Error analysis for the deep Kolmogorov method

TL;DR

This paper provides an error analysis and convergence rates for the deep Kolmogorov method, a neural network approach for solving heat PDEs, highlighting how architecture size, data points, and optimization errors affect accuracy.

Contribution

It offers the first rigorous convergence analysis for the deep Kolmogorov method applied to heat PDEs, detailing how various factors influence approximation errors.

Findings

Convergence rates depend on neural network architecture size.

Error bounds relate to the number of sample points used.

Optimization errors impact the overall approximation accuracy.

Abstract

The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.