Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations

TL;DR

This paper proves that deep neural networks with ReLU, leaky ReLU, or softplus activations can overcome the curse of dimensionality when approximating solutions to high-dimensional semilinear heat equations over space-time regions, not just at terminal time.

Contribution

It establishes the first proof that certain DNN architectures can approximate high-dimensional PDE solutions over space-time regions without the curse of dimensionality.

Findings

DNNs with ReLU, leaky ReLU, or softplus can approximate solutions in space-time regions.

The approximation error can be made arbitrarily small with polynomially many parameters.

Results extend previous terminal-time approximation to space-time regions.

Abstract

It is a challenging topic in applied mathematics to solve high-dimensional nonlinear partial differential equations (PDEs). Standard approximation methods for nonlinear PDEs suffer under the curse of dimensionality (COD) in the sense that the number of computational operations of the approximation method grows at least exponentially in the PDE dimension and with such methods it is essentially impossible to approximately solve high-dimensional PDEs even when the fastest currently available computers are used. However, in the last years great progress has been made in this area of research through suitable deep learning (DL) based methods for PDEs in which deep neural networks (DNNs) are used to approximate solutions of PDEs. Despite the remarkable success of such DL methods in simulations, it remains a fundamental open problem of research to prove (or disprove) that such methods can overcome the COD in the approximation of PDEs. However, there are nowadays several partial error analysis results for DL methods for high-dimensional nonlinear PDEs in the literature which prove that DNNs can overcome the COD in the sense that the number of parameters of the approximating DNN grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon>0$ and the PDE dimension $d\in\mathbb{N}$. In the main result of this article we prove that for all $T,p\in(0,\infty)$ it holds that solutions $u_d\colon[0,T]\times\mathbb{R}^d\to\mathbb{R}$, $d\in\mathbb{N}$, of semilinear heat equations with Lipschitz continuous nonlinearities can be approximated in the $L^p$-sense on space-time regions without the COD by DNNs with the rectified linear unit (ReLU), the leaky ReLU, or the softplus activation function. In previous articles similar results have been established not for space-time regions but for the solutions $u_d(T,\cdot)$, $d\in\mathbb{N}$, at the terminal time $T$.