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

2024-10-28

AI Summary

This work addresses the reliability of Physics-Informed Neural Networks (PINNs) for solving the Fokker–Planck partial differential equation (FP-PDE). We establish, for the first time, a tight, verifiable, and computable theoretical error bound for PINN approximations to FP-PDEs. Methodologically, we integrate variational analysis with the PINN framework to derive a general error estimation paradigm applicable to high-dimensional linear PDEs, and design a practical, training-friendly error bound. Our contributions are threefold: (1) the first rigorous error theory specifically tailored for PINNs solving FP-PDEs; (2) an error bound that is explicitly computable and numerically verifiable—enabling quantitative assessment of approximation accuracy; and (3) empirical validation across nonlinear, high-dimensional, and chaotic stochastic systems, demonstrating substantial improvements in both efficiency and accuracy for probability density function (PDF) estimation over Monte Carlo methods.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

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.

Problem

• Analyzing error bounds for PINNs in solving Fokker-Planck PDEs.
• Developing a theoretical framework for tight error bounds using PINNs.
• Validating PINNs' scalability and computational speedup over Monte Carlo methods.

Innovation

• Physics-informed neural networks approximate Fokker-Planck PDE solutions.
• Theoretical framework constructs tight error bounds for PINNs.
• Error-bound framework generalizes to other linear PDEs.