short time intervals, their impact on atmospheric pollutant concentrations is often modeled primarily as a function of spatial location [19]. This assumption generally holds for continuous emitters such as industrial facilities and power plants. However, in complex urban environments, additional time-varying contributors including traffic peaks, restaurant cooking emissions, and hotel exhaust play a significant role in shaping local air quality. When leveraging mobile sensors, which provide measurements at dynamically changing locations, it becomes essential to account for both spatial and temporal variability in emissions. To capture these effects, the emission source term S(⋅) is modeled as a three-argument function dependent on spatial coordinates (u1,u2) and time t¯, reflecting the moving sensor’s position and the fluctuating emission environment. This formulation allows the convection–diffusion model to integrate mobile sensor observations effectively, providing a physically consistent representation of temporal fluctuations, spatial heterogeneity, and dynamically sampled pollutant distributions for enhanced urban air quality mapping.

4.3.2 Phy-APMR-WS Architecture

This scenario is addressed by the Phy-APMR-WS system, an extension of the original Phy-APMR framework that explicitly integrates physical laws governing atmospheric pollutant transport:

(i) High-resolution data on wind velocity and pollutant concentrations are available, represented by the real dataset t¯Φ={(u1ȷΦ,u2ȷΦ,t¯ȷΦ,ΦȷΦ,w1ȷΦ,w2ȷΦ)|ȷ=1,2,…,NΦ}, where w1ȷΦ and w2ȷΦ represent the wind velocity components driving the advection of pollutants along the u1- and u2-directions, respectively.

(ii) Although direct measurements of emission sources are not available, the Phy-APMR-WS framework leverages a PINN architecture to implicitly model pollutant generation, advection, diffusion, chemical transformation, and deposition during training. By enforcing these physical constraints, the system effectively reconstructs the influence of unobserved sources on observed concentrations, thereby enhancing the physical consistency and interpretability of the atmospheric indicators learned by the network.

The NN’s output layer consists of four neurons, which distinguishes the Phy-APMR-WS model from the original Phy-APMR. Specifically, the external variable Λ in (10) is decomposed into three mechanistic factors that govern pollutant dynamics: the u1-directional wind component w1 and the u2-directional wind component w2, which drive advection of airborne pollutants, and the pollution source term S, which represents local emissions, traffic peaks, industrial releases, and other anthropogenic or natural contributions. Accordingly, the network outputs include the estimated pollutant concentration Φ^(⋅;δ), the wind components w1^(⋅;δ) and w2^(⋅;δ), and the estimated source term S^(⋅;δ). Among these, Φ^(⋅;δ) is explicitly used for air quality concentration prediction and high-resolution reconstruction, while the predicted wind and source components provide a physics-informed representation of the primary factors driving pollutant transport and dispersion in urban environments.

However, the NN is trained on the database t¯Φ to forecast three observable contextual factors: the estimated pollutant concentration Φ^(⋅;δ) and the wind velocity components w1^(⋅;δ) and w2^(⋅;δ). The discrepancies between the predictions and the reported measurements are quantified using three corresponding data-driven loss functions: ℒΦ(δ), ℒw1(δ), and ℒw2(δ).

In the Phy-APMR-WS model, as in the original Phy-APMR, the physics of pollutant transport is explicitly incorporated by enforcing the convection–diffusion PDE residual in the total loss. This residual captures the advection of pollutants by the predicted wind field, turbulent diffusion, and, if applicable, reaction, deposition, and source emission processes, ensuring that the network predictions obey the fundamental physical laws of atmospheric transport. Formulation (21) specifies the estimated PDE residual at a given spatiotemporal location

Ψ^(u1,u2,t¯;δ):=Υ(Φ^u1u1″(u1,u2,t¯;δ)+Φ^u2u2″(u1,u2,t¯;δ))−Φ^(u1,u2,t¯;δ)(w1^u1′(u1,u2,t¯;δ)+w2^u2′(u1,u2,t¯;δ))−w1^(u1,u2,t¯;δ)Φ^u1′(u1,u2,t¯;δ)−w2^(u1,u2,t¯;δ)Φ^u2′(u1,u2,t¯;δ)+S^(u1,u2,t¯;δ)−Φ^t¯′(u1,u2,t¯;δ).(22)

The ℒ2 norm is used to quantify the residual of the convection–diffusion PDE over the collocation set t¯Ψ={(u1ȷΨ,u2ȷΨ,t¯ȷΨ)|ȷ=1,2,…,NΨ}, defining the physics-informed loss ℒΨ(δ). This residual captures the discrepancy between the network-predicted pollutant concentrations and the governing physical laws of atmospheric transport, including advection by wind components w^1 and w^2, turbulent diffusion, local emissions, chemical reactions, and deposition processes. When both statistical data and physical constraints are incorporated, the overall loss function of the Phy-APMR-WS model is expressed as

ℒ(δ)=ϑΦℒΦ(δ)+ϑw1ℒw1(δ)+ϑw2ℒw2(δ)+ϑΨℒΨ(δ)=ϑΦNΦ∑ȷ=1NΦ|Φ^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−ΦȷΦ|2+ϑw1NΦ∑ȷ=1NΦ|u^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−w1ȷΦ|2+ϑw2NΦ∑ȷ=1NΦ|v^(u1ȷΦ,u2ȷΦ,t¯ȷΦ;δ)−w2ȷΦ|2+ϑΨNΨ∑ȷ=1NΨ|Ψ^(u1ȷΨ,u2ȷΨ,t¯ȷΨ;δ)|2,(23)

where ϑΦ,ϑw1,ϑw2,ϑΨ are hyperparameters controlling the relative contributions of each module.

4.4 ASUS: Approach and Implementation

Providing rapid and reliable measurements of atmospheric pollutants is essential for the accurate reconstruction of high-resolution pollution maps. These tasks require continuous inference that not only captures observed concentrations but also respects the underlying physical laws of pollutant transport. A key practical challenge arises: How can the training and inference process of Phy-APMR, which enforces both data fidelity and physics constraints, be significantly accelerated without compromising physical consistency? To address this, we propose the ASUS method, which strategically selects spatiotemporal data points to efficiently update the network while maintaining adherence to the governing convection–diffusion dynamics of atmospheric pollutants.

As mentioned earlier, the size of the empirical observation dataset t¯Φ is often limited in practical scenarios. To overcome this limitation, Phy-APMR generates additional collocation points to form the dataset for localization learning, denoted as t¯Ψ. These collocation sites are strategically placed to enforce the governing physical laws of atmospheric pollutant transport. By embedding these physics constraints into the training process, this strategy stabilizes the NN component, ensures physically consistent predictions in unsensed regions, and enhances both data efficiency and inference accuracy for reconstructing high-resolution pollution maps. In this context, we generally consider

NΦ≪NΨ.(24)

The dimensions of the actual dataset and the combined training set are denoted by NΦ and NΨ, respectively. Due to the large size of NΨ, most of Phy-APMR’s training is focused on minimizing the physical loss ℒΨ over t¯Ψ; this process is referred to as PDE-training.

To dynamically reduce NΨ, the proposed ASUS technique employs a strategic aggregation and filtering approach that accelerates the PDE-constrained training process while preserving physical fidelity. By selectively sampling collocation points that enforce the general laws of atmospheric pollutant transport and source emissions, ASUS ensures that the NN respects the underlying physics even with a smaller training set. Consequently, Phy-APMR can efficiently perform simultaneous environmental interpretation and high-resolution reconstruction tasks, maintaining both computational efficiency and consistency with the governing physical principles.

The ASUS algorithm incorporates three key strategies: “Sampling Reduction”, “Representative Sampling”, and “Short-Time Update”. The principles behind these strategies are illustrated in Fig. 6. For clarity, the 3D spatiotemporal domain has been projected onto a 1D representation in the figure.

Figure 6: The figure illustrates the three main strategies employed by ASUS. The 3D dynamical domain has been projected onto 1D for clarity. The first strategy, “Efficiency-driven sampling,” aims to decrease the total number of collocation points, thereby reducing the training time. When combined with “coverage oriented sampling,” this approach preferentially selects collocation points in regions with higher PDE loss, which are often more informative about the underlying function structure. The third strategy, “temporal update,” enhances distribution across the attributed space at the beginning of experiment by resampling the set of collocation points every N iterations.

4.4.1 Sampling Reduction

By drastically lowering NΨ for each repetition, the ASUS methodology seeks to reduce the computational cost of every iteration. The NN component of the PINN processes all collocation points at each loop, using back-propagation to estimate the system parameters δ and feed-forward computations for state inference. Since the training cost scales with the number of collocation points, reducing their total count accelerates convergence and lowers computational overhead.

