Bounds on the approximation error for deep neural networks applied to dispersive models: Nonlinear waves

TL;DR

This paper develops a rigorous framework to bound the approximation error of deep neural networks applied to dispersive PDE models like waves and Schrödinger equations, analyzing both linear and nonlinear cases in multiple dimensions.

Contribution

It introduces a probabilistic approach to derive bounds on neural network approximation errors for complex dispersive PDEs, covering linear and nonlinear wave models in various dimensions.

Findings

Rigorous error bounds for neural network approximations in dispersive PDEs.

Analysis of computational costs for different wave scenarios.

Application to both linear and nonlinear wave models in 1-3 dimensions.

Abstract

We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schr\"odinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labord\`ere and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions $d=1,2,3$, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions.