  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.

Numerical Analysis;Numerical Analysis