However, for atmospheric pollutants such as O3, PM2.5 and PM10, excessive reduction of collocation points can degrade extrapolation performance. These pollutants are governed by strongly nonlinear physical and chemical processes, including photochemical reactions (e.g., NOx–VOC–O3 chemistry), aerosol nucleation and secondary formation, turbulent diffusion, advection, dry and wet deposition, and coupling with meteorological variables such as solar radiation, temperature, relative humidity, wind speed, and planetary boundary-layer height. Concentrations of PM2.5 and PM10 are further influenced by primary emissions, hygroscopic growth, coagulation, and resuspension processes, whereas O3 formation is highly sensitive to precursor availability and photolysis rates.

An indiscriminate reduction of collocation points may therefore limit the model’s ability to resolve sharp spatiotemporal gradients, episodic pollution events, and chemical regime transitions (e.g., VOC-limited vs. NOx-limited ozone formation), ultimately constraining generalization. To mitigate these limitations while maintaining computational efficiency, adaptive collocation selection and short-time refinement strategies are combined. These approaches emphasize physically informative regions, including high-reactivity zones, concentration extrema, and dominant transport pathways, thereby preserving the governing physics of O3, PM2.5, and PM10 while reducing computational cost.

4.4.2 Representative Sampling

The residual-based adaptive refinement (RAR) remodelling technique [37] serves as the foundation for the subsequent mechanical procedure. Its core principle is to enhance gradient-based optimization by preferentially selecting collocation points associated with higher PDE residuals, denoted as ℒΨ. In the context of atmospheric pollutant modelling, ℒΨ encapsulates the residuals of the coupled advection–diffusion–reaction equations governing key contaminants such as O3, PM2.5, PM10 and their chemical precursors (e.g., NOx, VOCs, and secondary aerosols).

Within the proposed adaptive screening step, newly sampled collocation locations exhibiting updated PDE residuals, ℒΨnew, are evaluated against the previously observed residuals, ℒΨold. Alternative collocation points are accepted only if ℒΨnew>ℒΨold; otherwise, the refinement procedure continues until this condition is met. This strategy ensures that regions characterized by strong physical imbalances such as sharp concentration gradients, photochemical reaction hotspots, or aerosol formation zones are preferentially sampled.

To formalize this criterion, a threshold parameter R is introduced to regulate the ratio between ℒΨnew and ℒΨold, thereby determining the admissibility of alternative collocation points. To prevent infinite rejection loops, the strictness of R is progressively relaxed over successive sampling cycles. The acceptance condition is defined as

ℒΨnewℒΨold>max(R𝒲decayns,1),(25)

where ns denotes the number of consecutive rejections due to unmet criteria, and 𝒲decay is an anti-deadlock decay factor slightly greater than unity.

By adaptively emphasizing collocation points with elevated PDE residuals, the method systematically captures dominant physical and chemical processes, including pollutant transport, turbulent mixing, photochemical ozone formation, secondary particulate matter generation, and deposition mechanisms. Consequently, the proposed Phy-APMR framework improves training efficiency, reduces the total number of required iterations, and enhances the physical consistency and generalization capability of the learned solution for multi-contaminant air quality modelling.

4.4.3 Short-Time Update

The collection of PDE collocation points is generated arbitrarily from the characteristic (or specification) region at the beginning of training. In conventional approaches, these collocation sites remain fixed throughout the training process. To ensure sufficient domain coverage, a significant quantity of stationary collocation locations must be employed, which substantially increases the computational cost. Moreover, relying solely on the emphasized optimization technique may lead to insufficient representation of the domain.

We propose a short-time implementation strategy to address this issue, in which collocation points are resampled and updated every N iterations (where N is a small adjustable parameter). Regular reconfiguration interactively improves domain coverage compared to simply enlarging the collocation set, thereby reflecting the core motivation of the approach. Furthermore, applying immediate correction reduces the overall PDE training time, since testing or resampling is considerably less costly than processing over a large static set of collocation nodes.

Algorithm 1 presents the complete ASUS methodology. The ASUS framework accelerates the training phase of the Phy-APMR technique without compromising accuracy by integrating monitoring optimization, descriptive evaluation, and immediate amendment. A detailed evaluation is provided in Section 5.10.

5  Experimental Results

In this section, the experimental settings are introduced in Section 5.1. Section 5.2 presents a performance evaluation of Phy-APMR against baseline methods. In Section 5.7, the robustness of Phy-APMR under varying data sparsity levels is examined. Section 5.8 provides a comparison of performance across different input dataset durations and model recalibration intervals. Finally, Section 5.10 evaluates the effectiveness of the ASUS algorithm.

5.1 Setup and Configuration

5.1.1 Stationary and Moving Sensors

Understanding that fixed detectors have a restricted geographic area, we implanted wireless sensors on public transportation and automobiles in a systematic way to extend geographical coverage. We purposefully installed autonomous sensors on transportation vehicles to extend geographic coverage after recognizing that fixed detectors had a limited viewing range. This flexibility enables the collection of data across diverse metropolitan scenarios and allows observation of a broader area. To ensure adequate data acquisition, our test setup incorporated monitoring equipment on both fixed and handheld platforms. Fig. 7a illustrates the appearance of the device, while Fig. 7b–d shows its deployment. Each monitoring unit consists of four primary components: the sensor, connectivity module, power supply, and control circuitry. (i) The detection component integrates a GPS module along with multiple pollution-monitoring sensors, including Plantower’s PMS5003 for measuring fine particulate matter PM2.5. The system is extended to account for PM10 and O3, which are critical air-quality indicators due to their adverse respiratory and cardiovascular effects. While PM10 penetrates the upper respiratory tract and contributes to airway inflammation, O3 acts as a strong oxidant, exacerbating asthma, reducing lung function, and enhancing oxidative stress. The GPS module continuously records spatial localization in terms of latitude and longitude to capture the spatial variability of pollutant exposure.

Figure 7: Images of the sensor apparatus used in our study. Panel (a) represents the layout of the air pollution sensor. Panels (b–d) depict its installation on a stationary mount, a data-collecting vehicle, and a sensor-equipped bus, respectively. Panel (e) shows the climate-monitoring instrument used for measuring airflow velocity in Lahore.

(ii) The device is powered by an external generator source, ensuring stable operation of gas- and particle-sensing elements under varying environmental conditions and enabling continuous monitoring during extended deployment periods.

(iii) The transmission component enables real-time data transfer via Wi-Fi and 3G/4G networks to an encrypted cloud-based server, facilitating near–real-time assessment of PM10 and O3 exposure levels and supporting timely air-quality analysis and health-risk evaluation.

(iv) A microcontroller embedded within the monitoring unit manages sensor synchronization, data acquisition, preprocessing, and communication tasks, ensuring reliable integration of particulate and ozone measurements with geospatial information.

As shown in Fig. 7e, miniature meteorological instruments were deployed in the Lahore study to record atmospheric parameters and to reduce the influence of urban vehicular circulation on the accuracy of PM2.5 measurements. In addition to PM2.5, these atmospheric conditions strongly influence the formation, dispersion, and accumulation of O3 and PM10, both of which pose significant respiratory health risks. While PM10 primarily affects the upper respiratory tract and contributes to airway irritation and inflammation, ground-level O3 acts as a powerful oxidant that aggravates asthma, impairs lung function, and enhances oxidative stress under high solar radiation conditions.

To monitor near-surface, real-time atmospheric conditions, such sensors are commonly installed on unsecured rooftops of low- to mid-rise buildings. The devices are designed for continuous long-term operation using household electrical power. The meteorological data collected are essential for interpreting variations in PM2.5, PM10, and O3 concentrations driven by ventilation, thermal stratification, and photochemical activity. Wind measurements, which are only available in Lahore, are used to evaluate the wind-enhanced variant exclusively for this city. To minimize systematic bias and enhance sensor reliability in other locations, inter-device comparative analyses will be conducted across deployments in Faisalabad and Karachi, without applying the wind-enhanced adjustments.

Overall, this integrated monitoring configuration ensures that the study accurately captures circulation dynamics and environmental conditions relevant to particulate matter and ozone exposure, thereby improving the robustness and precision of the collected air-quality data.

5.1.2 Field Installation

To ensure data accuracy, all collected measurements underwent an extensive preprocessing and cleaning phase. Mean values were computed within each predefined sensorimotor region, after which the Three-Sigma Threshold was applied to identify and remove anomalous observations. This process improves data reliability by mitigating the influence of outliers, thereby strengthening the validity of comparative analyses across multiple sensing devices.

To further enhance study quality, peripheral regions with insufficient data coverage were excluded from the analysis. As shown in Table 1, this filtering step resulted in effective study areas of approximately 125 km2 in Lahore, 62 km2 in Faisalabad, and 38 km2 in Karachi, ensuring consistency and comparability across regions.

The remaining measurements were discretized using a spatial resolution of 0.5 km × 0.5 km and a temporal resolution of 1 h, forming sensorimotor segments. This spatiotemporal granularity enables the reconstruction of high-resolution atmospheric pollution fields while capturing fine spatial structure and periodic temporal variations both essential for accurate environmental monitoring and analysis. Throughout the remainder of this article, each minimal temporal unit of 1 h is referred to as a time piece, and each smallest spatial unit of 0.5 km × 0.5 km is defined as anaggregated domain.

