Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs

Chun-Wei Kong, Luca Laurenti, Jay McMahon, Morteza Lahijanian
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:2291-2324, 2025.

Abstract

Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-kong25b, title = {Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs}, author = {Kong, Chun-Wei and Laurenti, Luca and McMahon, Jay and Lahijanian, Morteza}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {2291--2324}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/kong25b/kong25b.pdf}, url = {https://proceedings.mlr.press/v286/kong25b.html}, abstract = {Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.} }
Endnote
%0 Conference Paper %T Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs %A Chun-Wei Kong %A Luca Laurenti %A Jay McMahon %A Morteza Lahijanian %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-kong25b %I PMLR %P 2291--2324 %U https://proceedings.mlr.press/v286/kong25b.html %V 286 %X Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the Fokker-Planck partial differential equation (FP-PDE). However, it is generally infeasible to solve the FP-PDE in closed form. In this work, we show that physics-informed neural networks (PINNs) can be trained to approximate the solution PDF. Our main contribution is the analysis of PINN approximation error: we develop a theoretical framework to construct tight error bounds using PINNs. In addition, we derive a practical error bound that can be efficiently constructed with standard training methods. We discuss that this error-bound framework generalizes to approximate solutions of other linear PDEs. Empirical results on nonlinear, high-dimensional, and chaotic systems validate the correctness of our error bounds while demonstrating the scalability of PINNs and their significant computational speedup in obtaining accurate PDF solutions compared to the Monte Carlo approach.
APA
Kong, C., Laurenti, L., McMahon, J. & Lahijanian, M.. (2025). Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:2291-2324 Available from https://proceedings.mlr.press/v286/kong25b.html.

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