Solving parametric elliptic interface problems via interfaced operator network

TL;DR

This paper introduces IONet, a neural network framework designed to solve parametric elliptic interface PDEs by capturing discontinuities across interfaces, reducing data needs, and outperforming existing methods in accuracy and versatility.

Contribution

The paper proposes a novel interfaced operator network (IONet) that effectively handles discontinuities in elliptic interface problems using domain decomposition and physics-informed training.

Findings

IONet accurately captures interface discontinuities.

It requires fewer training samples due to physics-informed loss.

Outperforms existing deep operator networks in tests.

Abstract

Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. To capture the discontinuities in both the input functions and the output solutions across the interface, IONet divides the entire domain into several separate subdomains according to the interface and uses multiple branch nets and trunk nets. Each branch net extracts latent representations of input functions at a fixed number of sensors on a specific subdomain, and each trunk net is responsible for output solutions on one subdomain. Additionally, tailored physics-informed loss of IONet is proposed to ensure physical consistency, which greatly reduces the training dataset requirement and makes IONet effective without any paired input-output observations inside the computational domain. Extensive numerical studies demonstrate that IONet outperforms existing state-of-the-art deep operator networks in terms of accuracy and versatility.