Deep neural network approximation for high-dimensional parabolic partial integro-differential equations

TL;DR

This paper demonstrates the theoretical existence of deep neural networks capable of approximating solutions to high-dimensional parabolic partial integro-differential equations, potentially overcoming the curse of dimensionality.

Contribution

It establishes, through theoretical analysis, that DNNs can approximate solutions to complex equations using stochastic representations, without relying on numerical experiments.

Findings

Existence of DNNs approximating solutions to high-dimensional PDEs

Circumventing the curse of dimensionality in approximation

Theoretical framework based on Feynman-Kac theorem

Abstract

In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we express the solution in terms of stochastic differential equations (SDEs). Based on several properties of classical estimators, we establish the existence of a DNN that satisfies the necessary assumptions. The results are theoretical and don't have any numerical experiments yet.