Learning solution operators of PDEs defined on varying domains via MIONet

TL;DR

This paper introduces MIONet, a meshless neural network approach for learning solution operators of PDEs on varying domains, with theoretical justification and experimental validation on 2D Poisson equations.

Contribution

The paper extends MIONet's approximation theory to metric spaces and demonstrates its ability to learn PDE solutions with all parameters varying, including domain shape and boundary conditions.

Findings

MIONet can approximate solution mappings for PDEs on complex, varying domains.

The method performs well on convex polygons and polar regions with smooth boundaries.

It is a flexible, meshless solver applicable to various PDE problems.

Abstract

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces, establishing that MIONet can approximate mappings with multiple inputs in metric spaces. Subsequently, we construct a set consisting of some appropriate regions and provide a metric on this set thus make it a metric space, which satisfies the approximation condition of MIONet. Building upon the theoretical foundation, we are able to learn the solution mapping of a PDE with all the parameters varying, including the parameters of the differential operator, the right-hand side term, the boundary condition, as well as the domain. Without loss of generality, we for example perform the experiments for 2-d Poisson equations, where the domains and the right-hand side terms are varying. The results provide insights into the performance of this method across convex polygons, polar regions with smooth boundary, and predictions for different levels of discretization on one task. We also show the additional result of the fully-parameterized case in the appendix for interested readers. Reasonably, we point out that this is a meshless method, hence can be flexibly used as a general solver for a type of PDE.