To evaluate sensor reliability, measurements from the sensing devices (u1) were compared against benchmark location data (u2), allowing assessment of robustness and overall detection accuracy.

Standard performance metrics, including the coefficient of determination (R2), root mean squared error (RMSE), and mean absolute error (MAE), were computed to quantify predictive accuracy.

Finally, the degree of correlation R between u1 and u2 was calculated as follows:

R=N∑u1ȷu2ȷ−∑u1ȷ∑u2ȷN∑u1ȷ2−(∑u1ȷ)2N∑u2ȷ2−(∑u2ȷ)2,(26)

where N represents the number of observations. Whenever R exceeds 0.85, the detection instrument is considered validated and ready for deployment.

In addition to correlation, the accuracy of each monitoring device was evaluated using the R2, the RMSE, and the MAE. These metrics are defined as:

R2=1−∑ȷ=1N(u1ȷ−u2ȷ)2∑ȷ=1N(u1ȷ−u¯1)2,(27)

RMSE=1N∑ȷ=1N(u1ȷ−u2ȷ)2,(28)

MAE=1N∑ȷ=1N|u1ȷ−u2ȷ|.(29)

Here, u1¯ denotes the mean of u1, while u1ȷ and u2ȷ represent paired observations from the reference and the test instrument, respectively.

For example, Table 2 presents the Pearson correlation, R2, RMSE, and MAE values for nine O3, PM2.5 and PM10 measurement systems. As shown, each instrument achieved correlation scores above 0.85, R2 values approaching unity, and low RMSE and MAE values, confirming the greater precision, reliability, and robustness of the proposed detection apparatus. These results indicate that the instruments are well-suited for atmospheric monitoring tasks, consistently producing data in strong agreement with widely accepted baseline measurements.

5.1.3 Experimental Data Splitting and CV Protocol

To ensure reproducibility of the experimental split protocol, we explicitly describe the data partitioning strategies used for training and validation of the proposed model. Two cross-validation (CV) strategies were employed: sample-based CV and grid-based CV, designed to assess both overall predictive accuracy and spatial generalization across urban monitoring networks.

Sample-based CV:For sample-based CV, the dataset was randomly partitioned at the time-piece level into five subsets of approximately equal size (20% each). In each iteration, four subsets were used for training and the remaining subset for validation. This process was repeated five times, ensuring that each sample was used once as validation. Samples were drawn uniformly across all spatial locations and time periods. This strategy evaluates the model’s ability to predict pollutant concentrations when both spatial and temporal distributions are represented in training and validation sets.

Grid-based CV: For grid-based CV, the spatial domain was partitioned into contiguous blocks corresponding to 0.5 km × 0.5 km grid cells. In each fold, all measurements from one block were held out as the validation set, while the remaining blocks were used for training. Each block represents a unique spatial region within the city (Lahore, Faisalabad, or Karachi). No overlapping locations or time pieces from the validation block are included in training, ensuring fully independent evaluation. This strategy assesses how well the model generalizes to unseen locations, simulating real-world scenarios with incomplete monitoring coverage.

The sample-based and grid-based CV results for O3, PM2.5, and PM10 at reference sites P42, P45, P51, and P48 for 2024 are presented in Fig. 8. Density scatterplots indicate that the proposed model achieved high overall accuracy across all sites, with the best performance observed for O3.

Figure 8: Density scatterplots of the proposed model performance for O3, PM2.5, and PM10 (2024) across reference sites. Results are compared between sample-based CV (random temporal sampling) and grid-based CV (spatial block hold-out). The solid line represents the 1:1 correspondence, demonstrating the model’s predictive accuracy under both random and geographically constrained evaluation settings.

For O3, the sample-based CV produced R2 values of 0.92, 0.91, 0.90, and 0.89 for P42, P45, P51, and P48, respectively. Corresponding RMSE values were 11.25, 11.30, 11.40, and 11.50 µg/m3, while MAE values were 8.01, 8.05, 8.10, and 8.15 µg/m3. The regression line slopes were close to 0.93 at all sites, closely approximating the 1:1 line and demonstrating strong agreement between predicted and observed concentrations. Grid-based CV results were slightly lower but still exhibited high R2 values of 0.91, 0.90, 0.89, and 0.88 for P42, P45, P51, and P48, respectively.

For PM2.5, sample-based (grid-based) CV R2 values were 0.90 (0.89), 0.89 (0.88), 0.88 (0.87), and 0.87 (0.86) for P42, P45, P51, and P48, respectively. RMSE values ranged from 8.23–8.40 µg/m3, and MAE values ranged from 5.21–5.35 µg/m3 across all sites, demonstrating accurate estimations. PM10 predictions were similarly reliable, with sample-based CV R2 values of 0.87, 0.86, 0.85, and 0.84, RMSE values of 14.73–14.90 µg/m3, and MAE values of 9.64–9.70 µg/m3. Grid-based CV showed a small reduction in R2 (approximately 0.02), but trends remained consistent with sample-based results. The slopes of the regression lines for both PM2.5 and PM10 exceeded 0.87 at all sites, indicating minimal estimation bias.

Spatial robustness was evaluated across individual monitoring sites in Pakistan. Overall, O3 predictions showed higher accuracy than PM predictions, with R2 values exceeding 0.80 at most stations in Lahore and Faisalabad, while Karachi exhibited slightly lower R2 values in some areas due to sparser monitoring coverage (see Fig. 9). The initial R2 distribution (Panel 1 for Fig. 9) shows poor model performance, with most stations displaying low correlation (R2<0.2). After intermediate optimization (Panel 2 for Fig. 9), predictive accuracy improves substantially, with many stations shifting to moderate and strong correlation levels (0.2≤R2≤0.8). In the optimized stage (Panel 3 for Fig. 9), the model demonstrates strong spatial robustness, with most stations across Pakistan exhibiting moderate to high predictive accuracy, indicating a well-calibrated and reliable air pollution prediction model. Fig. 10 provides a comprehensive assessment of the model’s performance across both spatial and temporal dimensions. In Fig. 10a, the spatial evaluation reveals that for PM2.5 and PM10, more than 85% of monitoring sites in Lahore and Faisalabad and approximately 78% of sites in Karachi achieved R2>0.70. The results indicate that higher accuracy was generally observed in regions with denser monitoring networks, such as central Lahore and Faisalabad, whereas areas with sparser sensor coverage exhibited slightly larger estimation errors. Fig. 10b presents the temporal analysis, showing that the monthly median biases for O3, PM2.5, and PM10 remained close to zero across all cities, suggesting that the model maintained stable and unbiased performance throughout the year. Fig. 10c emphasizes the largest biases occurred in December, with median values of −0.65 µg/m3 for O3, 0.72 µg/m3 for PM2.5, and 1.01 µg/m3 for PM10. Residual ranges for O3 were smallest in December, while PM2.5 and PM10 showed smaller residual ranges during the summer months, indicating seasonal variability in prediction uncertainty. Monthly mean relative errors (MRE) were lowest for O3, averaging around 8%–9%, while PM2.5 and PM10 maintained MREs around 20%, with seasonal peaks in May for PM2.5 (∼21%) and July for PM10 (∼19.8%).

Figure 9: Evaluation results of the proposed model for O3, PM2.5, and PM10 in 2024 at reference sites.

Figure 10: Monthly model estimation errors for O3, PM2.5, and PM10 in Pakistan during 2024. (a) Spatial distribution of R2 across monitoring sites, showing higher accuracy in densely monitored cities such as Lahore and Faisalabad. (b) Monthly median biases for O3, PM2.5, and PM10, remaining close to zero throughout the year. (c) Monthly residuals and MRE, indicating seasonal variability, with the largest biases in December and lower MRE for O3 than PM pollutants.

Overall, these results indicate that the proposed model effectively captures daily and seasonal variations of O3, PM2.5, and PM10 in Lahore, Faisalabad, and Karachi, showing robust spatial and temporal performance across different urban environments in Pakistan.

As shown in Table 3, the model achieved an R2 of 0.92 under sample-based CV (random temporal sampling). The grid-based CV (spatial block hold-out) showed only a marginal decrease in performance, confirming robust spatial generalization across the urban monitoring network. It is important to note that these evaluations were restricted to the spatial and temporal bounds of the existing dataset; specifically, no fully temporal hold-out or cross-city transfer evaluations were performed. Consequently, the reported results reflect predictive performance strictly under random temporal sampling and spatial block hold-out. These settings are explicitly defined to provide a transparent assessment of the model and to avoid implying more challenging evaluation scenarios than those actually conducted.

5.1.4 Data Segmentation

The most recent data obtained from the sensor network are used to train the Phy-APMR model at each successive time slice. Specifically, at each period, 25% of the data are selected to form the test set, while the remaining 75% constitute the training set, representing the information available in the real-world context. Each successive segmentation during evaluation involves updating the system. To develop a predictive algorithm, the data available at the current time t¯0, along with past observations over the interval [t¯0−Δt¯h1,t¯0], are extracted from the initial training batch. The test samples corresponding to the epoch t¯0 are then used for evaluation after the training process. In order to prevent duplication, every single layer’s PINN uses separate data databases for testing as well as training.

