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

arXiv: https://arxiv.org/abs/2605.00760 · HTML: https://mlsanity.com/paper/2605.00760/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 19 references that are in the map.
- First posted on arXiv 1 May 2026.

## Builds on (most cited first, 10 of 10)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries](https://mlsanity.com/paper/2308.12939/) (2023)
- [Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows](https://mlsanity.com/paper/2501.01934/) (2025)
- [Point-DeepONet: Predicting Nonlinear Fields on Non-Parametric Geometries under Variable Load Conditions](https://mlsanity.com/paper/2412.18362/) (2024)
- [Physics and geometry informed neural operator network with application to acoustic scattering](https://mlsanity.com/paper/2406.03407/) (2024)
- [Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs](https://mlsanity.com/paper/2512.14596/) (2025)

## Abstract (arXiv)

This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network, the DeepONet framework. We consider a 2D square domain with an inclusion of arbitrary boundary geometry at its center. It 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 in several points on the domain is used to encode its geometry. It serves as input for the branch part of the DeepONet architecture and local information as the 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 was compared to 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.