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

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.

💡 Research Summary

The paper addresses the challenging problem of propagating state uncertainty for stochastic differential equations (SDEs) by directly approximating the probability density function (PDF) that satisfies the associated Fokker‑Planck partial differential equation (FP‑PDE). While traditional numerical schemes (finite elements, finite differences) become intractable beyond three dimensions, and Monte‑Carlo methods are computationally expensive, the authors propose to use physics‑informed neural networks (PINNs) to learn the PDF without any data on the true solution.

The core contribution is a rigorous error‑bounding framework. Starting from the linear FP‑PDE operator (D), they note that the residual of a trained PINN (\hat p) directly defines a new PDE for the error (e_1 = p - \hat p). By recursively defining higher‑order error functions (e_i) and approximating each with its own PINN (\hat e_i), the total error can be expressed as a finite series plus a remainder term: \