Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
2026-05-01 • Machine Learning
Machine Learning
AI summaryⓘ
The authors developed a neural network method to solve how waves scatter off shapes inside a square area, without needing to describe those shapes with simple formulas. They use a special function to represent the shape's outline and feed it into a neural network that learns the connection between the shape and the scattered wave pattern. Their model can handle new, unseen shapes and is faster than traditional methods, without having to break the area into small pieces each time. This approach also allows updating the model for new situations without starting over.
2D Helmholtz equationphysics-informed neural operatorDeepONetsigned distance functionwave scatteringfinite element method (FEM)generalizationnon-parametric geometrysurrogate modeldomain remeshing
Authors
Rodolphe Barlogis, Ferhat Tamssaouet, Quentin Falcoz, Stéphane Grieu
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.