A Review on Emerging Unified Information–Physics Frameworks for Structural Design: Toward Topology Optimization Informatics

1  Introduction

Over the past several decades, scientific research and engineering practice have undergone a sequence of paradigm shifts, evolving from empirical observation and theoretical deduction to computational science, and more recently toward data-intensive discovery. According to Jim Gray’s classification [1], many engineering domains are situated at the intersection of the third paradigm (computational, physics-based modeling) and the fourth paradigm (data-driven and information-centric discovery). Structural design occupies a distinctive position within this transition. On the one hand, it must rigorously satisfy physical laws and governing equations, which characterizes third-paradigm modeling. On the other hand, modern structural design is increasingly driven by information-rich requirements, including functional intent, manufacturability, cost, reliability, and accumulated design experience, which follows the fourth paradigm. As a result, structural design naturally emerges as a bridge problem at the intersection of physics-based computation and information-centric discovery, where neither paradigm alone is sufficient to fully characterize optimal design.

Structural optimization combines computational mechanics with optimization methods to enable systematic and performance-driven structural design. Among its various formulations, topology optimization (TO) [2] is regarded as one of the most influential approaches, as it allows free exploration of structural configurations within a prescribed design domain without assuming an initial layout. Owing to this generality and efficiency, TO has been extensively adopted in aerospace [3–5], automotive [6–8], advanced manufacturing industries [9–12], architecture [13,14] and carbon footprint [15–18]. Among these methods, some focus exclusively on achieving high physical performance, while others incorporate various subjective factors; together, they provide a rich set of approaches for engineering design (Fig. 1). Although performance improvements and application extensions of TO have been widely discussed, its influence on the modeling of structural design itself has received less attention. From a broader methodological perspective, the development of TO can be interpreted as a gradual process in which the structural design workflow itself becomes increasingly formalized and computationally modeled.

Figure 1: Comparison between preference-driven and performance-driven structural design. Preference-driven design examples (a–c): wheel design, tall building design, and conceptual chair design [19–21]. Performance-driven design examples (d–f): aircraft structure, mandibular prosthesis scaffold, and rocket engine [3,22,23].

In early TO frameworks, engineering design was implicitly modeled through a staged CAD–CAE pipeline [24]. Geometric modeling and physics-based analysis were treated as separate steps, and design improvement was achieved through repeated human-in-the-loop iterations between geometry modification and performance evaluation. While effective in practice, this separation leads to well-known issues such as geometry–analysis inconsistency, repeated remeshing, and limited ability to directly encode design intent into the optimization process. To address these limitations, isogeometric analysis (IGA) was introduced as a unifying framework in which the same spline-based representations are used for both geometry and analysis. Its extension to TO, known as IGA-TO [25–28], marked an important conceptual shift: design representation and physical analysis were no longer loosely coupled but jointly embedded within a single computational model. From a design perspective, IGA-TO represents an important step toward modeling not only individual design stages in a unified manner, but also their coupling within a single mathematical framework.

With the rapid adoption of artificial intelligence (AI), TO is no longer confined to purely physics-centered computational design. Learning-based representations and data-informed models are increasingly incorporated into the design loop, giving rise to AI-aided TO, generative design, and inverse design paradigms. AI-aided design [29–32] methods can generate informed prior geometries that reflect human requirements, substantially reducing manual effort during early-stage design exploration. Generative design approaches further enable the efficient production of diverse candidate structures, facilitating the exploration of unconventional and innovative design solutions. From a design perspective, the integration of AI with TO represents a further decisive step toward the systematic coupling of design information and geometric construction within a unified computational design framework.

From the viewpoint of engineering design theory, the design process is commonly decomposed into several interrelated stages, including requirement specification, geometric representation, physical analysis, and post-processing. The evolution of TO can be understood through this lens. IGA-TO contributes to the unification of geometric modeling and physical analysis by enforcing design–analysis consistency, while AI-aided and AI-integrated TO approaches increasingly couple design requirements and preference information with geometric construction. Taken together, these developments reveal a clear trend toward the progressive integration of the entire design workflow, in which requirements, representations, and physical evaluations are no longer treated as isolated components, but are instead embedded within increasingly unified computational frameworks. In summary, TO has evolved from ensuring design–analysis consistency toward achieving intent–realization consistency (Fig. 2).

Figure 2: Structural design evolves from separate stages toward a unified representation, such as AI-aided design and IGA-TO. TOI shifts structural design from stage-wise processing to a unified representation by modeling the design itself.

In this review, we refer to this emerging trend as Topology Optimization Informatics (TOI). TOI does not denote a single method or established theory, but rather a unifying perspective that views TO as an information–physics integrated framework for structural design. Under this perspective, optimal structures are obtained through the systematic modeling and co-optimization of physical laws and design-relevant information. TO is thus interpreted as evolving from a physics-centered computational tool into a broader design paradigm, capable of supporting demand-oriented and increasingly fully modeled structural design processes. The novelties of this review paper can thus be summarized as follows:

•   A new field termed as TOI is defined as the joint modeling and optimization of physical laws and design-relevant information.

•   It moves beyond merely integrating individual design states modeling optimal structural design itself as a unified object. It shifts the focus from achieving design–analysis consistency to intent–realization consistency.

•   A central novelty is the development of a unified information–physics optimization formulation (Eq. (5)), which treats information variables as co-determining factors of optimal design alongside physical design variables and state variables.

•   The framework explicitly incorporates historical data, user preferences (images, text, sketches), and real-world constraints directly into the optimization loop. This allows for meaningful design even when physical processes are only partially modeled.

•   The TOI provides a coherent view that bridges traditionally isolated tasks—such as computational acceleration, design control, and fidelity enhancement—interpreting them as different manifestations of the same information–physics co-modeling principle.

•   It introduces the perspective of constructing a high-dimensional manifold where information acts as an intrinsic coordinate that interacts with physical states to shape the geometry of the admissible design space.

