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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.
The paper proposes a spectral connectivity-regularized graph learning framework (SCoGL) that incorporates Laplacian spectral priors to improve graph recovery and downstream tasks like graph signal denoising when data is scarce.
This paper introduces a nonlinear Laplacian operator for signed and directed graphs (NLSD) and a spectral GNN framework (NLSD-GNN) that achieves superior performance on node classification and link prediction by aligning message passing with edge direction.