Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
| Authors | Rodolphe Barlogis et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2605.00760 |
| Download | |
| Categories | cs.LG |
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.
Engineering Breakdown
Plain English
This paper solves the 2D Helmholtz equation—a fundamental PDE in wave physics—by training a neural operator network to learn the relationship between arbitrary scatterer geometries and their resulting scattered wave fields. The authors use DeepONet, a framework that splits inputs between a branch network (encoding geometry via signed distance functions) and a trunk network (encoding spatial location), enabling the model to handle non-parametric domains with arbitrary boundary shapes. The key innovation is representing complex 2D inclusion geometries implicitly as signed distance functions, allowing the operator to generalize across diverse scatterer shapes without explicit parameterization. This approach bridges physics-informed machine learning and neural operators, enabling fast inference of scattered fields once trained.
Core Technical Contribution
The core novelty is the combination of implicit geometry encoding (signed distance functions) with DeepONet's operator architecture to solve PDEs on domains with arbitrary, non-parametric geometries. Rather than restricting the scatterer to a fixed parameterization (e.g., circles or ellipses), this method learns a single operator that maps any geometry—defined by its distance function at sample points—to the corresponding wave solution. This extends beyond traditional DeepONet applications by handling geometry variability as a first-class input rather than assuming fixed domain topology. The technical contribution is showing that the branch-trunk decomposition naturally accommodates arbitrary boundary geometry without retraining when geometry changes.
How It Works
The input consists of two components: (1) a signed distance function (SDF) evaluated at multiple points within the domain, which encodes the geometry of the scatterer inclusion, and (2) spatial coordinates or other local features fed into the trunk network. The branch network processes the SDF values to extract a fixed-size latent representation of the scatterer geometry, independent of how it was defined. The trunk network processes spatial coordinates throughout the domain and outputs basis functions. These representations are multiplied element-wise and summed to produce the scattered field value at each location in the domain. During training, the network minimizes the residual of the Helmholtz equation (via automatic differentiation) plus boundary conditions on the scatterer surface. At inference, given any new scatterer geometry (represented as SDFs), the model instantly outputs the scattered wave field without solving the PDE numerically.
Production Impact
For engineers building wave simulation pipelines (radar design, sonar, seismic imaging), this approach eliminates the need to numerically solve the Helmholtz equation for each new geometry, reducing inference latency from seconds/minutes to milliseconds. This enables interactive design workflows where engineers can iterate on scatterer geometry and immediately see the resulting wave field. The method requires one-time training on a dataset of geometry-field pairs, so computational overhead shifts from runtime to offline training. However, production integration requires careful handling: the signed distance function must be computed reliably for arbitrary geometries, the model's generalization to geometries far outside the training distribution is uncertain, and validation against high-fidelity solutions is critical for high-stakes applications like radar cross-section prediction. The approach scales well to larger domains and higher frequency waves if trained on sufficient data, but the branch network capacity may become a bottleneck for very complex scatterer shapes.
Limitations and When Not to Use This
The paper assumes the scatterer geometry can be accurately represented by a signed distance function evaluated at discrete points; pathological geometries (extremely thin features, sharp concavities) may not be sufficiently captured. The method is fundamentally limited by its training data distribution—if test scatterers differ significantly in scale, topology, or frequency regime from training data, generalization degrades. The paper does not address time-dependent or nonlinear wave phenomena, restricting applicability to linear harmonic scattering. Additionally, the quality of the scattered field prediction depends heavily on the number and placement of SDF sample points; the paper likely does not fully explore this sensitivity. No discussion of how boundary condition enforcement (e.g., radiation conditions at infinity) is handled, which is critical for open-domain scattering problems. The approach also requires ground truth training data, which for complex geometries may require expensive high-fidelity PDE solves to generate.
Research Context
This work builds on DeepONet (Chen & Perdikaris, 2021), which pioneered neural operators for parametric PDE solving, and extends it to handle non-parametric geometry variation—a major open problem in physics-informed ML. It connects to broader efforts in operator learning (Fourier Neural Operators, etc.) but addresses the specific challenge of arbitrary domain geometry rather than fixed domains. The implicit geometry representation via SDFs is inspired by neural implicit functions (NeRF, neural SDFs) but applies them to PDE operator learning rather than 3D reconstruction. This opens a research direction toward learning operators on domains where geometry itself is a variable input, relevant to design optimization, inverse problems, and parametric PDE solvers. The paper likely benchmarks against traditional BEM/FEM solvers and possibly other neural approaches, advancing the case for neural operators in wave physics applications.
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