Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries

TL;DR

This paper introduces a DeepONet-based neural operator to solve the 2D Helmholtz equation on arbitrary geometries, enabling efficient and generalizable modeling of scattered fields without remeshing or FEM data.

Contribution

It presents a physics-informed neural operator that encodes arbitrary geometries and generalizes to unseen cases, reducing computational costs compared to traditional FEM methods.

Findings

The model accurately predicts scattered fields for unseen geometries.

It generalizes well when trained on diverse geometries.

The approach is computationally lighter than FEM.

Abstract

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network based on the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. This inclusion acts as a scatterer for an incoming harmonic wave. The aim is to learn the operator linking the geometry of the scatterer to the resulting scattered field. A signed distance function to the boundary of the inner inclusion, evaluated at several points in the domain, is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture, while local information is used as input for the trunk part. This approach enables the encoding of arbitrary geometries, whether they are parameterized or not. The evaluation of the model on unseen geometries is compared with its finite element method (FEM) equivalent to test its generalization capabilities. The trained network weights implicitly embed the local physics and their interaction with the domain geometry. If the training space sufficiently covers the target evaluation space, the model can generalize accordingly. Furthermore, it can be refined to extend to another region of interest without retraining from scratch. This framework also avoids the need to remesh the domain for each geometry. The proposed approach delivers a computationally lighter surrogate model than FEM alternatives and avoids relying on FEM-generated training data.