•   It highlights an operational convergence where information and physics play comparable functional roles as operators or constraints, bridging the third paradigm (physics-driven) and fourth paradigm (data-driven) of scientific discovery.

To illustrate the evolutionary trajectory of topology optimization in the era of artificial intelligence—from purely physics-based formulations, to AI-enhanced modules targeting localized tasks such as acceleration and control, and ultimately toward comprehensive information–physics integration—the remainder of this review is organized as follows. First, classical TO methods are briefly reviewed to establish the physical foundations of structural optimization in Section 2. Next, emerging approaches that integrate TO with AI are surveyed and categorized in Section 3. The concept and framework of TOI are then introduced as a unifying perspective that connects these developments in Section 4. Based on this framework, key features and emerging opportunities enabled by information–physics unification are discussed in Section 5. Finally, conclusions and future research directions are presented in Section 6.

2  Topology Optimization Methods

In TO, optimal design is formulated as a performance-driven optimization problem subject to physical constraints [2]. The objective is to determine an optimal material distribution that maximizes or minimizes a prescribed structural performance metric under given loads, boundary conditions, and material constraints. As a representative example of the third paradigm, TO defines optimality entirely in a physical sense, and design activity is primarily expressed as the formal mapping of design intent and requirements into computable objective functions, constraints, and governing equations.

Mathematically, a generic TO problem can be written as:

minxJ(x,u)s.t.R(x,u)=0g(x,u)<0(1)

where x denotes the design variables describing the material distribution, u represents the state variables characterizing the physical response of the structure, J(x,u) is the performance objective, R(x,u)=0 denotes the governing equations of the system, and g(x,u)<0 represents a set of inequality constraints. These governing equations are typically derived from continuum mechanics and discretized using the finite element method (FEM), ensuring that the structural response is physically consistent with the design. This formulation constitutes the common mathematical foundation shared by most TO methods, regardless of their specific geometric or numerical representations.

The origin of TO can be traced back to Michell’s pioneering work on optimal truss structures [33], which established fundamental limits on material efficiency under given loads. The first general numerical framework for TO was later developed by Cheng and Olhoff for elastic plates [34], where the material distribution problem was transformed into an equivalent microstructural sizing problem. This idea enabled continuous optimization techniques to be applied to originally discrete 0–1 material layout. Building on this foundation, Schmit integrated mathematical programming with the FEM, establishing a computational framework capable of handling complex geometries and loading conditions. A decisive milestone in the formal development of TO was achieved by Bendsøe through the homogenization method [35], which replaced the discrete material distribution problem with a continuous optimization of equivalent microstructural properties. This approach marked the formal birth of modern TO and laid the groundwork for a wide range of subsequent methods. Since then, several representative approaches have been proposed, including density-based methods such as the Solid Isotropic Material with Penalization (SIMP) [36], evolutionary methods such as Evolutionary Structural Optimization (ESO) [37] and Bidirectional ESO (BESO) [38], boundary-based methods such as the level-set method [39–41], and explicit geometry approaches such as the Moving Morphable Components (MMC) framework [42,43]. (Fig. 3) Although these methods differ in geometric representations and optimization strategies, they all ultimately conform to the same optimization formulation in Eq. (1), in which optimal design is defined through explicit physical objectives, governing equations, and constraint enforcement.

Figure 3: Topology optimization is a physics-driven optimal design method, and four representative approaches are shown.

Classical TO faces several intrinsic challenges associated with its iterative, density- or boundary-based formulation. From a computational perspective, the repeated solution of large-scale finite element systems and sensitivity analyses leads to high computational cost, which has motivated extensive research on improving computational efficiency, such as advanced numerical solvers [44], parallelization strategies [45,46], and hardware acceleration [47,48]. In addition to efficiency issues, density-based methods often suffer from numerical artifacts such as checkerboard patterns, gray-scale elements, and mesh dependency, which necessitate filtering, projection, and post-processing techniques to obtain physically meaningful and manufacturable designs. Furthermore, although TO naturally allows complex structural layouts to emerge, practical applications often require explicit control over topological features such as the number of holes [49,50], connectivity [51], and minimum feature size [52,53]. These efforts are primarily concerned with developing more efficient means to obtain solutions to a physics-defined optimal design problem, within a modeling framework that is fixed a priori by the chosen tools and formulations.

Beyond its methodological challenges, TO has been extensively extended to a wide range of physical scenarios, demonstrating its generality as a physics-driven optimization framework. Originally developed for linear elastic structures, TO has since been applied to nonlinear mechanics [54,55], multi-material systems [56], and microstructural design [57]. Moreover, it has been successfully generalized to various multi-physics problems, including thermal conduction [58], fluid flow [59], hyperelastic materials [60], frictional contact problem [61,62], and electromagnetic system [63]. In the context of mechanical metamaterials [64], TO provides a powerful tool for tailoring effective material properties through microstructural design. These extensions highlight that the core strength of TO lies in its ability to encode and exploit governing physical laws across diverse domains, reinforcing its characterization as a fundamentally physics-based optimization paradigm.

In summary, classical TO has achieved remarkable success by recasting design problems as physics-constrained optimization tasks and identifying optimal design with physically optimal solutions. Building upon traditional manual design practices, this framework introduces computational mechanics as a central driving force, substantially advancing engineering design methodologies and laying the foundation for subsequent paradigm evolution in design.

3  The Emergence of AI in TO

AI is widely regarded as a representative of the fourth paradigm of scientific discovery. The integration of AI with TO has significantly improved the efficiency of TO-based design workflows and extended their application scope [29,65–67]. In general, the fusion of AI and TO can be categorized into two main approaches: AI-based one-shot TO [68] and AI-enhanced iterative TO [29]. The former reformulates TO as a probabilistic learning problem to enable end-to-end design generation according to the given dataset, while the latter follows the classical physics-driven optimization framework and incorporates learning-based components to enhance performance and applicability. Representative network architectures are illustrated in Fig. 4.

