Three Ways to Solve Partial Differential Equations with Neural Networks -- A Review

TL;DR

This review compares three neural network-based methods for solving partial differential equations, highlighting their simplicity and effectiveness for high-dimensional problems, supported by educational software and comprehensive references.

Contribution

It provides a clear comparison of physics-informed neural networks, Feynman-Kac based methods, and backward stochastic differential equations approaches, with practical tutorials and extensive bibliography.

Findings

Three neural network approaches are effective for high-dimensional PDEs.

Software tutorials facilitate understanding and experimentation.

The methods vary in complexity and applicability.

Abstract

Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability for high-dimensional problems: physics-informed neural networks, methods based on the Feynman-Kac formula and methods based on the solution of backward stochastic differential equations. The article is accompanied by a suite of expository software in the form of Jupyter notebooks in which each basic methodology is explained step by step, allowing for a quick assimilation and experimentation. An extensive bibliography summarizes the state of the art.