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

Chun-Wei Kong, Luca Laurenti, Jay McMahon, Morteza Lahijanian

Keywords: Robustness of PINNs, Fokker-Planck partial differential equation, physics-informed learning, stochastic differential equation, uncertainty propagation, learning error quantification

TL;DR: Bounding the approximation error of physics-informed neural networks for solving the Fokker-Planck equation

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.

Code Link: https://github.com/aria-systems-group/pinn_pde/tree/release/uai2025

Readers: auai.org/UAI/2025/Conference, auai.org/UAI/2025/Conference/Area_Chairs, auai.org/UAI/2025/Conference/Reviewers, auai.org/UAI/2025/Conference/Submission779/Authors, auai.org/UAI/2025/Conference/Submission779/Reproducibility_Reviewers

Submission Number: 779