Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

TL;DR

This paper demonstrates that deep neural networks can efficiently approximate solutions to certain high-dimensional semilinear PDEs, with the number of parameters growing polynomially with dimension and accuracy, extending previous results beyond heat equations.

Contribution

It proves that neural networks can approximate solutions of semilinear Kolmogorov PDEs with polynomial complexity in dimension and accuracy, generalizing prior work on heat equations.

Findings

Neural networks approximate solutions with polynomial parameter growth.

Applicable to high-dimensional PDEs with Lipschitz nonlinearities.

Extends previous results from heat equations to more general PDEs.

Abstract

We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven in the case of semilinear heat equations.