Space-time deep neural network approximations for high-dimensional partial differential equations

TL;DR

This paper proves that deep neural networks can approximate solutions to certain high-dimensional Kolmogorov PDEs over space-time regions without suffering from the curse of dimensionality, extending previous fixed-time results.

Contribution

It establishes the first result showing DNNs can approximate entire PDE solutions over space-time regions without exponential complexity in dimension.

Findings

DNNs can approximate PDE solutions on space-time regions without curse of dimensionality.

Previous results only addressed fixed-time approximations, not entire solutions.

Theoretical proof applies to solutions of certain Kolmogorov PDEs.

Abstract

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision $\varepsilon>0$ grows exponentially in the PDE dimension and/or the reciprocal of $\varepsilon$. Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension $d\in\mathbb{N}$ and the reciprocal of the prescribed accuracy $\varepsilon>0$. There are now also a few rigorous results in the scientific literature which substantiate this conjecture by proving that DNNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that DNNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point $T>0$ and on a compact cube $[a,b]^d$ in space but none of these results provides an answer to the question whether the entire PDE solution on $[0,T]\times [a,b]^d$ can be approximated by DNNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every $a\in\mathbb{R}$, $ b\in (a,\infty)$ that solutions of certain Kolmogorov PDEs can be approximated by DNNs on the space-time region $[0,T]\times [a,b]^d$ without the curse of dimensionality.