Figure 4: AI-enhanced iterative TO iteratively optimizes geometric topology representations, where optimal designs are searched within a predefined geometric space; AI-based one-shot TO learns a distribution of high-quality structures via network parameters, from which optimal structures can be approximated through sampling.

3.1 AI-Based One-Shot TO

Following a statistical learning paradigm, AI-based one-shot TO can be formulated as follows:

minθE(x∗,u∗)∼Ω[ℓ(xθ(u∗),x∗)](2)

where xθ denotes the design variables describing the material distribution generated by a parametric design model with learnable parameters θ. x∗ represents reference design variables drawn from the dataset Ω:

Ω={(x∗,u∗)|R(x∗,u∗)=0,g(x∗,u∗)<0}(3)

The dataset consists of near-optimal designs, which are typically generated by classical TO and satisfy the prescribed physical governing equations and boundary conditions. ℓ(⋅,⋅) is a loss function measuring the discrepancy between the predicted and reference designs. E(x∗,u∗)∼Ω[⋅] denotes the expectation over the data distribution Ω. Although AI methods are commonly categorized into supervised, semi-supervised, and unsupervised learning, and differ in terms of model architectures, data construction approaches, and training strategies, from the perspective of data-driven modeling, most AI-based one-shot TO approaches can be abstracted within the learning framework defined in Eq. (2), where the objective is to learn a parametric mapping or distribution that approximates optimal designs from data.

Within this framework, optimal designs are derived from the analysis and recombination of existing design data, rather than from explicit physics-based optimization. Owing to the fact that neural network inference is significantly faster than finite element analysis, and that design candidates are generated through probabilistic sampling, AI-based one-shot TO offers notable advantages in terms of computational efficiency and design diversity [69]. However, in practice, the acquisition of high-quality design data is often expensive and limited. Under such data-scarce conditions, issues related to performance degradation, limited generalization to unseen scenarios, and weak physical interpretability are frequently attributed to insufficient expressive capacity of neural network models, dataset shift, boundary condition variability, and constraint satisfaction [70]. Consequently, a substantial body of research has focused on improving structural generation performance by enhancing network expressiveness. Early studies primarily relied on predictive or discriminative models, while subsequent efforts have increasingly adopted generative models, such as generative adversarial networks (GAN) [71,72] and diffusion models (DM) [69,73–75], Kolmogorov-Arnold Network (KAN) [76]. Reinforcement learning (RL) [77–79], graph neural networks (GNN) [80,81], and implicit representations [82,83] have also been widely regarded as promising architectures due to their ability to model sequential design processes and complex relational structures (Fig. 5).

Figure 5: AI-based one-shot topology optimization requires neural models with strong expressive capacity, such as GANs [84], diffusion models [69], GNNs [77], RL [77], and KAN [76]. The evolution of learning and representation capabilities in neural networks therefore constitutes a key driving force for advancing one-shot topology optimization.

As implied by Eq. (2), AI-based one-shot TO incorporates physical information into neural networks primarily through the training dataset, where physical feasibility is implicitly encoded in the data distribution. In addition, physical knowledge can be introduced explicitly by designing network architectures or loss functions that embed physical principles. For example, Yoo et al. [85] developed a GAN-based wheel design framework, in which two-dimensional geometric representations are directly coupled with three-dimensional CAE analyses. In this framework, the generated 2D designs are evaluated not only based on geometric features but also with respect to the global mechanical performance of the corresponding 3D skeletal structures, thereby incorporating three-dimensional physical constraints into the learning process. Nevertheless, in most one-shot approaches, physical consistency is not strictly guaranteed during inference, but rather encouraged through data-driven regularization or soft constraints.

Beyond rapid modeling and reproduction of existing designs, AI-based one-shot TO has also expanded the practical capabilities of TO by enabling efficient design exploration. For example, large-scale predictive models for mechanical metamaterials can be used to rapidly identify promising candidates from vast design spaces [86], while simultaneously providing multiple feasible alternatives for subsequent combinatorial or hybrid design. Qian and Li propose a Fourier and latent modulated neural network, in which human design intent and high-performance design data are embedded into a shared latent space, enabling intent-aware and physically consistent TO [87]. In this sense, AI-based one-shot approaches primarily enhance the exploration and reuse of design knowledge, rather than directly replacing physics-based optimality guarantees.

3.2 AI-Enhanced Iterative TO

AI-enhanced iterative TO improves the efficiency and extensibility of classical TO by embedding AI into the iterative optimization pipeline (Fig. 6). Given the rich algorithmic structure of TO, such as geometric representations, finite element analysis, and numerical optimization, many AI-enhanced iterative approaches operate at the algorithmic core of TO. These methods specifically target long-standing obstacles that are difficult to address within the third-paradigm, physics-driven framework, including high computational cost, limited scalability, and restricted design diversity. By introducing learning-based components into selected stages of the TO workflow, AI-enhanced iterative TO has achieved substantial improvements in computational efficiency, robustness, and design diversity, while preserving the underlying physics-based formulation of optimality.

minxJ(x,u)s.t.R(x,u)=0g(x,u)<0u≈u^θ(x)(4)

where x denotes the design variables, u represents the physical state variables, and u^θ(x) is a learning-based surrogate model that approximates the physical response to accelerate or enhance the iterative optimization process. Here, the governing equations and constraints remain unchanged from classical TO and are omitted for brevity.

Figure 6: AI-enhanced TO enhances topology optimization by introducing learning-based components into individual optimization stages, enabling improved efficiency or robustness without altering the fundamental physics-based optimization framework. Implicit representation enhancement [88], Substructure calculation enhancement [89], and Optimization convergence enhancement [90].

In AI-enhanced iterative TO, the most representative approach is surrogate models. A classical TO workflow typically consists of several key stages, including geometric representation, parameter initialization, FEM, sensitivity evaluation, and iterative optimization updates. Learning-based surrogates have been introduced at nearly all of these stages. For example, neural implicit representations have been employed for geometry construction, data-driven design priors have been used to replace random initialization [91,92], neural networks have been developed to approximate finite element responses or accelerate matrix operations within FEM, and learning-based models have been used to directly predict sensitivities [93] or update directions. In addition, learning-based optimizers have been explored as alternatives to traditional constrained optimization solvers. Collectively, these approaches aim to approximate or replace computationally expensive and repetitive components in the TO pipeline while retaining the overall iterative structure.

Although computational efficiency remains one of the primary performance metrics in TO, AI-enhanced iterative TO has also been leveraged to improve other important properties, such as fidelity [69,94], optimization stability [95,96], and convergence behavior [97]. Different TO formulations exhibit distinct strengths and weaknesses—for instance, density-based methods are prone to checkerboard patterns, while boundary-based methods often suffer from slow convergence. Learning-based enhancements enable targeted improvements by addressing method-specific limitations. As a result, AI-enhanced iterative TO contributes to a more comprehensive improvement of optimization performance beyond raw computational speed.

Beyond algorithmic performance gains, AI-enhanced iterative TO also extends the practical applicability of physics-based TO to more challenging physical scenarios. By alleviating the computational burden of high-fidelity simulations [98] and enhancing numerical stability, learning-based surrogates make it feasible to tackle large-scale [99], high-resolution [100,101], strongly nonlinear [102,103], and multi-physics optimization problems that are difficult to handle using classical TO alone. These include problems involving complex material behavior, coupled thermo-fluid-mechanical systems, and three-dimensional structures with fine geometric features. In this sense, AI-enhanced iterative TO expands the range of physical problems accessible to TO, while preserving its underlying physics-driven formulation of optimality.

4  Topology Optimization Informatics

4.1 From Design Integration to Optimal Design Modeling

The development of TO reflects a gradual trend toward embedding increasing parts of the structural design process into formal mathematical and computational frameworks. Representative advances illustrate this tendency from different perspectives. IGA-TO establishes a tighter coupling between geometric representation and physical analysis by enforcing design–analysis consistency within a unified formulation. More recently, the integration of AI with TO has further connected design requirements with geometric construction, enabling data-informed representations of design intent to enter the optimization loop. Importantly, these advances signal more than improved algorithmic integration. They indicate a paradigm shift in what is being modeled. Rather than focusing solely on integrating individual stages of the design workflow, recent developments increasingly point toward the need to model optimal structural design itself as a unified object.

In the traditional CAD–CAE workflow, the optimal structure is implicitly defined through human–machine interaction, where designers iteratively modify geometric models and evaluate performance via analysis tools. In TO the optimal structure is treated as the solution to a well-defined physics-based optimization problem, governed by objective functions and constraints. Generative design [65] reframes the problem as a selection process over a large set of automatically generated candidates, while AI-aided design [29] largely inherits the CAD–CAE paradigm and focuses on improving efficiency through intelligent assistance (Table 1). Although these paradigms differ in tools and representations, they share a common but often implicit assumption about what constitutes “optimality” and how it should be discovered within a given framework. This naturally raises a fundamental question: can optimal design itself be modeled directly, rather than being inferred indirectly through specific tools and methods?

From a more fundamental standpoint, structural designs can be understood as being governed by two inseparable components: physics and information. Physics defines the feasibility and performance of structures through governing equations, constitutive relations, and boundary conditions. These components form the foundation of physics-based optimization. In principle, an optimal design would require a complete and accurate modeling of the entire physical process underlying structural behavior. However, in practical engineering design, such complete physical modeling is often infeasible due to complex geometries, multi-physics coupling, scale separation, manufacturing uncertainties, and limited computational budgets. As a result, purely physics-based formulations may provide an incomplete or overly idealized description of the design space, leaving important details of the design process underrepresented. Information, in contrast, encapsulates requirements, preferences, historical design knowledge, functional intent, manufacturability constraints, cost considerations, and implicit human expertise. Rather than replacing physical models, information acts as a complementary component that compensates for aspects of the design process that are difficult to fully capture through governing equations alone. TOI is introduced in this review to denote a unified information–physics optimization perspective, in which optimal structural design is obtained through the joint modeling and optimization of physical laws and information requirements. In this framework, information is explicitly coupled with physics within a unified formulation, enabling meaningful design optimization even when parts of the physical process are only partially modeled [115,120]. Consequently, optimal designs can be systematically explored that are both physically admissible and consistent with design intent and practical requirements.

4.2 Unified Process and Formulation of Topology Optimization Informatics

The unified information–physics framework of TOI can be summarized as three tightly coupled stages (Fig. 7): unified information–physics representation, unified information–physics modeling, and unified information–physics optimization. By modeling the design problem within a unified framework, TOI enables the systematic identification of structural designs that satisfy both physical performance requirements and human preference–driven design intent.

Figure 7: Process of TOI includes unified information-represnetation, unified information-physics modeling and unified information-physics optimization. Density presentation [21], Latent presentation [118].

A central challenge in information–physics unified representation lies in the absence of a canonical form for representing design-relevant information. To enable information–physics co-optimization, TOI requires that both physical relations and information-related relations act on a shared set of design variables, so that variations in these variables simultaneously influence physical behavior and information-driven design attributes. Under this requirement, existing information–physics representations in TO can be broadly summarized into two representative categories. The first category establishes an explicit equivalence between material distribution variables and information descriptors [21,116,121]. Representative studies adopt this paradigm by treating material density fields and image grayscale values as equivalent design representations. For example, Zhang et al. [116] formulated material distribution and pixel intensity within a unified density-based framework, and employed the method of moving asymptotes (MMA) to jointly optimize structural performance and visual style, enabling style transfer in bridge design. Similarly, Liang et al. [21] introduced a UDF-weighting strategy within the BESO framework to couple sensitivity information from physical objectives and information-driven criteria, thereby achieving intelligent conceptual design through coordinated material removal and information guidance. The second category achieves unified representation by embedding material distributions into a shared latent space constructed for information processing [117,118]. In this paradigm, high-dimensional material layouts are compressed into a low-dimensional latent representation, and optimization is performed directly in the latent space. Variations in latent variables simultaneously influence both physical performance and perceptual or functional attributes encoded in the learned representation. For instance, Ijaz et al. [117] employed a SAGNet encoder to encode material distributions into a latent space, and demonstrated that optimization in this latent space can generate chair designs with both high structural performance and increased design diversity. In this sense, the latent space serves as a unified information–physics representation, bridging material distribution and design semantics within a single optimization domain.

