Practical Aspects on Solving Differential Equations Using Deep Learning: A Primer

TL;DR

This paper provides a practical introduction to using deep learning, specifically the Deep Galerkin method, for solving various differential equations, including PDEs, ODEs, and integral equations, with accessible implementation guidance.

Contribution

It offers a comprehensive primer on the Deep Galerkin method, including step-by-step examples and code, making deep learning approaches to differential equations more accessible.

Findings

Successfully applied to the 1D heat equation

Extended to systems of ODEs and integral equations

Code is executable on a standard computer without GPU

Abstract

Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep Galerkin method, which uses deep neural networks to solve differential equations. This primer aims to provide technical and practical insights into the Deep Galerkin method and its implementation. We demonstrate how to solve the one-dimensional heat equation step-by-step. We also show how to apply the Deep Galerkin method to solve systems of ordinary differential equations and integral equations, such as the Fredholm of the second kind. Additionally, we provide code snippets within the text and the complete source code on Github. The examples are designed so that one can run them on a simple computer without needing a GPU.