arxiv: 2604.21431 · v1 · submitted 2026-04-23 · 💻 cs.CE · physics.comp-ph

Recognition: unknown

JAX-BEM: Gradient-Based Acoustic Shape Optimisation via a Differentiable Boundary Element Method

James Hipperson, Jonathan Hargreaves, Trevor Cox

Pith reviewed 2026-05-08 13:33 UTC · model grok-4.3

classification 💻 cs.CE physics.comp-ph

keywords differentiable boundary element methodacoustic shape optimizationgradient-based optimizationJAX automatic differentiationcomputational acousticsboundary integral equationwave scattering

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The pith

JAX-BEM provides a differentiable boundary element method that matches standard BEM accuracy while enabling gradient-based acoustic shape optimization.

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

The paper introduces an implementation of the boundary element method inside the JAX automatic-differentiation framework. This implementation tracks gradients through the discretization and the linear solve, so that geometry parameters can be adjusted directly by gradient descent. The authors verify that the resulting solver produces errors comparable to conventional BEM codes on a standard acoustic benchmark. Because the same code also supplies exact gradients, it removes the need for finite-difference approximations or derivative-free optimizers when designing shapes that control sound fields.

Core claim

A boundary-element solver written in JAX produces acoustic fields whose accuracy matches that of existing non-differentiable BEM codes while automatically supplying gradients with respect to surface geometry, thereby permitting gradient-based shape optimization for acoustic objectives.

What carries the argument

Automatic differentiation applied to the boundary-integral discretization and the resulting dense linear system for time-harmonic acoustic scattering.

If this is right

Acoustic geometries with dozens of free parameters can be refined by gradient descent instead of derivative-free search.
Inverse problems that recover shape from measured acoustic data become directly solvable by the same code path.
The same differentiable formulation can be reused for electromagnetic or other linear wave problems without new derivation.
Optimization loops can be embedded inside larger machine-learning pipelines that treat geometry as trainable parameters.

Where Pith is reading between the lines

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

The approach could be combined with neural-network surrogates to accelerate repeated forward solves inside long optimization trajectories.
Extension to three-dimensional or topologically changing surfaces would test whether the current discretization remains stable under differentiation.
Because JAX supports hardware acceleration, the method may lower the cost of high-frequency acoustic design studies relative to finite-element alternatives.

Load-bearing premise

Automatic differentiation through the BEM discretization and linear solve yields accurate, stable gradients for the acoustic boundary conditions.

What would settle it

On a simple scattering benchmark, finite-difference gradients and JAX-BEM gradients differ by more than a few percent, or an optimization run guided by JAX-BEM gradients fails to reduce the objective function.

Figures

Figures reproduced from arXiv: 2604.21431 by James Hipperson, Jonathan Hargreaves, Trevor Cox.

**Figure 1.** Figure 1: Error vs. analytic solution for increasing mesh refinement. (Scattering from a rigid sphere) 10 0 10 1 k [m^-1] 10 4 10 3 10 2 10 1 10 0 MAE bempp JAX-BEM view at source ↗

**Figure 2.** Figure 2: Error vs. analytic solution for increasing wavenumber k. (Scattering from a rigid sphere) wavenumbers. The mean absolute error of JAX-BEM relative to bempp ranged from 8 × 10−4 (N = 128) to 8.2×10−6 (N = 8192). Mean absolute error compared to analytic solutions ranged from 4 × 10−2 (N = 128) to 4 × 10−4 (N = 8192) for both JAX-BEM and bempp, demonstrating good agreement (Figs. 1, 2). Although bempp uses nu… view at source ↗

**Figure 4.** Figure 4: Intitial horn mesh view at source ↗

**Figure 5.** Figure 5: Optimised horn mesh. is that the optimiser has arrived at a complex mouth shape. A smooth mouth termination has been added, and this varies from a very large flare at 45◦ to a sharper termination in the horizontal and vertical planes. A smooth edge termination contributes the most to the improvements seen in the directivity plot view at source ↗

