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

Jan Blechschmidt, Oliver G. Ernst
2/23/2021
math.NA

Official code: janblechschmidt/PDEsByNNs (189 stars, MIT, last commit 2021-02-25).

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.

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Code Implementations(1)

janblechschmidt/PDEsByNNsOfficial100%
MIT

This repository contains a number of Jupyter Notebooks illustrating different approaches to solve partial differential equations by means of neural networks using TensorFlow.

18970Jupyter NotebookJan 26, 20215 years agoMIT

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Cite this paper

@article{blechschmidt2021three,
  title  = {Three Ways to Solve Partial Differential Equations with Neural Networks -- A Review},
  author = {Jan Blechschmidt and Oliver G. Ernst},
  year   = {2021},
  eprint = {2102.11802},
  archivePrefix = {arXiv},
  url    = {https://arxiv.org/abs/2102.11802}
}

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