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In this paper, we study the statistical limits of deep learning techniquesfor solving elliptic partial differential equations (PDEs) from random samplesusing the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs).To simplify the problem, we focus on a prototype elliptic PDE: theSchr\"odinger equation on a hypercube with zero Dirichlet boundary condition,which has wide application in the quantum-mechanical systems. We establishupper and lower bounds for both methods, which improves upon concurrentlydeveloped upper bounds for this problem via a fast rate generalization bound.We discover that the current Deep Ritz Methods is sub-optimal and propose amodified version of it. We also prove that PINN and the modified version of DRMcan achieve minimax optimal bounds over Sobolev spaces. Empirically, followingrecent work which has shown that the deep model accuracy will improve withgrowing training sets according to a power law, we supply computationalexperiments to show a similar behavior of dimension dependent power law fordeep PDE solvers.

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality