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

TL;DR

This paper develops a theoretical framework for error bounds in physics-informed neural networks approximating the probability density functions governed by Fokker-Planck PDEs, validated through empirical results on complex systems.

Contribution

It introduces a novel error analysis framework for PINNs applied to Fokker-Planck PDEs, including practical bounds and scalability insights.

Findings

PINNs can accurately approximate solution PDFs of Fokker-Planck PDEs.

Theoretical error bounds are validated empirically on complex systems.

PINNs offer significant computational speedup over Monte Carlo methods.

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.