Neural networks for spectral optimization

2026-09-28

AI Summary

This study addresses the prohibitive computational cost of domain optimization based on spectral functionals of differential operators, which typically relies on expensive PDE solvers. To overcome this limitation, we propose two neural network surrogate models that integrate Fourier coefficient encoding, landscape function representations, and Gram-Schmidt orthogonalization to directly learn spectra from geometric descriptions, enabling efficient optimization of star-shaped and landscape domains. Furthermore, coefficient scaling is introduced to satisfy eigenvalue scaling laws, while output averaging ensures rotational and reflectional invariance. The proposed approach achieves 0.2% accuracy for star-shaped domains with a 1% error across the first ten eigenvalues, significantly outperforming Fourier Neural Operators (FNO). By successfully reproducing classical spectral optima, this work establishes a novel paradigm for efficient spectral shape optimization.

Abstract

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.

Problem

• spectral optimization
• shape optimization
• eigenvalues
• differential operator
• domain optimization

Innovation

• spectral optimization
• neural surrogate models
• landscape function
• shape optimization
• eigenvalue prediction