For unified information–physics modeling, although the governing equations of specific physical problems and the models used to represent information may differ, the underlying mathematical formulation remains common. In TOI, optimal design is understood as being jointly determined by physical constraints and information, where information represents human preferences, functional intent, manufacturability, cost, and physics-related quantities that are difficult to model explicitly but can be learned or represented through AI. By formulating design as a unified information–physics co-optimization problem, TOI provides a systematic way to express human requirements and translate them into corresponding structural forms. This can be mathematically expressed as follows:

minxJ(x,u,I)s.t.R(x,u,I)=0g(x,u,I)<0(5)

where x denotes the design variables, u represents the physical state variables, and I denotes information variables that encode both explicit physical knowledge and implicitly learned representations relevant to design optimality. In this formulation, information and physics are treated as co-determining factors of optimal design, rather than being separated into predefined physical objectives and auxiliary constraints.

R(x,u,I)=0 in TOI can be interpreted from multiple complementary perspectives (Fig. 8).

Figure 8: Four interpretations of TOI [114].

(1)   Optimal design as an information–physics unified entity.

In TOI, optimal design is not determined solely by physical variables, but jointly by physical states and information. The equation R(x,u,I)=0 represents a unified modeling framework in which physical laws and information are treated as co-determining components of optimal design. Rather than prescribing physical equations alone, TOI formulates optimality as a coupled information–physics consistency condition.

(2)   High-dimensional design manifolds shaped by information

From an information-centric perspective, TOI can be viewed as constructing an information–physics coupled manifold defined on the physical design space, in which information and physical variables are jointly embedded and co-determined by governing relations. In this formulation, information I is not treated as an external guide but as an intrinsic coordinate that interacts with physical states to shape the geometry of the admissible design space. Optimization therefore corresponds to navigating this coupled manifold, where optimal designs emerge from the joint structure of information and physics rather than from purely physics-defined criteria.

(3)   Information-assisted compensation of physical modeling and discretization errors.

Governing equations are idealized abstractions that are commonly simplified through modeling assumptions, linearization, and numerical discretization, which inevitably introduce errors when applied to real-world systems. Within TOI, the information term I provides an auxiliary modeling mechanism that captures residual discrepancies, implicit physical effects, or empirical regularities not explicitly represented in conventional formulations. By incorporating such information into governing relations, TOI enables systematic compensation of modeling and discretization errors while preserving the physics-based structure of the optimization problem.

(4)   Toward an operational convergence between information and physics in governing relations.

At the level of computational operations, information and physics can play increasingly comparable functional roles. Both may enter the formulation as operators or constraints that regulate admissible designs and influence optimization trajectories. In this sense, information is not merely an auxiliary input, but participates directly in the governing relations alongside physical laws. The formulation R(x,u,I)=0 therefore suggests a potential operational convergence between information and physics in defining optimal design, rather than a strict equivalence. This emerging operational convergence may provide a conceptual pathway toward integrating the third paradigm of physics-driven modeling with the fourth paradigm of data- and information-driven approaches.

By jointly modeling physics and information, structural design gains enhanced capability not only in achieving high physical performance but also in satisfying diverse and evolving design requirements.

Within TOI, the unified condition R(x,u,I)=0 admits multiple complementary interpretations, each highlighting a distinct way in which information acts on physical design problems. This unified condition can be systematically constructed through a Lagrangian formulation of constrained optimization, providing a common mathematical foundation for information–physics co-optimization.

For information–physics co-optimization algorithms in TOI, optimization is carried out on a unified information–physics representation. As a result, optimization mechanisms originating from physics-based TO and information-driven learning can act simultaneously on the same coupled representation. Physics-based optimizers enforce deterministic physical consistency, while information-based optimizers guide statistical or data-informed adaptation within the same optimization process. Through this joint action, TOI enables true information–physics co-optimization, in which physical performance and information-driven requirements are optimized in a coordinated and scalable manner across high-dimensional design spaces.

In summary, TOI formalizes information and physics within a unified governing framework, enabling collaborative computation through the extended formulation R(x,u,I)=0. Under this perspective, multiple topology optimization tasks are no longer treated as isolated procedures, but as different manifestations of a single information–physics coupled system (Fig. 9). The fundamental relationship between information and physics thus becomes the central theoretical focus of the next stage.

Figure 9: Development of information-physics optimization function. Here, ID represents the information from dataset, and IR represents the information from runtime human-machine interaction. Ijaz et al. (2025) [117]; Zhan et al. (2025) [118]; Nobari et al. (2025) [73]; Liang et al. (2025) [21]; Jeong et al. (2023) [122]; Zhao et al. (2025) [123]; Huang et al. (2023) [89]; Vijayakumaran et al. (2025) [124]; Hao et al. (2025) [86]; Zhang et al. (2023) [116]; Zhang et al. (2024) [125].

4.3 Unified Interpretation across Design Tasks

In the following, we show that the introduction of TOI enables a more unified view of structural design problems that are traditionally treated in isolation. Under this perspective, seemingly distinct tasks, such as computational acceleration, design control, and fidelity enhancement, can be interpreted as different manifestations of information–physics co-modeling within a single optimization framework, thereby providing a coherent understanding of their underlying connections and design implications. The framework is shown in Fig. 10.

Figure 10: Unified interpretation of different tasks in the view of TOI.

