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The paper proposes two neural network surrogate models that learn the spectrum of the Laplace operator directly from domain geometry, enabling fast differentiable shape optimization of eigenvalue functionals. A Fourier-coefficient MLP reaches 0.2% precision on star-shaped domains, while a landscape-function model achieves 1% mean relative error on the first ten eigenvalues, outperforming FNO baselines and recovering classical spectral optima.
This paper argues that the bulk of near-zero Hessian eigenvalues in neural networks arises from weakly broken symmetries of the network parametrization, showing that high-curvature directions are orthogonal to the symmetry subspace while the bulk lies within it.