arxiv: 2605.00760 · v1 · submitted 2026-05-01 · 💻 cs.LG

Recognition: unknown

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

Rodolphe Barlogis , Ferhat Tamssaouet , Quentin Falcoz , St\'ephane Grieu

Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:58 UTC · model grok-4.3

classification 💻 cs.LG

keywords DeepONetHelmholtz equationoperator learningsigned distance functionscattered fieldnon-parametric geometriesphysics-informedsurrogate model

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The pith

DeepONet learns the operator mapping arbitrary scatterer geometries to 2D Helmholtz scattered fields

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates a DeepONet-based approach to learn the solution operator for the 2D Helmholtz equation on domains with arbitrary central inclusions acting as scatterers. Geometry is encoded using the signed distance function to the inclusion boundary, sampled at points throughout the domain, and fed into the branch network alongside local coordinates in the trunk. This setup allows handling non-parameterized geometries and testing generalization on unseen shapes against finite element solutions. The resulting model acts as a fast surrogate that implicitly captures physics and geometry interactions without requiring domain remeshing for each new geometry.

Core claim

A physics-informed DeepONet learns the operator that links the geometry of an arbitrary scatterer, encoded via signed distance function, to the scattered field solution of the Helmholtz equation in 2D, generalizing to new geometries provided the training distribution is adequate and avoiding the need to remesh for each instance.

What carries the argument

DeepONet with branch network input as signed distance function values at domain points encoding the scatterer boundary, and trunk network input as local coordinates, to learn the geometry-to-field operator.

If this is right

If the training space covers the target evaluation space, the model generalizes to unseen geometries.
The network weights implicitly embed the local physics and their interaction with the domain geometry.
The approach provides a computationally lighter surrogate model compared to traditional FEM methods.
It avoids the need to remesh the domain for each new geometry.
The model can be refined for new regions of interest without full retraining.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar signed distance encoding might enable operator learning for time-dependent or nonlinear wave equations.
Optimized scatterer designs could be found more rapidly by querying the learned operator repeatedly.
This surrogate could support real-time simulations in applications like acoustic design or radar cross-section prediction.

Load-bearing premise

The signed distance function evaluated at several points in the domain is enough to encode arbitrary boundary geometries so the branch network can learn the full operator mapping.

What would settle it

A direct numerical comparison between the DeepONet prediction and a high-resolution FEM solution for an unseen scatterer with a highly irregular or non-convex boundary, checking if errors exceed acceptable tolerances across the domain.

Figures

Figures reproduced from arXiv: 2605.00760 by Ferhat Tamssaouet, Quentin Falcoz, Rodolphe Barlogis, St\'ephane Grieu.

**Figure 2.** Figure 2: Schematic representation of the physics-informed DeepONet solving an Helmholtz [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: 2D domain with an inclusion. ϕ is evaluated in specific points around to distinguish different geometries. It serves as input of the branch net. Future work may investigate how both the number and the spatial distribution of these points can be optimized in order to cover a broader class of shapes. The trunk network receives local information at the query point (x, y). Its input vector includes (x, y), the… view at source ↗

**Figure 4.** Figure 4: 2D Domain used for experiment. Highlight the incident wave field [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

**Figure 5.** Figure 5: Symmetrical geometry encoded with ϕ evaluated in ten different points. It serves as input of the branch net. 2,000 points per geometry are distributed along the corresponding boundaries characterized by ϕ = 0. The numbers of collocation points and boundary points were chosen empirically so as to maintain a numerical balance among the different loss terms being enforced. More optimal configurations are like… view at source ↗

**Figure 6.** Figure 6: Evolution of the training losses over iterations. Individual contributions from [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: FEM solution of the Helmholtz problem (Incidence angle: [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: Absolute pointwise error between the trained model predictions and the FEM [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗

**Figure 9.** Figure 9: Normalized pointwise error between the trained model predictions and the FEM [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗

**Figure 10.** Figure 10: Subdomain of interest D2 used to compute the localized error near the concavity. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies DeepONet to the 2D Helmholtz scattering problem on arbitrary non-parametric inclusions by encoding geometry with sampled signed distance functions, but the abstract and setup leave the actual accuracy and boundary resolution unproven.

read the letter

The core move is straightforward: they encode an arbitrary central inclusion via its signed distance function evaluated at a fixed set of points, feed that into the branch network of a DeepONet, and let the trunk handle local coordinates to output the scattered field for the Helmholtz equation. Training uses a physics-informed loss rather than FEM-generated labels, and they check performance on held-out shapes against traditional FEM solves. This removes the remeshing step that normally appears when the geometry changes. That part is practical and follows the DeepONet template without obvious invention. The claim that the weights implicitly carry the local physics plus geometry interaction is the usual operator-learning pitch, and the idea of refining the model on a new region without full retraining is a minor but useful note. What the paper does cleanly is show how to drop the usual parametric assumption on the scatterer shape. The soft spot is the missing evidence on whether the fixed SDF sampling actually works. The abstract gives no error values, no count of sample points, no description of the loss weighting between PDE residual and boundary enforcement, and no information on how the training geometries were sampled. Without those, it is impossible to judge if the finite projection of the boundary is dense enough for wave scattering on non-smooth or high-curvature inclusions. The stress-test concern about under-resolved boundaries therefore lands as a real open question rather than a minor quibble. If the full paper contains quantitative tables or plots that close this gap, the contribution becomes more solid; right now it reads as a plausible setup that still needs its central assumption tested. This is for groups already working on neural operators for wave or scattering problems who want a surrogate that handles variable domains. A reader looking for a drop-in replacement for FEM on complex 2D scatterers would need the numbers first. I would send it to peer review so the authors can supply the missing metrics and sampling details; the architecture is standard enough that referees can evaluate the implementation quickly once the results are shown.