Structural control objectives such as controlling the number of holes, enforcing minimum length scales, and improving aesthetic qualities can be formulated as a coupled modeling problem, in which human preference information, including images, sketches, and textual prompts, explicitly interacts with physics-based performance criteria to determine admissible and desirable designs. Within the TOI framework, constructing an information–physics coupled design space can be broadly realized through two complementary paradigms, depending on how physical constraints are enforced during optimization: a weakly constrained information–physics space and a strongly constrained information–physics space (Fig. 11). In weakly constrained information–physics spaces, physical consistency is enforced in a relaxed manner while prioritizing the satisfaction of human preference information. Such formulations allow the optimization process to explore smoother and more continuous coupled spaces, which facilitates convergence in practice. From a design perspective, physically near-optimal solutions can be efficiently obtained by slightly tightening performance requirements at an earlier stage. Representative studies adopt this paradigm by embedding preference information into optimization processes with softened physical enforcement. For example, Vulimiri et al. [126] replaced conventional TO optimizers such as OC and MMA with Adam, and employed Gram-matrix-based similarity measures to transfer texture information from reference images to cantilever beam designs. In this formulation, preference consistency is emphasized during exploration, while physical performance is subsequently refined. In contrast, strongly constrained information–physics spaces require strict satisfaction of physical constraints throughout the optimization process, while preference information is incorporated in a secondary but explicit manner. Owing to the tighter coupling between preference and physics, these formulations are generally more challenging to optimize and computationally more demanding. Gradient-based coupling strategies are commonly adopted to construct such spaces. Building on Gram-matrix-based style representations, Zhang et al. [116] explicitly coupled gradients derived from style loss and MMA-based physical optimization, thereby forming a unified information–physics optimization space. This approach enabled image-driven style transfer under strong physical constraints, producing bridge designs with diverse aesthetic characteristics. Related methodologies have also been extended to sketch-guided information–physics coupling [125], enabling interactive control during topology evolution. Similar ideas were further applied to shell structure design to achieve style-aware yet physically consistent configurations [127]. To reduce hyperparameters in the Gram-matrix-based method, Liang et al. proposed a UDF-weighting strategy to construct a semantic–physics coupled design space [21], allowing efficient conceptual design generation. In addition to explicit feature coupling, latent-space-based fusion provides an alternative mechanism for information–physics integration. For example, semantic and performance features can be jointly embedded into a latent space using CLIP [128], where pre-designed latent priors guide the generation of cantilever beam designs with multiple stylistic variations. From the perspective of input information types, design control is evolving from relatively simple and single-form inputs, such as images [129], sketches [125,130], design skeletons [131], and textual prompts [128,132], toward more complex forms of information input, for example those informed by biological inspiration [133,134].

Figure 11: Preference information, such as sketch [135], image [128], prompt [12] and VR [136] interaction, can improve the controllability of TO through information-physics co-optimization.

Acceleration can be understood in TOI as an information–physics co-modeling process in which historical information actively reshapes the effective design manifold explored by TO (Fig. 12). All intermediate design data produced during iterative TO can be regarded as historical information, because they encode accumulated knowledge about the structure–performance relationship along the optimization trajectory. Traditional TO acceleration methods typically extract empirical patterns from optimization histories, such as characteristic features of matrix computations or convergence behavior, to improve computational efficiency within a fixed physics-based formulation. In contrast, TOI conceptually incorporates it into a unified optimization formulation that co-defines both the admissible design space and the search dynamics. In this view, acceleration is not merely computational speed-up; it reflects how past states deform the geometry of the optimization landscape so that subsequent search is guided toward physically meaningful, computationally efficient regions. This interpretation is supported by an observable methodological evolution. Early “iteration-free” one-shot approaches represent an extreme form of acceleration, in which historical datasets are used to learn direct mappings from problem specifications to final topologies [84,137]. In these methods, explicit traversal of the design space is replaced by sampling from a learned distribution of high-quality designs. Owing to the limited size and diversity of available training datasets, such approaches typically rely on low-capacity or low-dimensional representation networks, which in turn leads to restricted expressiveness and poor generalization across problem settings [82,138]. As a result, while one-shot methods achieve remarkable speed-ups, their applicability is often confined to narrowly defined design scenarios. Subsequent research recognized that historical information can be more robustly exploited within iterative TO frameworks, allowing acceleration to be achieved without sacrificing optimization accuracy or physical consistency. Rather than bypassing iterations entirely, these methods incorporate algorithm-internal historical information—such as intermediate designs, sensitivities, and state variables—into the optimization loop. This shift redirected attention toward reducing iteration counts [139,140] and alleviating the most computationally expensive stages of the process, particularly repeated finite element analyses. A wide range of acceleration strategies emerged from this perspective, including physics informed neural network (PINN) [88,122], deep energy method (DEM) [123,141], Operator learning [89,142], all of which preserve the underlying physics-based optimality formulation. More recent studies have further abstracted acceleration by analyzing the influence of neural representations on the geometry of the optimization landscape itself. Sanu et al. introduce neural topology optimization framework [114], this work demonstrates that acceleration can arise from a fundamental reshaping of the decision space. In this view, neural parameterizations alter the effective design manifold explored by the optimizer, changing optimization trajectories and convergence behavior even when the governing physics and objective functions remain unchanged.

Figure 12: Historical information accumulated from prior design attempts [87], including intermediate physical quantities (e.g., sensitivities, displacement fields, and stiffness matrices) [142], design results [143], and optimization-stage information [144], can be reused to significantly accelerate the TO process.

