Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Deep neural network approximation for high-dimensional elliptic PDEs
  with boundary conditions   In recent work it has been established that deep neural networks are capable
of approximating solutions to a large class of parabolic partial differential
equations without incurring the curse of dimension. However, all this work has
been restricted to problems formulated on the whole Euclidean domain. On the
other hand, most problems in engineering and the sciences are formulated on
finite domains and subjected to boundary conditions. The present paper
considers an important such model problem, namely the Poisson equation on a
domain $D\subset \mathbb{R}^d$ subject to Dirichlet boundary conditions. It is
shown that deep neural networks are capable of representing solutions of that
problem without incurring the curse of dimension. The proofs are based on a
probabilistic representation of the solution to the Poisson equation as well as
a suitable sampling method.

22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph