XI-DeepONet: An operator learning method for elliptic interface problems

TL;DR

XI-DeepONet is a novel operator learning framework that incorporates interface geometry and level set functions as inputs, enabling mesh-free, accurate, and robust solutions to parametric elliptic interface problems without requiring training data.

Contribution

The paper introduces XI-DeepONet, an extension of DeepONet that models interface geometries and level set functions as inputs, allowing mesh-free learning of solutions to elliptic interface PDEs.

Findings

Accurately predicts solutions for complex interface geometries.

Operates without mesh or input-output data pairs.

Demonstrates robustness across various numerical experiments.

Abstract

Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges when dealing with problems involving complex geometries. Machine learning-based methods, on the other hand, are mesh-free, thus providing a promising alternative. In particular, operator learning methods have been proposed to learn the mapping from the input space to the solution space, enabling rapid inference of solutions to partial differential equations (PDEs) once trained. In this work, we address the parametric elliptic interface problem. Building upon the deep operator network (DeepONet), we propose an extended interface deep operator network (XI-DeepONet). XI-DeepONet exhibits three unique features: (1) The interface geometry is incorporated into the neural network as an additional input, enabling the network to infer solutions for new interface geometries once trained; (2) The level set function associated with the interface geometry is treated as the input, on which the solution mapping is continuous and can be effectively approximated by the deep operator network; (3) The network can be trained without any input-output data pairs, thus completely avoiding the need for meshes of any kind, directly or indirectly. We conduct a comprehensive series of numerical experiments to demonstrate the accuracy and robustness of the proposed method.