Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof that rectified deep neural networks overcome the curse of
  dimensionality in the numerical approximation of semilinear heat equations   Deep neural networks and other deep learning methods have very successfully
been applied to the numerical approximation of high-dimensional nonlinear
parabolic partial differential equations (PDEs), which are widely used in
finance, engineering, and natural sciences. In particular, simulations indicate
that algorithms based on deep learning overcome the curse of dimensionality in
the numerical approximation of solutions of semilinear PDEs. For certain linear
PDEs this has also been proved mathematically. The key contribution of this
article is to rigorously prove this for the first time for a class of nonlinear
PDEs. More precisely, we prove in the case of semilinear heat equations with
gradient-independent nonlinearities that the numbers of parameters of the
employed deep neural networks grow at most polynomially in both the PDE
dimension and the reciprocal of the prescribed approximation accuracy. Our
proof relies on recently introduced multilevel Picard approximations of
semilinear PDEs.

7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics