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

Understand

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.