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Neural networks for spectral optimization

Alexis de Villeroché, Beniamin Bogosel, Stéphane Breuils, Dorin Bucur, Jacques-Olivier Lachaud

arXiv:2609.36047v1cs.LGcs.AImath.OC

TL;DR

Shape optimization based on differential-operator spectra is computationally demanding, and stable learning depends on how domains are represented. The paper proposes neural surrogates using Fourier coefficients or landscape-based inputs; its reported optimization experiments recover classical spectral shapes, including a near-disk for the first eigenvalue.

  • Problem

    Finite-element spectral calculations are accurate but require an expensive new solve for each shape, while learning stable spectral dependence on geometry requires addressing representation sensitivity.

    Source: §1 · p. 1, §2 · p. 3

  • Method

    The paper proposes Fourier-coefficient and landscape-based neural surrogates that predict spectra from domain geometry for shape optimization.

    Source: §3.1 · pp. 4, 6, §5 · p. 11

  • Results

    The λ1 optimization reached aspect ratio a/b = 1.02, with the predicted eigenvalue 1.43% above the exact value; λ2 optimization converged to a size ratio of 0.99 and ellipse aspect ratios 1.08 and 1.05.

    Source: §4.2 · p. 10

  • Takeaways & Limitations

    The authors conclude that the landscape function is an appropriate indicator representation for sets when studying quantities linked to the Dirichlet Laplacian.

    Source: §5 · p. 11

    An SDF can retain a zero near a vanishing hole even as the Dirichlet spectrum converges to that of the domain without the hole.

    Source: §2 · p. 3

Abstract

from arXiv · show

Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2\%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1\% mean relative error on the first ten eigenvalues, compared with 4\% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.

1. Introduction

The paper targets shape optimization from spectral quantities, where accurate finite-element solves are computationally expensive and neural-surrogate accuracy depends on domain representation. It introduces Fourier-coefficient and landscape-based surrogates for differentiable spectral prediction and optimization.

1. Introduction

  • Finite-element spectrum calculations are accurate but require a new computationally expensive solve for each shape, motivating fast neural surrogates.The accuracy of those surrogates depends on how the domain is represented.Source: §1 · p. 1
  • SDF representations are topologically sensitive: as an internal hole vanishes, the spectrum converges to that of the domain without the hole, while the SDF retains a nearby zero value.The passage frames this as a representation issue for spectral prediction.Source: §1 · pp. 1–2
  • FNOs use global frequency transformations and can struggle with sharp localized geometry, complex boundary conditions, and high-frequency features relevant to upper Laplacian eigenvalues.The passage also notes artifacts near edges.Source: §1 · p. 2
  • V-PINNs use the Rayleigh quotient as a loss, but are limited to the first few eigenvalues and require a separate optimization for every shape.This contrasts with the paper's learned spectral surrogates.Source: §1 · p. 2

1.1. Our contributions.

  • The paper proposes two differentiable spectral surrogates for shape optimization: a Fourier-coefficient model and a landscape-based model predicting eigenvalues and eigenfunctions.The landscape model takes the landscape function, indicator, and landscape gradient as input and applies a differentiable Gram-Schmidt step.Source: §1.1 · p. 2

2. Background and related work

This section frames spectral shape optimization as a search over domains for functionals of Laplace eigenvalues, while emphasizing the challenge of representing geometry in a way that captures spectral sensitivity. The paper motivates a unified neural approach and uses the landscape function as its main representation for Dirichlet spectral quantities.

2. Background and related work

  • Shape optimization seeks admissible domains minimizing or otherwise optimizing a functional of eigenvalues; the fixed-area first-eigenvalue problem is the classical disk-minimization problem.The section notes that such optimization procedures require expertise across PDEs, calculus of variations, optimization, and scientific computing.Source: §2 · p. 3
  • The paper proposes a unified neural method combining eigenvalue computation, shape derivatives, and optimization for a general spectral functional.The stated aim is to avoid readapting separate algorithms for each functional.Source: §2 · p. 3
  • Stable learning of spectra from geometry requires understanding whether small geometric perturbations can strongly affect eigenvalues and choosing a representation that captures this dependence.The section identifies this as a question affecting computational robustness.Source: §2 · p. 3
  • Indicator functions and signed-distance representations can misrepresent spectral sensitivity: measure-zero changes preserve the indicator in L1, while point changes in an SDF can matter geometrically despite having no spectral influence in dimension two.The text motivates the landscape function as the main input representation.Source: §2 · p. 3
  • The landscape function is presented as a representation that controls the spectrum, and related torsion-based work is distinguished from approaches requiring per-shape re-optimization or a domain mesh.The paper states that its methods predict the first K = 10 eigenvalues in one differentiable forward pass for raster-image domains.Source: §2 · p. 4

3. Fourier coefficients based model

The Fourier-based surrogate represents star-shaped domains through finite Fourier coefficients and predicts normalized eigenvalues for shape optimization. Its symmetry averaging and scaling-aware input support optimization across admissible shapes, with optimized results close to established shapes for higher eigenvalues.

