An exploratory study on machine learning to couple numerical solutions of partial differential equations

TL;DR

This paper explores a novel machine learning-based approach to couple numerical solutions of PDEs, using neural networks to connect subdomain solutions at interfaces, showing preliminary promising results for complex coupled PDEs.

Contribution

It introduces a new ML paradigm for coupling PDE solutions via neural networks at interfaces, diverging from traditional methods.

Findings

Feasibility demonstrated with coupled Poisson and advection-diffusion equations.

Preliminary results show promising performance of ML coupling.

The approach offers a potential alternative to conventional PDE coupling methods.

Abstract

As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it has become highly desired to develop new methods for such computation. In deviation from conventional approaches, this short communication paper explores a computational paradigm that couples numerical solutions of PDEs via machine-learning (ML) based methods, together with a preliminary study on the paradigm. Particularly, it solves PDEs in subdomains as in a conventional approach but develops and trains artificial neural networks (ANN) to couple the PDEs' solutions at their interfaces, leading to solutions to the PDEs in the whole domains. The concepts and algorithms for the ML coupling are discussed using coupled Poisson equations and coupled advection-diffusion equations. Preliminary numerical examples illustrate the feasibility and performance of the ML coupling. Although preliminary, the results of this exploratory study indicate that the ML paradigm is promising and deserves further research.