From a TOI perspective, the incorporation of physical information into TO extends beyond improving numerical accuracy or computational efficiency (Fig. 13); it fundamentally reconstructs the notion of physics-based optimality by embedding physical fidelity directly into the optimization model itself. Rather than treating physical laws as fixed and exact governing constraints, TOI recognizes that practical physical models are inherently approximate and incomplete. Physical information—derived from experiments, multiscale simulations, or constitutive observations—therefore plays a critical role in calibrating, correcting, and enriching physics-based formulations, leading to a fidelity-aware definition of optimality that more faithfully reflects real material behavior and structural response. A representative application of this paradigm is multiscale TO based on architected cellular structures. Vijayakumaran et al. [124] replaced conventional RVE-based homogenization with a neural constitutive model that satisfies hyperplastic constraints, embedding TO within a finite-strain framework to enable accurate, differentiable, and efficient multiscale structural design. By encoding physically admissible material behavior directly into the constitutive representation, this approach allows physical information at the microscale to be propagated consistently through the optimization process, thereby redefining optimality in terms of both structural performance and constitutive fidelity. Similarly, the DL-MSTO+ framework [145] incorporated a positive-definiteness–guaranteed deep material representation network into multiscale TO, achieving a tight coupling between microstructural parameterization and macroscopic structural optimization while simultaneously improving computational efficiency. Ihuaenyi et al. [146] proposed a mechanics informatics paradigm that quantifies the information content of mechanical test data using stress state entropy, and embeds this metric into a Bayesian optimization framework to enable information-driven specimen design for accurate constitutive model learning. Building upon the mechanics informatics paradigm, subsequent studies extended the information-driven framework from parameterized specimen shapes to TO [115], enabling a much richer design space for informative test specimen design. In summary, these methods exemplify how physics-informed neural representations can act as fidelity-preserving intermediaries between different physical scales, rather than as black-box accelerators.

Figure 13: Physics information, such as material response [124] and manufacturing process [147], can improve fidelity and reliability of structural designs.

5  Opportunities in Topology Optimization Informatics

5.1 General Representations and Optimization Frameworks for Information–Physics Unified Design

Recent advances in large-scale AI have shown that unifying representation, modeling, and optimization within a single computational framework can fundamentally change how complex systems are learned and generated. From a TOI perspective, the central issue is therefore no longer how to couple TO with specific neural architectures, but how to establish more general and expressive forms of information–physics unification.

A critical aspect of this unification concerns the form of representation. Existing density-based descriptions, grayscale fields, and their learned latent embeddings have enabled compact encoding and one-shot generation of design families, yet they remain limited in their ability to explicitly encode physical meaning, manufacturability, and multi-scale structural semantics. This highlights the need for representations in which geometric, physical, and informational attributes are jointly parameterized, rather than hierarchically decoupled.

Closely related challenges arise at the levels of modeling and optimization. When information is treated as an intrinsic component of the governing equations instead of an external modifier, fundamental questions emerge regarding how preference information, historical optimization trajectories, experimental observations, and multi-fidelity knowledge can be consistently incorporated into physics-based formulations. In this context, learning-based generation, surrogate-assisted acceleration, and classical physics-driven optimization may be better viewed not as competing approaches, but as different manifestations of a more general information–physics optimization principle.

5.2 Semantic Structures

At present, structural design remains constrained by existing design methodologies and manufacturing techniques, as it still relies on geometric parameterization within CAD models. Traditional design approaches have consequently reached a developmental bottleneck. Nowadays, we are confronted with the need to develop a representation of semantic design—that is, to define the inherent meaning of structures. This semantic essence encompasses both the objective physical characterization of a structure and its contextual significance within human society. AI has demonstrated the ability to transcend linguistic barriers by enabling translation across diverse natural languages; in this regard, design language can likewise be viewed as a means of describing objective entities. The semantics of structure thus require systematic construction and clarification [148–150] (Fig. 14).

Figure 14: Load identification provides the missing semantic-to-physics bridge. Design semantics define a prior over boundary conditions, while inverse identification assimilates sensing/response data to obtain a posterior, enabling closed-loop semantic structural design under uncertainty [148,149].

Building semantic structural representations requires not only large-scale datasets but also the establishment of a unified ontology that encodes structural functions, physical behaviors, fabrication constraints, material characteristics, and usage contexts in a machine-interpretable manner. Such an ontology functions as the semantic backbone of intelligent design systems, bridging low-level geometric and material attributes with high-level engineering intents and social meanings. Recent advances in multimodal foundation models suggest that structural semantics can be learned through cross-domain alignment—linking geometry, performance metrics, textual descriptions, manufacturing rules, and even environmental or ergonomic considerations into a shared latent space. Within this unified representation, structures are no longer treated merely as geometric configurations but as carriers of functional purposes, physical laws, and human-centered values. This shift lays the groundwork for AI systems capable of understanding why a structure exists, not just how it is shaped, thereby enabling more interpretable, adaptive, and context-aware design generation [151]. Such a semantically enriched framework can be naturally extended to diverse physical domains, including fluid dynamics, fatigue [152] and durability analysis, thermal transport, and multiphysics coupling problems, where functional intent and physical behavior must be coherently integrated.

5.3 Information-Based Acceleration Algorithms

With the rapid development of AI, the paradigm of computational acceleration is shifting from traditional numerical optimization toward an “information–physics integrated” framework. Recent studies increasingly attempt to combine the representational capacity of deep learning with the differentiable solving mechanisms of physical equations, enabling computational procedures to retain the interpretability of classical numerical methods while benefiting from the efficiency of data-driven models. In tasks such as finite element analysis, TO, and multiscale modeling, learning-based surrogate models have been progressively embedded into the physical solving pipeline to accelerate matrix assembly, predict iterative updates, approximate constitutive relations, or provide reduced-order representations of solution spaces. These approaches no longer rely on neural networks as black-box predictors; instead, they integrate them into critical steps of traditional solvers, enforcing structured and differentiable constraints to ensure numerical stability and thereby achieving genuine reductions in computational cost for complex physics-based problems.

More importantly, information-driven acceleration is reshaping the workflow of computational design. As learning-augmented solvers become capable of producing approximate solutions within milliseconds, human–computer collaborative design is significantly enhanced, allowing designers to explore much broader design spaces with real-time feedback. This interactive and instant-response design paradigm not only increases the diversity and creativity of structural innovation but also accelerates the translation of conceptual layouts into engineering-ready solutions.

