Deep Learning Schemes For Parabolic Nonlocal Integro-Differential Equations

TL;DR

This paper develops neural network-based numerical methods for solving nonlocal parabolic integro-differential equations, which are relevant in various scientific fields, by extending existing stochastic schemes to the nonlocal context.

Contribution

It introduces a neural network approach for approximating nonlocal terms in parabolic equations, generalizing stochastic schemes to handle nonlocal operators effectively.

Findings

Neural network schemes successfully approximate solutions to nonlocal equations.

The method extends stochastic Euler schemes to nonlocal integro-differential equations.

Results demonstrate improved accuracy in modeling nonlocal phenomena.

Abstract

In this paper we consider the numerical approximation of nonlocal integro differential parabolic equations via neural networks. These equations appear in many recent applications, including finance, biology and others, and have been recently studied in great generality starting from the work of Caffarelli and Silvestre. Based in the work by Hure, Pham and Warin, we generalize their Euler scheme and consistency result for Backward Forward Stochastic Differential Equations to the nonlocal case. We rely on L\`evy processes and a new neural network approximation of the nonlocal part to overcome the lack of a suitable good approximation of the nonlocal part of the solution.