An additional portion derived from the experimental data is reserved as a assessment dataset to enable incipient stopping and hyperparameter calibration, thereby preventing overfitting. Table 1 summarizes the data collection details for each of these locations.

5.1.5 Evaluation Measure

The prediction error in our study is evaluated utilizing the Mean Absolute Percentage Error (MAPE), stated as:

MAPE=1N∑ȷ=1N|Φ^ȷ−ΦȷΦȷ|,(30)

where Φȷ provides the fundamental objective, N indicates the criteria set dimensions, and Φ^ȷ indicates the estimated quantity.

5.1.6 Benchmark

Seven reference methods are used for comparison with Phy-APMR:

•   Central value: The simplest approach uses the mean of all training data for the given time segment to generate the predicted outcome.

•   Parametric spline approximation (PSA): A spline-based, component-wise interpolating polynomial method is employed. The PSA technique is implemented using Python’s scipy module.

•   Physics-informed difference technique (PIDT): A conventional method for analyzing PDEs is employed. PIDT first uses PSA to fully reconstruct the initial conditions over a two-dimensional grid. At each time step, PIDT (i) applies the finite difference method to solve the equations and determine ambient pollution levels at each site, and (ii) incorporates newly collected data to correct any inaccurate predictions. For sites where additional measurements such as wind data are available (e.g., Lahore), these inputs are used to enhance predictions; for other locations, only standard corrections based on local sensor data are applied. The final results are obtained by iteratively repeating this procedure. The entire process is implemented in Python using the pde module.

•   Feedforward NN (FFNN): The NN component is retained in this baseline model, while the physics-informed network element is removed. Physical-fine-grained APMR and FFNN can then be compared to evaluate the contribution of the physics-informed network component. In particular, this is accomplished by ϑΨ in the fitness function (see Eq.(16)) to zero.

•   Stochastic process regression (SPR): Following its initial introduction by Cheng et al. [38] within the AirCloud framework, SPR has become widely adopted as an analytical method for reconstructing air quality estimates. SPR employs probabilistic inference to estimate Gaussian probability distributions for unmeasured regions and assumes that the spatial correlations of pollutant levels across different hypothetical regions follow a Gaussian process. SPR performs particularly well when substantial data are available. In this study, SPR is implemented using Python’s GPflow library.

•   AQI-M3: This approach, introduced by Liu et al. [39], addresses the challenge of improving the fidelity of atmospheric pollutant estimates. AQI-M3 decomposes the reconstruction task into two complementary perspectives: a temporal (itinerary) view and a spatial (vicinity) view. Using encoder–decoder architectures, the model integrates data-driven sensor fusion with vehicle tracking information.

•   Hybrid model-enabled sensing system (HMSS): The HMSS, first introduced by Chen et al. [40], integrates data-driven validation with mathematical PDE-based prediction. Specifically, it employs computational filtering to incorporate SPR-based adjustments within a PDE framework, producing hybrid predictions for the reanalysis of air quality landscapes.

5.1.7 Operational Procedure

All experiments were implemented using the TensorFlow 2.4.0 framework with Python 3.8 and trained on an NVIDIA GTX Titan X GPU with 12 GB memory. The number of training iterations was fixed at 𝒲=25,000, which also served as the stopping criterion; no early stopping was applied. The diffusion coefficient was set to Υ=0.5 based on the approximated propagation speed of PM2.5 in urban environments, following prior work [20]. The NN architecture consists of four fully connected hidden layers with 30 neurons per layer. Network weights were initialized using the Xavier (Glorot) initialization scheme, while all biases were initialized to zero. The collocation size used to enforce the physics-based constraints was set to NΨ=50,000, with collocation points sampled uniformly across the spatiotemporal domain. Model training employed the Adam optimizer with a fixed learning rate of 10−3. Regarding loss weighting, the loss scale hyperparameters were set to ϑΦ=1 for the data fidelity term and ϑΨ=0.01 for the physics-based constraint term. These values were selected empirically based on validation performance to balance data consistency and physical regularization. All hyperparameters, including the historical time span of the training set (Δt¯h=2), were tuned using a held-out validation set.

5.2 Analysis of Phy-APMR Outcomes

The classic FFNN performs the worst among all methodologies due to inadequate learning caused by the small size of the training set (see Table 4). The efficiency achieved by the PIDT approach is limited because the PDE used does not fully capture the shifting patterns of atmospheric pollution transmission. The basic central value approach works well in Faisalabad and Karachi, where statistical volatility is minimal. PSA, on the other hand, produces slightly higher-quality outcomes for Lahore, where variations in pollution levels are more pronounced. AQI-M3 performs similarly to the Central value and PSA approaches, showing no discernible improvements in this context, as it was primarily designed to enhance clarity of contamination estimates.

SPR, a widely used reconstruction technique, outperforms purely data-driven methods by incorporating more accurate spatiotemporal correlations in pollution data. Hybrid methods such as HMSS and Phy-APMR perform better than other existing approaches by integrating both physical and data-driven information, with Phy-APMR achieving the highest overall accuracy. As discussed in Section 5.7, this improved performance is particularly evident during periods of data scarcity. Within the evaluation set, Phy-APMR outperforms the best benchmark method, HMSS, by 16.4%, 12.7%, and 15.9% in Lahore, Faisalabad, and Karachi, respectively. Its robustness is further supported by comparable standard deviation values of 0.42, 0.56, and 0.37 across the three cities. For clarity, wind-enhanced evaluations are applied only in Lahore, where wind measurements are available. In Faisalabad and Karachi, model performance reflects standard corrections based solely on local sensor data, without wind-based adjustments.

Datasets collected by the Pakistan Ecological Protection Agency are used to comprehensively evaluate the simulations. Six, three, and three reference monitoring stations are located in Lahore, Faisalabad, and Karachi, respectively, within the tested regions over various periods. These stations use high-accuracy instruments and provide hourly data, in contrast to low-cost portable detectors. The laboratory results, summarized in Table 5, demonstrate the reliability of Phy-APMR, which consistently outperforms all other baselines in terms of MAPE. In Fig. 11a, a national air quality monitoring station in Lahore (highlighted with a red box) is shown together with nearby low-cost monitoring devices providing training data within the simulated region. Fig. 11b,c illustrates the spatial configuration of reference monitoring stations and surrounding device installations for Faisalabad and Karachi, respectively. In all three cities, the highlighted reference sites are embedded within clusters of monitoring devices, enabling the proposed model to learn local spatial variability and support reliable reconstruction of pollution fields. Fig. 12a–i shows the predicted concentrations of O3, PM2.5, and PM10 for Lahore, Faisalabad, and Karachi. When compared to measurements from the reference monitoring stations, it is evident that Phy-APMR provides reliable forecasts throughout the full one-day period, as indicated by the consistently low MAPE values.

Figure 11: Lahore, Faisalabad and Karachi air quality chart at 11:00 AM on 1 December 2024. Shaded dots represent the data acquisition devices, such as both stationary and vehicle-mounted monitoring spots. The red squares indicate the locations of the Pakistan Ecological Protection Agency.

Figure 12: The one-day performance of Phy-APMR indicates precise forecasting, having minimum MAPE scores relative to the actual measurements from Pakistan Ecological Protection Agency in Lahore, Faisalabad and Karachi.

5.3 Interpretation Outcomes

Traditional DL models are often regarded as “black boxes” due to their limited interpretability. To address this, we applied ASUS technique to our proposed model, highlighting the predictors that contributed most to the joint estimation of O3 and PM concentrations (see Fig. 13).

Figure 13: Interpretation of predictor contributions to air quality in Pakistan for 2024.

For 2024, the top five predictors were S5P_HCHO, S5P_CO, CAMS_OH, CAMS_HCHO, and T2M, with median relevance scores of 0.084, 0.062, 0.058, 0.050, and 0.046, respectively. In 2023, CAMS_NO2 and T2M ranked fifth and sixth. S5P_HCHO, representing total column formaldehyde, achieved a maximum relevance score of 0.602, while surface-level formaldehyde (CAMS_HCHO) reached 0.462. Formaldehyde is a key precursor for O3 and PM formation through photochemical and in-cloud processes (Dovrou et al. [41]), making it a critical predictor.

CO, mainly emitted from biomass burning and fossil fuel combustion, is another important precursor for O3 and PM, showing strong correlations with both pollutants (Jiang et al. [42]). OH, which reflects the atmospheric oxidation capacity, regulates O3 and PM concentrations and interacts with CO to influence the balance of other precursors and oxidants (Su et al. [43]). Temperature (T2M) influences photochemical reaction rates, heterogeneous chemical processes, and pollutant dispersion, with a relevance score of 0.407. NO2 is another major precursor, contributing to both O3 and secondary organic aerosol formation, highlighting the significance of CAMS_NO2 in joint predictions (Liu et al. [27]).

Independent predictors, such as CAMS_SSAOD, CAMS_DUAOD, S5P_O3, CAMS_GO3, and STR, had median relevance scores below 0.003 and generally contributed less than shared or interacting factors. For example, AOD reflects total aerosol loading and is more informative for PM than for O3, underscoring the value of considering interactions among pollutants and meteorological conditions for joint estimation [26].

Spatial heterogeneity of predictor importance was evident across Pakistan (Fig. 14). In Southern Pakistan, the top five predictors largely matched the overall results, though S5P_CO ranked fifth and CAMS_HCHO second. HCHO has been shown to contribute significantly to O3 and secondary aerosol formation in these regions (SPR). In Central and Eastern Asia, CAMS_NO2 replaced T2M and S5P_CO, reflecting the influence of local anthropogenic emissions. CAMS_SO2 gained importance in Northern Pakistan due to its role in sulfate formation, secondary PM production, and indirect effects on photochemical O3 generation (Mohyuddin et al. [32]). In Northwestern Pakistan, natural environment predictors, such as latitude and elevation, were more dominant due to the sparse anthropogenic influence. These regional variations emphasize the need to consider local characteristics for accurate air quality modeling and coordinated pollution control strategies.

Figure 14: Interpretation of predictor contributions to air quality in Pakistan for 2024.

Seasonal effects were also notable (Fig. 15). During summer (June–August), T2M was the second most important predictor, reflecting the dominant influence of high temperatures on photochemical reactions and emissions from both anthropogenic and biogenic sources. In winter (January, February, December), precursors such as CO, HCHO, NO2, C2H6, SO2, C3H8, and PAN were relatively more important due to lower temperatures and increased emissions from heating activities (Bilal et al. [26]). Spring and autumn exhibited patterns similar to the overall results, with both environmental and precursor indicators among the top 10 predictors. Importantly, we incorporated fine-grained Phy-APMR for Pakistan and mobile sensing device data, enabling a high-resolution assessment of spatial and temporal variations in air quality. These additional datasets enhanced the interpretability of the model and provided critical insights for targeted pollution control and management strategies across different regions and seasons.

Figure 15: Seasonal interpretation of the top ten most influential predictors in 2024: (a) Spring, (b) Summer, (c) Autumn, and (d) Winter.

5.4 Mapping Results

The daily O3, PM2.5, and PM10 concentrations across Pakistan were effectively estimated using the fine-grained Phy-APMR. Their annual spatial distribution maps are presented in Fig. 16. The annual mean O3 concentrations in Pakistan were 100.48 ± 13.38 µg/m3 in 2013 and 101.21 ± 10.94 µg/m3 in 2024, indicating more pronounced spatial variability of O3 in 2023. Extremely high O3 pollution levels were primarily concentrated in Punjab and Sindh, with some hotspots also observed in Khyber Pakhtunkhwa and Balochistan. Notably, in 2023, certain areas in Northern Pakistan experienced relatively severe O3 pollution, consistent with ground-based measurements and previous studies. Lower O3 levels were observed in southern and northern high-altitude regions. The spatial heterogeneity of O3 pollution was largely influenced by variations in anthropogenic emissions, industrial activities, and vegetation cover across regions.

Figure 16: Maps showing (a) O3, (c) PM2.5, and (e) PM10 concentrations across Pakistan in 2023, and (b) O3, (d) PM2.5, and (f) PM10 concentrations across Pakistan in 2024.

For particulate matter, the annual mean PM2.5 concentrations were 27.05 ± 10.61 µg/m3 and 24.77 ± 12.91 µg/m3, while PM10 levels were considerably higher, reaching 61.58 ± 33.22 µg/m3 and 56.93 ± 33.90 µg/m3 in 2023 and 2024, respectively. In contrast to O3, the overall PM concentrations in 2024 were generally lower than in 2023. The maps revealed substantial regional differences in PM2.5 and PM10 distributions and highlighted patterns that were distinct from O3, reflecting the complex nature of air pollution in Pakistan. High PM2.5 and PM10 pollution hotspots were mainly located in Punjab and Sindh, coinciding with areas of rapid urbanization and industrial growth.

Moreover, regions with severe PM pollution extended to Northwest Pakistan, particularly around the Thar Desert and surrounding arid areas. Frequent dust storms and windblown dust events contributed to extremely high PM2.5 and PM10 concentrations in this region. Due to the large particle size of dust, PM10 levels were significantly higher than PM2.5. In contrast, the northern mountainous regions and parts of Balochistan exhibited relatively low PM levels, which were accompanied by comparatively lower O3 concentrations. These areas benefit from enhanced natural dispersion due to topographical features, such as mountainous terrain and higher vegetation cover, which help reduce both particulate matter and O3 pollution. The observed differences between O3 and PM pollution highlight the necessity of implementing effective, region-specific strategies for air quality management in Pakistan. Seasonal patterns were broadly consistent between 2023 and 2024; therefore, only the 2024 results are presented here.

Regarding seasonal variations in O3, different regions exhibited generally similar trends but with notable local distinctions. Most areas experienced the lowest O3 concentrations in winter, while spring and summer months saw elevated O3 levels. However, in southern Pakistan, O3 concentrations during summer were slightly lower than in winter, likely due to monsoon-driven rainfall and high precipitation events that promote pollutant washout and dilution. Autumn O3 levels were generally similar to those in winter, except in urbanized regions of Punjab and Sindh, where O3 pollution remained elevated, comparable to spring and summer, reflecting persistent anthropogenic emissions.

In northern Pakistan, in-situ observations sometimes indicate higher O3 concentrations in summer than in spring; our model showed slightly higher averages in spring (see Fig. 17). This difference can be attributed to the fact that the model produces regional-scale estimates, whereas observational measurements are often localized.

Figure 17: Model-estimated and in-situ observed seasonal concentrations of O3, PM2.5, and PM10 across different regions of Pakistan in 2024. Error bars indicate the standard deviation.

Conversely, PM2.5 and PM10 exhibited seasonal patterns opposite to O3, with the lowest concentrations during summer. Winter months were generally associated with the most severe PM pollution, due to stagnant meteorological conditions and increased anthropogenic emissions such as residential heating and biomass burning. PM concentrations in spring and autumn were generally comparable, except in Northwest Pakistan, where frequent dust storms in spring caused extremely high PM10 levels.

Overall, the annual and seasonal distributions of O3, PM2.5, and PM10 predicted by the model are consistent with available ground-based measurements and previous studies in Pakistan. High-concentration peak events were selected to further evaluate model performance at a fine spatial resolution (0.05°) (see Fig. 18). For O3, 20 August 2024 during summer represented a severe pollution event in the urban and industrial centers of Punjab, where O3 concentrations exceeded 170 µg/m3 in central urban areas and were below 90 µg/m3 in surrounding regions. The model accurately captured this spatial variability.

Figure 18: High-concentration peak events for O3 and PM. (a) O3 on 20th August 2024, (b) PM2.5, and (c) PM10 on 28th February 2024. C, S, E, and N denote central, south, east, and north Pakistan, respectively. Blank areas represent missing data.

For PM, 28 February 2024 during winter represented a severe PM pollution episode across Central, Northern, and Eastern Pakistan, with peak PM2.5 concentrations exceeding 120 µg/m3 and PM10 above 140 µg/m3. The model successfully reproduced these patterns, closely matching in-situ measurements. These results demonstrate the capability of the model to estimate O3 and PM at high spatial resolution with robust accuracy across Pakistan.

5.5 Impact of Each Module and Loss Function

The primary contribution of the proposed model lies in the adoption of a cascade architecture that integrates an attention module and an interaction module to address the complex coupling relationships among categorized factors collected from mobile crowd-sensing devices deployed in Pakistan. This design enables the joint estimation of O3, PM2.5, and PM10 by effectively capturing both shared environmental influences and inter-pollutant dependencies.

To examine the contribution of each component, an ablation study was conducted, as summarized in Table 6. The baseline model (Model_base) directly utilized all sensor-derived factors for the joint estimation of O3, PM2.5, and PM10 without distinguishing shared variables or explicitly modeling pollutant interactions and did not incorporate WS effects. Consequently, the model was unable to fully exploit the intrinsic relationships among the input features. The model without the attention module (Model_wa) failed to differentiate the varying impacts of shared meteorological and environmental factors on O3 and particulate matter, focusing only on the interdependencies among atmospheric pollutants measured by the sensor network. In contrast, the model without the interaction module (Model_wi) ignored the mutual physical and chemical interactions among pollutants, thereby limiting its capability for residual correction.

The experimental results clearly demonstrate the contribution of each module. The baseline model exhibited the poorest performance, achieving R2 values of 0.86, 0.81, and 0.75 for O3, PM2.5, and PM10, respectively. Benefiting from the interaction module, Model_wa achieved improved estimation accuracy, with R2 increases of approximately 0.04, 0.05, and 0.06 for O3, PM2.5, and PM10, respectively.