5.4 Human-Aided Design

As intelligent systems continue to mature, the role of human designers is progressively shifting from direct geometric manipulation toward higher-level semantic steering and strategic decision-making. In this emerging paradigm, designers formulate intentions, constraints, and value judgments, while AI systems execute large-scale exploration, optimization, and real-time evaluation across vast design spaces that far exceed human cognitive limits. This division of labor transforms design from a deterministic procedure into a co-evolutionary process, in which human intuition and domain knowledge guide the search direction, and machine intelligence provides rapid, data-informed feedback that refines and reshapes the evolving solution. Such synergy encourages the discovery of unconventional yet feasible structures, enabling breakthroughs that may not arise through traditional human-only or machine-only workflows.

Furthermore, human-aided design [153] frameworks increasingly rely on multimodal interaction mechanisms, such as natural language instructions, sketch-based prompts, gesture interfaces, and semantic constraints, to bridge the cognitive gap between human intent and machine interpretation (Fig. 15). These interaction paradigms redefine how design knowledge is expressed, making structural concepts more accessible and enabling non-expert stakeholders to participate meaningfully in the design process. From an engineering standpoint, the integration of explainable AI techniques and interactive simulation further enhances trust, transparency, and accountability, ensuring that designers can interrogate machine-generated outcomes, understand physical rationale, and validate design decisions within a continuous feedback loop.

Figure 15: Structural design evolves from human-centered requirement analysis and structural realization toward computer-centered requirement analysis and structural practice.

5.5 Reliable Optimization through Theory–Data Synergy

The acquisition of high-quality design data is often expensive and limited in practice. Rather than representing a weakness of the TOI framework, this constraint highlights a crucial direction for its development. TOI does not aim to replace physics-based modeling with data-driven components; instead, it seeks to establish a structured synergy between theoretical models and real-world data under unified optimization principles. Within this framework, information is not treated as an independent or arbitrary input, but as a controlled interface through which empirical knowledge, historical records, or preference data are systematically embedded into the governing relations while preserving physical consistency.

When available data are sparse, noisy, or low-fidelity, the physics-based model provides a structural anchor that stabilizes the optimization process and prevents over-reliance on incomplete information. Strategies such as multi-fidelity integration, physics-guided data correction, and uncertainty-aware modeling enable informational enhancement while maintaining computational reliability and convergence properties. In this sense, information serves to refine and augment, rather than override, the underlying physical structure.

Therefore, the synergy between theory and real data in TOI is not a simple aggregation of modeling paradigms, but a coherent and interpretable optimization framework that balances physical rigor with empirical adaptability. This direction not only addresses practical data limitations but also offers a concrete pathway toward the integration of physics-based and data-driven paradigms in structural design.

6  Conclusion

TOI shifts structural design from a purely physics-centric process to a paradigm that equally emphasizes needs and physical realization. Structural design is essentially the externalization of internal requirements into manufacturable artifacts. Traditionally, this translation has relied on prevailing manufacturing conditions and individual experiential knowledge. With increasing computational power and richer material systems, demand-side specification and satisfaction have become more attainable. Within a unified optimization framework, explicitly integrating historical information, preference information, and real-world information enables joint modeling of objectives, constraints, manufacturability, and user intent. Consequently, the process evolves from merely solving a prescribed physical problem to a verifiable, multi-objective decision-making procedure that achieves complex requirements more economically.

Looking forward, the evolution of structural design patterns and TO is expected to move toward greater autonomy, integration, and semantic understanding. Future design systems will increasingly depart from the traditional workflow that treats modeling, analysis, and optimization as isolated procedures, and instead adopt holistic frameworks in which geometry, physics, manufacturing constraints, sustainability considerations, and user intent are jointly represented and reasoned about. Advances in differentiable simulation, multimodal generative modeling, and operator learning will allow TO to operate across scales and materials with unprecedented fidelity, enabling structures that adapt to evolving environments and multifunctional requirements. At the same time, interactive human–AI design interfaces will transform the role of engineers from manual decision-makers to strategic supervisors who shape design intent while delegating large-scale exploration and evaluation to intelligent computational agents. The integration of semantic representations, real-world data, and adaptive learning mechanisms will further support the creation of design systems capable of continuous self-improvement and contextual reasoning. Collectively, these developments point toward a future in which TO becomes not merely a computational tool but a foundational paradigm for general structural design—one that unifies creativity, performance, and manufacturability within a coherent, intelligent framework (Fig. 16).

Figure 16: General structural design represents the evolution of structural design toward an intelligent framework that unifies creativity, performance, and manufacturability.

Apart from the advantages of the approaches, some major drawbacks and challenges associated with the current state of TOI are addressed as follow:

(1)   The acquisition of high-quality design data is often expensive and limited, which hinders the training of robust models.

(2)   Current structural design still relies heavily on geometric parameterization within CAD models, which the sources describe as having reached a developmental bottleneck.

(3)   Establishing general and expressive forms of information–physics unification remains a central issue, as geometric, physical, and informational attributes are often still hierarchically decoupled rather than jointly parameterized.

(4)   Conventional physical models are often idealized abstractions that introduce errors through linearization and numerical discretization, requiring complex auxiliary mechanisms to compensate for these discrepancies.

(5)   Semantic design infrastructure remains immature. A canonical semantic representation of design intent is still missing, making design-relevant information hard to formalize and reuse beyond CAD geometry.

Acknowledgement: This research was supported by Meituan Academy of Robotics Shenzhen.

Funding Statement: This research was funded by the Guangdong Basic and Applied Basic Research Foundation (2024A1515011786 and 2025A1515010672).

Author Contributions: Study conception and design: Zelong Liang, Yingjun Wang; data collection: Zelong Liang; analysis and interpretation of results: Zelong Liang; draft manuscript preparation: Zelong Liang, Yingjun Wang, Tinh Quoc Bui, Zhichao Dong, Weihua Li. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Not applicable.

Ethics Approval: Not applicable.

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

Abbreviations

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