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

摘要

As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it is highly desired to develop new methods for such computation. In deviation from traditional approaches, this short communication paper explores a computational paradigm that couples numerical solutions of PDEs via machine-learning (ML) based methods. Particularly, it solves PDEs in subdomains as in a conventional method but develops and trains artificial neural networks (ANN) to couple the PDEs at their interfaces, leading to their solutions in the whole domains. The concepts and algorithms for the ML coupling are discussed via computation of coupled Poisson equations and coupled advection-diffusion equations. Unlike in a conventional method, Schwarz iteration may not be necessary for the ML coupling in computation of the latter. Numerical examples show that the coupling can generate solutions to these equations with acceptable accuracy. They also illustrate that it exhibits predictability; ML algorithms trained with a set of initial and boundary conditions still work in computation when the conditions are modified. Although preliminary, this exploratory study indicates that the ML paradigm is promising in terms of feasibility and performance and deserves further research.(c) 2021 Elsevier B.V. All rights reserved.