Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense

TL;DR

This paper demonstrates that multilevel Picard methods and certain deep neural networks can efficiently approximate solutions to high-dimensional semilinear parabolic PDEs in an $L^p$-sense, overcoming the curse of dimensionality.

Contribution

It establishes that these approximation techniques achieve polynomial growth in computational effort and network size with respect to dimension and accuracy, for a broad class of PDEs.

Findings

Polynomial growth in computational effort with dimension and accuracy

Neural networks with ReLU, leaky ReLU, and softplus effectively approximate PDE solutions

Overcomes curse of dimensionality in high-dimensional PDE approximation

Abstract

We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogorov PDEs in $L^\mathfrak{p}$-sense, $\mathfrak{p}\in [2,\infty)$, in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations and the required number of parameters in the neural networks grow at most polynomially in both dimension $d\in \mathbb{N}$ and reciprocal of the prescribed accuracy $\epsilon$.