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

arXiv: https://arxiv.org/abs/2410.22371 · HTML: https://mlsanity.com/paper/2410.22371/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 38 references that are in the map.
- First posted on arXiv 28 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [Deep Learning Method for Computing Committor Functions with Adaptive Sampling](https://mlsanity.com/paper/2404.06206/) (2024)
- [Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification](https://mlsanity.com/paper/2403.15626/) (2024)
- [On The Evaluation of Collision Probability along a Path](https://mlsanity.com/paper/2311.08204/) (2023)

## Built on by (most cited first, 1 of 1)

- [Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees](https://mlsanity.com/paper/2603.19165/) (2026)

## Abstract (arXiv)

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.