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This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrodinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension $d$, under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.
A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are $mathcal{L}mathcal{L}^*$ methods. In this wor
Data assisted reconstruction algorithms, incorporating trained neural networks, are a novel paradigm for solving inverse problems. One approach is to first apply a classical reconstruction method and then apply a neural network to improve its solutio
A novel orthogonalization-free method together with two specific algorithms are proposed to solve extreme eigenvalue problems. On top of gradient-based algorithms, the proposed algorithms modify the multi-column gradient such that earlier columns are
This paper provides an a~priori error analysis of a localized orthogonal decomposition method (LOD) for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the
Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $bar z_{ell+1}=bar z_ell + mathrm{Re}sum_{k=1}^Kbar b_{ell k}e^{mathrm{i}omega_{ell k}bar z_ell}+ mathrm{Re}sum_{k=1}^Kbar c_{ell