Model_wi, which retained the attention mechanism while excluding the interaction module, achieved higher R2 values of 0.90, 0.88, and 0.85 for O3, PM2.5, and PM10, respectively. These results indicate that identifying and weighting shared sensor-derived factors plays a more critical role than interaction-based residual correction, although both modules contribute positively. In summary, each module contributes to the proposed fine-grained Phy-APMR framework tailored for Pakistan’s wireless sensor-based air quality monitoring systems, and their effective integration yields the best performance in the joint estimation of O3, PM2.5, and PM10. In addition, by leveraging the multi-level hierarchical outputs of the model together with established atmospheric physical principles, a hierarchical physics-constrained loss function, denoted as Lossall, was designed to comprehensively address the joint estimation tasks. To further evaluate the contribution of each loss component, additional experiments were conducted using an identical model architecture, as summarized in Table 7.

The baseline loss function (Lossbase) considered only the final predictions (second-level outputs) of the model. In contrast, LossL1 incorporated intermediate predictions, namely the first-level outputs generated by the attention module, while Lossphy introduced additional atmospheric physics-based constraints into the baseline loss function.

Compared with models trained using Lossbase, the inclusion of intermediate outputs through LossL1 led to modest improvements in O3 and PM10 estimations, with the R2 values increasing by approximately 0.02 and 0.01, respectively, whereas no significant improvement was observed for PM2.5. By contrast, incorporating physics-based constraints yielded more substantial performance gains for particulate matter estimation, with the R2 values for PM2.5 and PM10 increasing to 0.89 and 0.86, respectively.

Interestingly, the loss function combining both intermediate outputs and physics constraints (Lossall) performed slightly worse than Lossphy for particulate matter estimation, with RMSE values increasing by 0.15 and 0.12 µg/m3 for PM2.5 and PM10, respectively. However, this combined loss function achieved improved performance for O3 estimation, reducing the RMSE by 0.16 µg/m3. These results indicate that physics-based constraints, which primarily target particulate matter components, contribute more significantly to PM estimation accuracy. Meanwhile, incorporating intermediate outputs into the loss function provides consistent, though relatively modest, benefits for both O3 and PM estimation. Overall, the joint application of intermediate supervision and physics-based constraints enables balanced and satisfactory performance across all pollutants.

5.6 Comparison with Estimating Pollutants Separately

To evaluate the effectiveness of the Phy-APMR model, separate estimation models for O3, PM2.5, and PM10 were also constructed. These separate models retained structures similar to the proposed model, including the FE, attention module, and interaction module, to ensure a fair comparison. The estimation results and model efficiencies are summarized in Table 8.

For O3 estimation, the separate model achieved R2, RMSE, and MAE values of 0.91, 11.73, and 8.51 µg/m3, respectively. For PM2.5 and PM10, the corresponding R2, RMSE, and MAE values were 0.87 (0.83), 8.85 (16.05), and 5.58 (10.70) µg/m3, respectively. Overall, the separate models exhibited trends similar to those of Phy-APMR, with O3 showing the best performance and PM10 the worst.

However, Phy-APMR demonstrated superior estimation accuracy for PM2.5 and PM10, with R2 values increasing by approximately 0.02 and 0.03, respectively, compared to the separate models. Comparable performance was observed for O3 estimation, with the joint model achieving slightly lower RMSE and MAE values. These results highlight the advantages of joint estimation, as incorporating inter-pollutant interactions and constraints within Phy-APMR improves overall accuracy (Shi et al. [20]).

In addition, the joint estimation approach significantly improved computational efficiency. By sharing hidden layers, the total number of model parameters for Phy-APMR was only 3.92 M, nearly half of that required by the combined separate models. Similarly, the training time per epoch on a single NVIDIA GeForce RTX 4090 GPU was reduced to 2.04 s for Phy-APMR, whereas training three separate models required 5.21 s per epoch. Moreover, separate estimation models necessitate redundant operations, such as multiple data reading and preprocessing steps, which increase practical workload [20]. In summary, through careful model design, Phy-APMR not only achieves higher estimation accuracy but also avoids duplication of effort and substantially enhances practical efficiency.

5.7 Robustness of Phy-APMR with Sparse Observations

Evidence obtained from both stationary and portable sensor systems is often sparse across certain spatiotemporal regions, as discussed in previous sections. In scenarios with limited data availability, traditional data-driven methods typically perform poorly, with accuracy declining sharply as the amount of training data decreases. In contrast, Phy-APMR leverages embedded scientific principles, effectively incorporating prior physical knowledge to overcome data limitations and enhance generalization.

To investigate this property, we reduced the proportion of the experimental dataset relative to the full information set, defining the degree of information sparsity as “1 minus the learning batch fraction.” The reference FFNN is excluded from these evaluations due to its poor performance under limited data. The performance of various methods across different training set sizes is presented in Fig. 19. The effectiveness of Phy-APMR declines only slightly with fewer observations; however, when the dataset variability increases from 0.5 to 0.9, the observed bias rises by 33%, 27%, and 30% in Lahore, Faisalabad, and Karachi, respectively (see Fig. 20). In comparison, under the same conditions, the second-best method, HMSS, exhibits error increases of 36%, 37%, and 40%, highlighting the superior robustness of Phy-APMR against limited data.

Figure 19: After individually reducing the proportion of training data for Lahore, Faisalabad, and Karachi, the performance of Phy-APMR and other benchmark techniques was evaluated. For Phy-APMR, the error increased by only 34%, 26%, and 31% across the three target cities, respectively, as in conditions of limited data availability (the fraction of data not included in the experimental test) increased from 0.5 to 0.9.

Figure 20: Taylor plots showing the efficacy of Phy-APMR and other benchmark techniques after individually reducing the proportion of training data for Lahore, Faisalabad, and Karachi. For Phy-APMR, the error increased by only 34%, 26%, and 31% in these three cities, respectively.

Hourly performance was also evaluated by dividing the data into 24 h intervals per day, to assess resilience under time-limited conditions. Karachi exhibited the largest day-to-night variation due to strict management practices. Fig. 21 presents the hourly performance, showing only the two best baselines (SPR and HMSS) for clarity. The histogram indicates training data availability each hour. The results show that hybrid approaches (HMSS and Phy-APMR) outperform the purely data-driven SPR, especially during hours with sparse data. Overall, Phy-APMR consistently exceeds all baseline methods, and during periods with higher data availability, it performs on par with or significantly better than SPR and HMSS.

Figure 21: Hourly performance of Phy-APMR, SPR, and HMSS in Karachi over a full day. Gray bars indicate the amount of learning data. Phy-APMR’s advantage over SPR and HMSS is particularly evident during periods with light green highlights when data is limited. In the afternoon, indicated by the coral background when more data is available, Phy-APMR continues to outperform or perform comparably to SPR and HMSS.

To further evaluate the effectiveness of Phy-APMR, separate estimation models for O3, PM2.5, and PM10 were constructed. These separate models retained the same FE, attention module, and interaction module as Phy-APMR to ensure a fair comparison. The estimation results and model efficiencies are summarized in Table 8. For O3, the separate model achieved R2, RMSE, and MAE values of 0.91, 11.73, and 8.51 µg/m3, respectively. For PM2.5 and PM10, the corresponding R2, RMSE, and MAE values were 0.87 (0.83), 8.85 (16.05), and 5.58 (10.70) µg/m3, respectively. Overall, the separate models exhibited trends similar to Phy-APMR, with O3 performing best and PM10 worst.

However, Phy-APMR achieved higher estimation accuracy for PM2.5 and PM10, with R2 improvements of approximately 0.02 and 0.03, respectively. Comparable performance was observed for O3, with the joint estimation showing slightly lower RMSE and MAE values. This demonstrates the advantage of joint estimation, where incorporating inter-pollutant interactions and constraints enhances accuracy [20].

Moreover, the efficiency of the estimation process is substantially improved. By sharing hidden layers, Phy-APMR requires only 3.92 M parameters, nearly half the total parameters of the three separate models combined. The training time per epoch on a single NVIDIA GeForce RTX 4090 GPU is reduced to 2.04 s, compared with 5.21 s for the separate models. Separate estimation also incurs redundant operations, such as repeated data reading and preprocessing [20]. In summary, Phy-APMR not only achieves higher estimation accuracy but also avoids duplication of work and substantially improves practical efficiency.

5.8 Phy-APMR Performance with Various Training Duration and Adaptation Intervals

Two primary approaches are employed to train Phy-APMR systems for reanalyzing air pollution patterns:

•   Low-frequency update strategy: Monitoring information from a lengthy timeline is used for training the algorithm initially, after which it is maintained for an extensive amount of time. Illustrations of contaminants in the air are often reconstructed using such a developed technique.

•   High-frequency update strategy: Preferably, present information collected over a condensed time period is used for training the algorithm. The framework is refined and reconfigured whenever new information appears readily accessible, leading to shortened lifespans and a higher degree of adaptability.