Referee Report

2 major / 2 minor

Summary. The paper proposes a DeepONet-based physics-informed neural operator to learn the mapping from arbitrary non-parametric 2D scatterer geometries (encoded via signed distance function values sampled at multiple points) to the scattered field solution of the Helmholtz equation in a fixed square domain. The branch network takes the SDF input while the trunk uses local coordinates; training uses a physics-informed loss, with claims of generalization to unseen geometries validated against FEM, computational efficiency, no remeshing requirement, and implicit embedding of physics in the weights.

Significance. If the quantitative results and generalization claims hold, this would provide a useful surrogate modeling framework for wave scattering on variable non-parametric domains, extending neural operators beyond parametric geometries and reducing the need for repeated FEM meshing. It correctly leverages DeepONet for operator learning and physics-informed training, which are established strengths in the field.

major comments (2)

[Abstract] Abstract: the central claim of generalization to unseen arbitrary geometries is presented without any quantitative error metrics, training hyperparameters, or implementation details for the physics-informed loss (PDE residual plus boundary conditions at the SDF zero level set), making it impossible to assess whether the operator is learned accurately or merely approximates the training distribution.
[Abstract and method description] Geometry encoding (as described in the abstract and method): the branch network receives only the signed distance function evaluated at a fixed finite set of points. For the physics-informed loss to enforce correct geometry-dependent scattering conditions on non-smooth or high-curvature inclusions, this fixed sampling must resolve all boundary features that affect the wave field; the manuscript provides no analysis or ablation showing that the chosen sampling density suffices, which is load-bearing for the non-parametric geometry claim.

minor comments (2)

[Abstract] Abstract: the number and distribution of points used to evaluate the signed distance function are not specified, which affects both reproducibility and the ability to judge whether boundary details are captured.
[Abstract] Abstract: the statement that the model 'can be refined to extend to another region of interest without retraining from scratch' is asserted but not supported by any description of the refinement procedure or supporting experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim of generalization to unseen arbitrary geometries is presented without any quantitative error metrics, training hyperparameters, or implementation details for the physics-informed loss (PDE residual plus boundary conditions at the SDF zero level set), making it impossible to assess whether the operator is learned accurately or merely approximates the training distribution.

Authors: We agree that the abstract would benefit from quantitative support. The main text (Section 4) and appendix already report relative L2 error metrics versus FEM on unseen geometries, along with training hyperparameters and the explicit form of the physics-informed loss (Helmholtz PDE residual plus boundary conditions at the SDF zero level set). We will revise the abstract to include key quantitative results and a brief description of the loss components so that the generalization claim can be assessed directly from the abstract. revision: yes
Referee: [Abstract and method description] Geometry encoding (as described in the abstract and method): the branch network receives only the signed distance function evaluated at a fixed finite set of points. For the physics-informed loss to enforce correct geometry-dependent scattering conditions on non-smooth or high-curvature inclusions, this fixed sampling must resolve all boundary features that affect the wave field; the manuscript provides no analysis or ablation showing that the chosen sampling density suffices, which is load-bearing for the non-parametric geometry claim.

Authors: We acknowledge that the current manuscript does not contain an ablation or sensitivity analysis on the SDF sampling density. The fixed-point sampling is indeed central to the non-parametric claim. In the revision we will add a dedicated analysis (including an ablation on sampling density for representative high-curvature and non-smooth inclusions) to demonstrate that the chosen density suffices to resolve the boundary features relevant to the wave field at the frequencies considered. revision: yes

Circularity Check

0 steps flagged

No circularity; standard physics-informed DeepONet operator learning

full rationale

The paper encodes non-parametric geometries via fixed-point samples of the signed distance function fed to the DeepONet branch network, uses coordinate inputs for the trunk, and trains with a physics-informed loss that enforces the Helmholtz PDE residual plus implicit boundary conditions at the SDF zero level set. Generalization is assessed by direct numerical comparison against independent FEM solutions on held-out geometries. No derivation step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing claim rests on self-citation chains or imported uniqueness theorems. The approach is self-contained against external FEM benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the neural operator's ability to approximate the solution operator for the Helmholtz equation using SDF geometry encoding, with standard assumptions about well-posedness of the PDE and sufficient coverage of geometry space in training.

free parameters (2)

DeepONet branch and trunk network hyperparameters
Architecture details such as layer counts and widths are chosen to enable learning the operator.
Training data distribution over geometries
Selection of training scatterer shapes determines generalization.

axioms (2)

standard math The 2D Helmholtz equation with appropriate boundary conditions admits unique solutions for the scattered field.
Invoked implicitly as the physical model being learned.
domain assumption Signed distance function evaluated at discrete points provides a complete encoding of arbitrary boundary geometry.
Central to the branch net input design.

pith-pipeline@v0.9.0 · 5557 in / 1329 out tokens · 67414 ms · 2026-05-09T18:58:45.573472+00:00 · methodology

discussion (0)

Reference graph

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