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

2021/02/23 by Jan Blechschmidt, Blechschmidt, Jan, Oliver G. Ernst +1 · 8 citations
Physics and Astronomy · #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2102.11802

openalex publication_date 2021/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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