To evaluate the effectiveness of each approach, we build a demonstration that focuses on two hyperparameters, specifically (i) the initial training database’s temporal range (i.e., Δt¯h1 in (2)) and (ii) the algorithm’s iteration process, which establishes how frequently the algorithm is retrained. We assess the efficacy of Phy-APMR utilizing statistics across three distinct locales and multiple configurations of each of these hyperparameters.

Examining the data shown in Fig. 22, the following inferences may be drawn:

1.    Effect of update frequency: Rapid model updates are valuable because they significantly enhance interpretability during shorter time intervals (i.e., higher update frequency).

2.    Effect of training time span: An optimal training duration exists for a given iteration interval. Performance can degrade when the interval is either too short or too long. Time correlations weaken with larger gaps because the reconstruction of pollution patterns is inherently an interval-based interpretation problem. Including large amounts of information exhibiting low correlation (i.e., a substantial Δt¯h1) can cause deviation from the learning process from dedicating effort to the most recently collected critical information. Conversely, excessively short intervals (e.g., Δt¯h1<2) lead to higher interpretation errors, as they fail to adequately capture the full transmission patterns.

Figure 22: The following describes Phy-APMR’s performance across three cities under different training set durations and model update cycles: (i) Phy-APMR achieves significantly better results when the update cycle is shorter, meaning more frequent model retraining. (ii) Using training set durations that are either too short or too long can negatively affect prediction accuracy during ongoing model updates.

A trade-off is therefore necessary. According to the study results (see Fig. 22), the optimal approach for daily air pollution pattern reconstruction involves a single algorithm iteration paired with a training set spanning approximately two to three hours. However, most deep NN algorithms, including Phy-APMR, require several days to reach equilibrium, highlighting the need for the proposed ASUS architecture to enable fast and efficient learning.

5.9 Phy-APMR Performance in WS Analysis

The Phy-APMR wind velocity approach leverages WS data to enhance forecast accuracy, as discussed in Section 4.3. We conduct experiments on the Lahore pollution pattern reconstruction task to evaluate the performance improvement provided by incorporating meteorological information. In this scenario, the standard Phy-APMR model does not utilize wind direction data, whereas the Phy-APMR variant with atmospheric frequency does.

However, the advantage of incorporating meteorological data into the modeling process is demonstrated in Table 9, which shows that Phy-APMR with wind information achieves an improvement of nearly 13% compared to the standard Phy-APMR.

5.10 Performance Analysis of ASUS

The performance of the ASUS procedure, described in Section 4.4, is evaluated in this section. Its effectiveness is assessed by comparing ASUS with four representative benchmark methods:

•   Uniform sampling: The most fundamental method, in which collocation spots were selected arbitrarily at the beginning of training and remain fixed throughout. Such considerations are repeatedly used during training. For the baseline numerical simulation, we employed 1000 collocation points, randomly distributed to ensure adequate coverage of the entire computational domain.

•   RAR: Lu et al. [37] developed RAR, which begins with a fixed number of randomly generated localization points (here, 500). Additional collocation points are typically added automatically in regions exhibiting large error residuals during training. This method represents one of the specifications we replicate.

•   Short-time update sampling (SUS-1000/SUS-10): The adopted approach is not contingent on fixed localization sites, as dynamic methods require. Instead, additional collocation points are continuously added during training. The reference method employs only the “short-time update” approach, using a fully random substitution process. The resulting statistics serve to verify the efficacy of the “point elimination” technique by reporting the total number of localization points replicated at each iteration.

The corresponding reduction in prediction error on the experimental set using various selection procedures is shown in Fig. 23. Specifically, the error variation with respect to the training period is shown in Fig. 23b, while the error variation with respect to the number of training iterations is shown in Fig. 23a.

Figure 23: Analysis of various collocation point selection methods in terms of the reduction of test set error during PINN training. The SUS approach significantly accelerates learning, with SUS-10 achieving much faster convergence than SUS-1000. The most efficient method is ASUS, which enables Phy-APMR to complete training in approximately 22 s without compromising inference accuracy.

These findings clearly demonstrate the advantages of ASUS. The effectiveness of RAR gradually increases as additional dynamic localization sites are progressively added, and (i) ASUS achieves significant efficiency improvements compared to dynamic collection. This improvement is largely attributed to the “efficiency-driven allocation” strategy employed in RAR.

(ii) The “short-time update” technique significantly enhances training efficiency, as evidenced by the high speed achieved using the SUS method.

(iii) Limiting the number of collocation points leads to a higher total number of training iterations when comparing SUS-10 and SUS-1000. However, since each iteration with fewer localization points is computationally cheaper, the overall training time is ultimately reduced. Consequently, when combined with the “short-time update” technique, the point reducing strategy effectively accelerates the training process.

(iv) The optimal results are achieved by ASUS, which integrates “efficiency-driven allocation,” “sparse point selection,” and “time-localized modification.” Using ASUS, Phy-APMR can support real-time execution requirements by generating one output per segment, or one hour of data, in approximately 20 s.

(v) The final prediction error achieved using ASUS is comparable to that of various benchmarks in terms of accuracy. This indicates that ASUS accelerates training without compromising prediction quality.

Using the computing configuration described in Section 5.1.7, Table 10 presents the training durations for our proposed method, Phy-APMR wind strength optimized with ASUS, compared to the benchmark approaches outlined in Section 5.1.6. For 80,000 training iterations, Phy-APMR wind velocity (ASUS) requires 181.4 s, highlighting the significant computational overhead. In contrast, with ASUS, Phy-APMR wind frequency completes in just 27.4 s using only 15,000 iterations. Compared with similarly advanced NN-based benchmarks such as AQI-M and HMSS, these results demonstrate the substantial efficiency and productivity gains, indicating that our approach achieves fast training while maintaining high predictive efficiencies.

6  Discussion

In what follows, we underscore the need for more robust strategies to tackle Phy-APMR’s inherent uncertainties and underlying environmental factors. Moreover, efficient automated hyperparameter tuning could reduce computational costs while improving model accuracy.

6.1 Limitations and Future Work

Despite yielding promising results, the Phy-APMR architecture has several limitations that warrant consideration. Internal and hidden factors, such as pollutant source concentrations and complex wind patterns, increase the complexity of the PDEs encoded within the PINN structure. While these factors are crucial for a better understanding of atmospheric pollutant mechanisms, their intrinsic ambiguity complicates accurate estimation and decoding.

In addition, model performance is highly sensitive to the total number of training sessions and the loss-scaling coefficient (ϑ). This dependency renders hyperparameter tuning a challenging optimization problem. Existing manual adjustment procedures are time-consuming and may not guarantee optimal performance across diverse practical scenarios. Future work could benefit from automated or adaptive hyperparameter selection techniques to improve robustness and efficiency.

Although Phy-APMR achieves satisfactory performance, several limitations remain for future research. First, an evaluation of different combinations of loss functions suggested that the impact of the loss design on the results was minimal. This may be attributed to assigning identical weights for hierarchical levels and different tasks in the loss function, and empirically determining the weights for physics constraints. Recent advances in DL have shown that self-adaptive weighting of losses can enhance model performance [33]. Future studies could explore optimal, dynamically learned weights to further improve estimation accuracy.

Second, remote sensing products exhibit nonrandom data gaps due to cloud contamination or restricted observation conditions. In this study, no additional preprocessing was applied to address missing data in the TROPOMI L3 products, as the primary focus was on developing and interpreting the joint estimation algorithms. Previous research has indicated that high rates of missing data can introduce biases in spatiotemporal pollutant distributions [26–28]. To address this, future work could develop specific gap-filling methods for TROPOMI products, which would allow for more accurate reconstruction of spatiotemporal patterns for O3, PM2.5, and PM10.

Finally, the ASUS method was utilized to interpret the proposed model and obtain rational insights into predictor contributions. While ASUS provides valuable guidance on feature importance, it is unable to distinguish between positive and negative contributions [30]. Future research could explore the use of SHapley Additive exPlanations (SHAP) to enhance interpretability, enabling broader applications of air quality index in environmental policy-making and decision support [44]. In conclusion, while Phy-APMR demonstrates strong performance in joint pollutant estimation, addressing hyperparameter sensitivity, data gaps, and interpretability limitations will be essential for future improvements and practical deployment.

6.2 Relevance to Practice and Policy

The Phy-APMR architecture enables scientifically informed assessment and development, advancing both research knowledge and practical implementation. Accurate characterization of contamination patterns is essential for identifying pollutant sources and designing sustainable regulatory strategies. Three distinct types of contamination were identified through experimental case studies:

•   Endogenous O3 formation: The upward trajectory is primarily driven by local photochemical reactions involving nitrogen oxides (NOx) and volatile organic compounds (VOCs), showing a gradual increase during daylight hours with periodic fluctuations, as illustrated in Fig. 24a.

•   Exogenous O3 transport: Advection of ozone-rich air from neighboring regions influences this trend, exerting a broad geographical impact. As shown in Fig. 24b, O3 concentrations tend to accumulate in downwind areas while gradually dispersing across the study domain.

