Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEs
This work provides a theoretical and practical error-bound framework for PINNs in solving linear PDEs, which is incremental but addresses a specific bottleneck in uncertainty quantification for stochastic systems.
The paper tackles the problem of solving Fokker-Planck PDEs for stochastic processes by using physics-informed neural networks (PINNs) to approximate probability density functions, achieving significant computational speedup compared to Monte Carlo methods while validating error bounds on nonlinear, high-dimensional, and chaotic systems.
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.