3.1. Approach.

  • The model encodes star-shaped domains by finite Fourier coefficients of a positive, periodic radial function, giving exact analytic shapes independent of numerical discretization.The area of shapes in U15 can also be computed explicitly.Source: §3.1 · p. 4
  • The training database contains 20,000 domains satisfying a coefficient-decay assumption used to ensure positivity of the truncated Fourier boundary function.The first ten Dirichlet eigenvalues are computed with P2 elements and a FreeFEM PDE solver.Source: §3.1 · pp. 4–5, Figure 1 · p. 5
  • Training on normalized eigenvalues computed from coefficient ratios makes the surrogate satisfy the eigenvalue scaling law directly by construction.The normalization uses j_k(Ω) = |Ω|λ_k(Ω), which is invariant under dilations.Source: §3.1 · p. 5
  • Aggregating three independently trained MLPs is used to increase the stability of surrogate derivatives.Each MLP has architecture 30 → 192 → 192 → 192 → 10.Source: §3.1 · pp. 5–6
  • Averaging predictions over reflections and twelve rotations makes the surrogate reflection-invariant and close to rotationally invariant without learning these properties in the network weights.The averaging is applied to the network output for each shape.Source: §3.1 · pp. 5–6

3.2. Experiments.

  • Across tested widths, SiLU aggregates are most accurate and Tanh aggregates least accurate; width 192 is retained because width 300 adds training time for marginal accuracy gain.Table 2 reports mean relative error by eigenvalue and training time by width and activation function.Source: §3.2 · p. 6, Table 2 · p. 7
  • For λ5–λ10, ReLU and SiLU produce similar PDE-checked optimized shapes, close in aspect to Antunes and Freitas.The resulting shapes are described as good candidates for initializing optimization with a traditional PDE solver.Source: §3.2 · pp. 6–7, Figure 2 · p. 7
  • The retained SiLU surrogate optimizes any C1 function of the first ten eigenvalues over area-1 shapes in U15, with examples for different functionals.SiLU is retained for slightly better accuracy and smoothness; the resulting model is also smooth.Source: §3.2 · p. 7, Figure 3 · p. 8

4. Topological aware model with the landscape function

The landscape-based model combines geometry and landscape information to predict eigenvalues and eigenfunctions, then uses its differentiable surrogate for shape optimization. Its experiments compare input representations and architectures, test generalization, and recover several spectral optima.

4.1. Approach.

  • The U-Net takes smoothed occupancy, normalized landscape, and squared landscape-gradient channels, then orthonormalizes ten candidate eigenfunctions and estimates eigenvalues with Rayleigh quotients.A pooled bottleneck head predicts the scaled first eigenvalue; the training loss combines ratio, scaled-first-eigenvalue, and Rayleigh-quotient terms.Source: §4.1 · p. 8
  • Because the 64 × 64 grid lacks rotation invariance, training uses rotations and horizontal flips.The dataset combines analytically computed landscapes for some domains with finite-difference landscapes for polygons.Source: §4.1 · p. 8
  • On 100,000 varied domains, the trained model achieved 1.15% mean relative error and 0.53% median relative error across all ten eigenvalues.The dataset includes near-circular, star-shaped, non-convex, multiply connected, multipart, polygonal, and triangular domains.Source: §4.1 · pp. 8–9

4.2. Experiments and ablation study.

  • After training on the same dataset of N = 100000 domains for 300 epochs, our model, FNO, and DeepONet are compared using landscape-based and indicator-function inputs by mean relative error over λ1 through λ10.The test set contains 15,000 shapes unseen during training.Source: §4.2 · p. 9
  • Against indicator-only inputs, landscape-based models reduced error by a factor of 6 to 10; replacing the landscape with SDF yielded 33.06% test relative error, versus 45% for DeepONet and 54.68% for FNO.The indicator-based FNO and DeepONet models also showed a train/validation gap after the first 20 epochs.Source: §4.2 · pp. 9–10
  • On 1,400 MPEG-7 silhouettes, mean relative error was 6.88% and median relative error was 4.83%.The evaluation used binary silhouettes from 70 animal and tool classes.Source: Table 6 · p. 10, §4.2 · p. 10
  • For λ1, an ellipse converged to aspect ratio 1.02 with predicted value 1.43% above exact; shrinking two holes lowered predicted λ1 monotonically across all 20 tested radii.The ellipse experiment also reported a 2.35% relative error for the resulting landscape function against the theoretical landscape.Source: §4.2 · p. 10
  • For λ2, eccentric ellipses converged toward equal disks with prediction 2.35% above theory; λ6 optimization from a near-circle produced a rounded triangle with 2.5 relative error.The λ2 prediction decreased from 111.52 to 74.38.Source: §4.2 · p. 10

5. Conclusion and perspectives

The paper closes by contrasting two spectral-optimization approaches with different methods and scopes. It reports optimization demonstrations, distinguishes the approaches’ intended uses, and identifies possible extensions.

5. Conclusion and perspectives

  • The Fourier model is a lightweight PDE-solver companion for conjecture testing, but its shape formalism limits generalization to larger classes of shapes.The authors describe it as supporting, rather than replacing, the PDE solver.Source: §5 · p. 10
  • 2.35% relative error for ˆλ2 at the two-equal-disk optimum, and 2.5% relative error for ˆλ6 versus FreeFem.After 100 iterations, the optimized domain had a1/b1 = 1.08, a2/b2 = 1.05, and R1/R2 = 0.99.Source: Figure 5 · p. 11
  • The landscape approach seeks a more fundamental account at some loss of precision, treating the landscape as an indicator for Dirichlet-Laplacian quantities and motivating capacitary-measure extensions.The authors identify learning spectra of capacitary measures as a possible extension.Source: §5 · p. 11

AI use statement.

  • The authors used generative AI to expand early code during exploration, not to draft the paper or prove mathematical claims; AI-assisted code was checked by at least two authors.Most AI-produced code was manually rewritten, and the authors take responsibility for the final content.Source: AI use statement · p. 11

Reproducibility statement.

  • Experiments use fixed-seed generated parameters and shapes and are reproducible; code, trained torsion-model weights, and the Fourier-model web app are available.The code repository, model weights, and web app are linked in the paper.Source: Reproducibility statement · pp. 11–12
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