Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th