Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective

TL;DR

This paper investigates the representational capacity of neural networks for approximating solutions to nonlinear elliptic PDEs, demonstrating they can overcome the curse of dimensionality through Barron norm bounds and a novel proof technique.

Contribution

It extends theoretical analysis of neural network approximations from linear to nonlinear PDEs, providing bounds on network size and Barron norm growth for high-dimensional problems.

Findings

Neural networks can approximate solutions to nonlinear PDEs with polynomial complexity in dimension.

The approach generalizes prior linear PDE results to a broader class of nonlinear elliptic PDEs.

The proof involves simulating gradient flows in a Hilbert space with exponential convergence.

Abstract

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as \emph{nonlinear elliptic variational PDEs}, whose solutions minimize an \emph{Euler-Lagrange} energy functional $\mathcal{E}(u) = \int_\Omega L(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $\epsilon$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ \epsilon), p^{\log(1/\epsilon)}\}}\right)$. By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, \epsilon, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.