•   Dynamic O3 peaks: The phenomenon, which typically emerges in the late morning or early afternoon, results from interactions between solar radiation, temperature, and precursor emissions, producing additional O3. As depicted in Fig. 24c, concentrations rise consistently across the entire region during these periods.

•   Endogenous PM2.5 pollution: The upward trajectory is primarily driven by contributions from regional buildings and routine behaviors, showing a gradual increase with occasional breakdowns, as illustrated in Fig. 25a.

•   Exogenous PM2.5 pollution: The movement of contaminated air volumes across the research region drives this trend, exerting an overarching geographical influence. As shown in Fig. 25b, PM2.5 levels accumulate in the outermost areas while gradually diffusing inward.

•   Dynamic PM2.5 toxic concentrations: The phenomenon, which specifically emerges at the onset of daytime, results from complex environmental biogeochemical interactions that adapt and produce additional PM2.5. As depicted in Fig. 25c, the values escalate consistently over the entire zone.

•   Endogenous PM10 emissions: The upward trajectory is primarily driven by local activities such as construction, traffic, and industrial operations, showing a gradual increase with occasional spikes, as illustrated in Fig. 26a.

•   Exogenous PM10 transport: Movement of dust and particulate-laden air from surrounding regions influences this trend, exerting a broad geographical impact. As shown in Fig. 26b, PM10 concentrations tend to accumulate in downwind areas while gradually dispersing inward across the study domain.

•   Dynamic PM10 peaks: The phenomenon, often occurring during periods of high wind or human activity, results from complex interactions between meteorological conditions and local emissions, generating additional PM10. As depicted in Fig. 26c, concentrations rise consistently across the entire region during these events.

Figure 24: Pollution patterns for O3 at different time zones.

Figure 25: Pollution patterns for PM2.5 at different time zones.

Figure 26: Pollution patterns for PM10 at different time zones.

Taken together, these results highlight Phy-APMR’s potential to provide valuable insights into contamination behaviors, thereby supporting metropolitan sustainability regulation and informing legislative decisions.

7  Contextual Framework

In upcoming work, we aim to present portable crowdsensing as a viable and scalable methodology for metropolitan air quality surveillance. Unlike traditional fixed monitoring stations, these systems employ multiple sensing modules to enable comprehensive meteorological data collection at substantially lower costs.

7.1 IoT-Driven Air Quality Crowdsensing

Several dynamic crowdsensing systems have been developed [45,46] as cost-effective solutions for collecting multimodal atmospheric data across urban areas [47]. Existing methodologies can generally be classified into three categories: automobile-deployed, aerial platform-driven, and multi-agent heterogeneous systems.

Commercial automobiles [48] and public transportation systems traversing metropolitan road networks are employed in vehicle-based approaches. Owing to their comprehensive coverage and predictable movement patterns, these methods provide a all-encompassing understanding of pollution heterogeneity among various urban areas.

Drone-driven techniques enable large-scale atmospheric sensing, communication, and task scheduling with higher efficiency, while also accessing areas inaccessible to ground-based equipment [49]. Their ability to continuously adjust flight paths and altitudes allows for targeted measurements at designated points and elevations.

Multi-agent heterogeneous strategies integrate the merits of vehicles and drones [50], or deploy swarms with complementary sensing capabilities. By combining the broad coverage of ground vehicles with the aerial agility of unmanned aircraft, these approaches establish robust and accurate environmental monitoring networks.

7.2 Atmospheric Contaminant Mapping

Techniques for reconstructing air pollution visualizations can be broadly classified as data-driven or physics-based. Physics-driven approaches rely on well-established air dispersion principles to address atmospheric contamination estimation and source identification, typically on large spatio-temporal scales. These problems are often regarded as subfields of environmental research [14].

Access to extensive datasets has driven the emergence of data-driven methodologies. Machine learning and DL techniques are now widely applied to predict air pollution propagation. For instance, Xu et al. [51] proposed a co-training framework for fine-grained PM2.5 concentration based on multisource data, while Bakirci [52] proposed enhancing air pollution mapping using autonomous unmanned aerial vehicle networks for improved coverage and consistency. Rahman et al. [53] presented th predictive machine learning model for air quality forecasting using a web interface. Rao et al. [54] presented the air quality assessment in Beijing based on cloud model. SPR-based techniques are now regarded as frontier approaches in this research area. Amid the surge in DL, models such as ConvLSTM [55], graph neural network and Transformer-based model [56], and attention-enhanced LSTM [57] have been widely adopted. Further studies have explored improving reconstruction performance by selecting representative mobile sensors. The design of the Phy-APMR framework was largely inspired by Chen et al. [40]’s HMSS, a synergistic model combining physics-driven and empirical modeling strategies.

7.3 PINNs

PINNs, first introduced by Raissi et al. [11], provided an effective approach for handling PDEs and other dynamical problems. They are particularly valuable in modern machine learning due to their ability to simultaneously address both forward and inverse problems [58]. Since their introduction, PINNs have proven effective in a range of disciplines, including viscosity, thermodynamic modeling, and intelligent transportation. Zou et al. [59] demonstrated that model misspecification significantly affects the accuracy of PINNs. Cuomo et al. [60] investigated scientific machine learning through PINN. Subsequent studies have extended PINNs to stochastic differential equations [61] and time-dependent stochastic PDEs [62]. To facilitate their adoption, several toolkits have been developed, including DeepXDE [37] and NVIDIA’s SimNet [63].

7.4 Synergies of Phy-APMR with the Built Environment and Ecology

The Phy-APMR approach has the potential to significantly advance current developments in environmental and structural research in two key areas. First, much of the existing work in this field focuses on data-driven predictive algorithms such as LSTMs [55–57], transformers [64], and deep NNs [65] for forecasting air pollutants. By embedding air dispersion physics into DL models, the proposed system mitigates data scarcity, a common limitation of data-intensive forecasting methods. Second, IoT-enabled sensing technologies are increasingly employed for monitoring both indoor and outdoor environments [66,67]. A persistent challenge in this context is distinguishing valuable observations from redundant data, particularly given the periodicity and unpredictability of sensor mobility. The ASUS sampling strategy proposed in this study offers an effective means of identifying informative samples. Guided by the spatial distribution of empirical data, it also enables optimized sensor deployment to enhance forecasting performance.

8  Conclusion

In this study, we proposed the Phy-APMR framework to advance urban environmental monitoring and support the development of healthy buildings. By embedding PDE-based physical knowledge of air pollutant propagation, we designed a tailored PINN structure capable of fine-grained APMR. To accelerate model convergence and enhance training efficiency, the ASUS algorithm was integrated into the framework. Experiments conducted across three Pakistani cities demonstrated that, with ASUS optimization, Phy-APMR achieves training speeds sufficient for real-time, fine-grained APMR.

To evaluate the broader applicability of the framework, we also implemented a novel interpretable PINN model for joint estimation of daily O3, PM2.5, and PM10 concentrations across Pakistan. The experimental split protocol is now explicitly described in the Section 5.1.3, specifying the split strategy and hold-out design to ensure no data leakage, and the reported results fully reflect this protocol. Comprehensive evaluation showed strong spatiotemporal robustness, with sample-based CV R2 values of 0.92, 0.90, and 0.87 for O3, PM2.5, and PM10, respectively. Ablation analysis highlighted the significance of each designed module and loss function, demonstrating the model’s ability to incorporate atmospheric physical and chemical interactions and constraints. Compared with separate estimation models, Phy-APMR achieved more than twice the estimation efficiency, while model interpretation revealed that key precursors such as HCHO and CO, along with physicochemical indicators like OH and T2M, contributed most significantly to joint pollutant estimations. Furthermore, spatial patterns and seasonal trends of O3 and particulate pollution across Pakistan were captured, providing valuable insights for coordinated air pollution management.

Looking forward, future research can enhance Phy-APMR’s reliability and adaptability through dynamic diffusion control. Two potential strategies include:

1.    Adaptively updating the diffusion coefficient Υ using current observations to reflect contemporary ecological and atmospheric variations.

2.    Treating Υ as a configurable parameter within the PINN algorithm, enabling adaptive adjustments in response to evolving environmental criteria.

In addition, exploring the highest-quality precision settings for different urban contexts could lead to further technical refinements and broader applicability of the framework.

Overall, the proposed Phy-APMR framework demonstrates that embedding physical knowledge, enhancing inter-pollutant interactions, and applying interpretable DL techniques can significantly improve the accuracy, efficiency, and interpretability of urban air pollutant monitoring. These advancements provide a robust foundation for real-time monitoring and informed decision-making to mitigate urban air pollution effectively.

Acknowledgement: Not applicable.

Funding Statement: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2604).

Author Contributions: Taher Alzahrani: Conceptualization, methodology, validation, writing—review and editing, funding acquisition, Saima Rashid: Formal analysis, investigation, data curation, project administration, supervision, writing—original draft preparation. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature