Open Access

ARTICLE

A Stochastic Ensemble Physics-Informed Neural Networks via Bagging and Monte Carlo Dropout

Thao Nguyen-Trang^1,2,*, Hiep Ha-Hoang³

1 Laboratory for Artificial Intelligence, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam
2 Faculty of Information Technology, Van Lang School of Technology, Van Lang University, Ho Chi Minh City, Vietnam
3 Department of Computer Science, University of York, York, UK

* Corresponding Author: Thao Nguyen-Trang. Email:

Computer Modeling in Engineering & Sciences 2026, 147(2), 33 https://doi.org/10.32604/cmes.2026.080808

Received 15 February 2026; Accepted 27 April 2026; Issue published 27 May 2026

Abstract

Solving differential equations (DEs), including ordinary differential equations (ODEs) and partial differential equations (PDEs), is fundamental to scientific computing and engineering. The development of deep learning has led to Physics-Informed Neural Networks (PINNs), in which physical laws are embedded directly into the loss function. However, PINNs inherit the intrinsic instability of deep neural networks (DNNs) and lack an effective mechanism for Uncertainty Quantification (UQ). This paper proposes a stochastic ensemble framework to address these limitations. The proposed method is a double-stochastic ensemble framework that combines bagging (via bootstrap resampling and randomized collocation points) with Monte Carlo dropout (MCDO) applied at inference time to provide consistent UQ. This double-stochastic design yields two main contributions. First, it reduces the intrinsic variance of the predicted solution and improves accuracy, particularly in small-data and/or noisy-data regimes. Second, the variance of the ensemble outputs serves as a reliable and computationally feasible proxy for the solution uncertainty. Numerical experiments on ODEs and nonlinear PDEs (e.g., Burgers’ equation) under impulse noise demonstrate that the proposed method outperforms standard baselines in predictive accuracy. Furthermore, the obtained Prediction Interval Coverage Probability (PICP) and Mean Prediction Interval Width (MPIW) confirm that our framework yields well-calibrated prediction intervals and effectively mitigates the severe performance degradation observed in single PINNs.

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Keywords

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