Solving differential equations via artificial neural networks

Findings and failures in a model problem

verfasst von

Tobias Knoke, Thomas Wick

Abstract

In this work, we discuss some pitfalls when solving differential equations with neural networks. Due to the highly nonlinear cost functional, local minima might be approximated by which functions may be obtained, that do not solve the problem. The main reason for these failures is a sensitivity on initial guesses for the nonlinear iteration. We apply known algorithms and corresponding implementations, including code snippets, and present an example and counter example for the logistic differential equations. These findings are further substantiated with variations in collocation points and learning rates.

Organisationseinheit(en)

Institut für Angewandte Mathematik
PhoenixD: Simulation, Fabrikation und Anwendung optischer Systeme

Externe Organisation(en)

Universität Paris-Saclay

Typ

Artikel

Journal

Examples and Counterexamples

Band

1

Publikationsdatum

11.2021

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Computational Mathematics, Angewandte Mathematik, Mathematik (sonstige)

Elektronische Version(en)

https://doi.org/10.1016/j.exco.2021.100035 (Zugang: Offen)