Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations

TL;DR

This paper proves that deep neural networks can efficiently approximate solutions to certain nonlinear PDEs with gradient-dependent nonlinearities, overcoming the curse of dimensionality, which was previously only established for gradient-independent cases.

Contribution

It provides the first rigorous proof that DNNs overcome the curse of dimensionality for a class of nonlinear PDEs with gradient-dependent nonlinearities.

Findings

DNNs approximate solutions with polynomial complexity in dimension and accuracy.

Numerical experiments support the theoretical results.

Overcomes limitations of previous methods for gradient-dependent PDEs.

Abstract

Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $\epsilon\in (0,1)$. The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities.