The effect of time discretization on the solution of parabolic PDEs with ANNs

TL;DR

This paper explores how different time discretization methods affect the accuracy and stability of solving parabolic PDEs using Extreme Learning Machine neural networks, demonstrating promising results with BDF techniques.

Contribution

It introduces a novel approach combining ELM neural networks with classical time discretization methods for solving parabolic PDEs efficiently.

Findings

ELM-based collocation effectively solves stationary elliptic problems

BDF methods enhance stability and accuracy for parabolic PDEs

Numerical experiments confirm high-accuracy solutions with proposed methods

Abstract

We investigate the resolution of parabolic PDEs via Extreme Learning Machine (ELMs) Neural Networks, which have a single hidden layer and can be trained at a modest computational cost as compared with Deep Learning Neural Networks. Our approach addresses the time evolution by applying classical ODEs techniques and uses ELM-based collocation for solving the resulting stationary elliptic problems. In this framework, the $\theta$-method and Backward Difference Formulae (BDF) techniques are investigated on some linear parabolic PDEs that are challeging problems for the stability and accuracy properties of the methods. The results of numerical experiments confirm that ELM-based solution techniques combined with BDF methods can provide high-accuracy solutions of parabolic PDEs.