**Figure 6.** Figure 6: Initial directivity (normalised to on-axis). Left: Horizontal, Right: Vertical. 5 6 8 10 12 16 Frequency (kHz) 50 0 50 Angle (degrees) ±35° 5 6 8 10 12 16 Frequency (kHz) 50 0 50 Angle (degrees) ±25° 10 8 6 4 2 0 Relative level (dB) 10 8 6 4 2 0 Relative level (dB) view at source ↗

**Figure 7.** Figure 7: Optimised directivity (normalised to on-axis). Left: Horizontal, Right: Vertical. Because each optimisation iteration is an incremental change to the mesh, it seems inefficient that the entire problem must be solved from scratch. Warmstarting GMRES with the previous solution provides a useful speed-up, but the forward algorithm remains expensive. Although the operator assembly does not benefit from GPU c… view at source ↗

read the original abstract

Engineering structures are increasingly designed using numerical optimisation. However, traditional optimisation methods can be challenging with multiple objectives and many parameters. In machine learning, stable training of artificial neural networks with millions or billions of parameters is achieved using automatic differentiation frameworks such as JAX and Pytorch. Because these frameworks provide accelerated numerical linear algebra with automatic gradient tracking, they also enable differentiable implementations of numerical methods to be built. This facilitates faster gradient-based optimisation of geometry and materials, as well as solution of inverse problems. We demonstrate JAX-BEM, a differentiable Boundary Element Method (BEM) solver, showing that it matches the error of existing BEM codes for a benchmark problem and enables gradient-based geometry optimisation. Although the demonstrated examples are for acoustic simulations, the concept could be readily extended to electromagnetic waves.

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

JAX-BEM gives a working JAX port of BEM for acoustic shape optimization, but the gradients lack the finite-difference checks needed to trust the results.

read the letter

JAX-BEM ports the boundary element method to JAX so that automatic differentiation can drive geometry changes for acoustic problems. That is the central new piece: a solver that stays inside the autodiff framework instead of requiring separate adjoint code or finite differences at each step. They report that the forward solver reproduces the error of standard BEM codes on a benchmark and then run a few shape-optimization examples to show the pipeline works end to end. The mention of possible extension to electromagnetic waves is a reasonable aside but not developed here. The implementation itself is the main deliverable and appears to be the first JAX-native version aimed at this use case. That is useful for anyone already working inside JAX who wants to optimize scatterers or enclosures without leaving the ecosystem. The soft spot is exactly the one flagged in the stress test. BEM kernels are singular and the quadrature rules are not automatically differentiable in a stable way; matching the forward error does not guarantee that the gradients are free of artifacts at the same tolerance. No finite-difference comparison or manufactured-solution test for the derivatives is described, so the optimization results rest on an unverified assumption. The paper is therefore more of a demonstration than a fully validated tool. Readers who need a ready-made differentiable acoustics solver will get value from trying the code. Readers looking for new theory or exhaustive numerical validation will find it thin. It is worth sending to peer review so that referees can examine the actual implementation and ask for the missing gradient checks. I would not desk-reject it.

Referee Report

2 major / 2 minor

Summary. The paper introduces JAX-BEM, a differentiable Boundary Element Method (BEM) solver implemented in the JAX framework for acoustic simulations. It claims that the implementation reproduces the forward-solution error of existing (non-differentiable) BEM codes on a benchmark problem and that the resulting gradients enable gradient-based shape optimization of acoustic geometries. The work is presented as a demonstration that automatic differentiation through BEM matrix assembly and linear solves can be used for engineering optimization, with a brief note on possible extension to electromagnetics.

Significance. If the gradients are shown to be accurate and stable, the approach would provide a practical route to gradient-based acoustic shape optimization without the need for adjoint derivations or finite-difference approximations. The use of an existing AD framework for a classical boundary-integral method is a useful engineering contribution, though the manuscript does not claim new theoretical results or parameter-free derivations.

major comments (2)

[Abstract / Results] Abstract and results section: the statement that JAX-BEM 'matches the error of existing BEM codes for a benchmark problem' is not supported by any reported quantitative error values, mesh sizes, quadrature orders, or direct comparison tables. Without these data the forward-solver accuracy claim cannot be evaluated and the subsequent optimization results rest on an unquantified foundation.
[Methods / Results] Methods / gradient verification: the central claim that automatic differentiation through the BEM discretization and linear solve produces usable gradients for shape optimization is not accompanied by any finite-difference gradient checks, manufactured-solution tests, or adjoint-consistency verification. Given the singular kernels and numerical quadrature required by BEM, the absence of such checks leaves open the possibility that reported optimization trajectories are driven by gradient artifacts rather than true sensitivities.

minor comments (2)

[Abstract] The abstract mentions possible extension to electromagnetic waves, yet the manuscript contains no discussion, implementation notes, or caveats for that extension.
[Methods] Notation for the acoustic boundary conditions and the differentiation of the single- and double-layer operators should be made explicit; current presentation leaves the precise form of the differentiated kernels unclear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important aspects of how the accuracy and reliability of JAX-BEM should be presented. We agree that additional quantitative evidence is needed to support the claims and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / Results] Abstract and results section: the statement that JAX-BEM 'matches the error of existing BEM codes for a benchmark problem' is not supported by any reported quantitative error values, mesh sizes, quadrature orders, or direct comparison tables. Without these data the forward-solver accuracy claim cannot be evaluated and the subsequent optimization results rest on an unquantified foundation.

Authors: We agree that the forward-solver accuracy claim requires explicit quantitative support. In the revised manuscript we will add a results subsection (or table) that reports concrete error metrics (e.g., L2-norm or pointwise errors) for the benchmark problem, together with the mesh sizes, quadrature orders, and direct numerical comparisons against a reference non-differentiable BEM code. This will allow readers to evaluate the accuracy claim directly and will provide a clearer foundation for the subsequent optimization examples. revision: yes
Referee: [Methods / Results] Methods / gradient verification: the central claim that automatic differentiation through the BEM discretization and linear solve produces usable gradients for shape optimization is not accompanied by any finite-difference gradient checks, manufactured-solution tests, or adjoint-consistency verification. Given the singular kernels and numerical quadrature required by BEM, the absence of such checks leaves open the possibility that reported optimization trajectories are driven by gradient artifacts rather than true sensitivities.

Authors: We acknowledge the referee's concern regarding the lack of explicit gradient verification, especially in the presence of singular kernels and quadrature. In the revised manuscript we will insert a dedicated verification subsection that (i) compares automatic-differentiation gradients against finite-difference approximations on a simple test geometry and (ii) presents a manufactured-solution test confirming consistency of the computed sensitivities. These checks will demonstrate that the reported optimization trajectories are driven by accurate gradients rather than numerical artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: implementation demonstration with independent validation

full rationale

The paper describes the implementation of a differentiable BEM solver in JAX and validates it by matching forward error on a benchmark problem before demonstrating gradient-based optimization. No derivation chain, uniqueness theorem, fitted parameter renamed as prediction, or self-citation load-bearing step is present in the provided text. The contribution is an engineering demonstration whose correctness can be checked externally against existing BEM codes and finite-difference gradients; it does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

This is a software implementation paper. It relies on the standard mathematical formulation of the boundary element method for the Helmholtz equation and on JAX's existing autodiff and linear algebra primitives. No new physical parameters, axioms, or entities are introduced.

axioms (2)

domain assumption The acoustic problem is governed by the Helmholtz equation with standard boundary conditions.
Invoked implicitly as the foundation for the BEM discretization.
domain assumption JAX's automatic differentiation correctly propagates gradients through the BEM matrix assembly and linear solve.
Required for the gradient-based optimization claim.

pith-pipeline@v0.9.0 · 5436 in / 1349 out tokens · 37752 ms · 2026-05-08T13:33:59.470639+00:00 · methodology

discussion (0)

Reference graph

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