Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: DGM: A deep learning algorithm for solving partial differential
  equations   High-dimensional PDEs have been a longstanding computational challenge. We
propose to solve high-dimensional PDEs by approximating the solution with a
deep neural network which is trained to satisfy the differential operator,
initial condition, and boundary conditions. Our algorithm is meshfree, which is
key since meshes become infeasible in higher dimensions. Instead of forming a
mesh, the neural network is trained on batches of randomly sampled time and
space points. The algorithm is tested on a class of high-dimensional free
boundary PDEs, which we are able to accurately solve in up to $200$ dimensions.
The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE
and Burgers' equation. The deep learning algorithm approximates the general
solution to the Burgers' equation for a continuum of different boundary
conditions and physical conditions (which can be viewed as a high-dimensional
space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is
similar in spirit to Galerkin methods, with the solution approximated by a
neural network instead of a linear combination of basis functions. In addition,
we prove a theorem regarding the approximation power of neural networks for a
class of quasilinear parabolic PDEs.

22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
8
The categories of this paper are: hep-ex