A Priori Generalization Analysis of the Deep Ritz Method for Solving
High Dimensional Elliptic Equations
This paper concerns the a priori generalization analysis of the Deep Ritz
Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for
solving high dimensional partial differential equations. We derive the
generalization error bounds of two-layer neural networks in the framework of
the DRM for solving two prototype elliptic PDEs: Poisson equation and static
Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we
prove that the convergence rates of generalization errors are independent of
the dimension $d$, under the a priori assumption that the exact solutions of
the PDEs lie in a suitable low-complexity space called spectral Barron space.
Moreover, we give sufficient conditions on the forcing term and the potential
function which guarantee that the solutions are spectral Barron functions. We
achieve this by developing a new solution theory for the PDEs on the spectral
Barron space, which can be viewed as an analog of the classical Sobolev
regularity theory for PDEs.