The Kolmogorov Infinite Dimensional Equation in a Hilbert space Via Deep Learning Methods

TL;DR

This paper introduces a novel deep learning approach using DeepOnets to approximate solutions of the nonlinear Kolmogorov equation in infinite-dimensional Hilbert spaces, extending finite-dimensional methods and stochastic approximation frameworks.

Contribution

It generalizes finite-dimensional deep learning schemes and stochastic methods to infinite-dimensional Hilbert spaces for solving the Kolmogorov equation, utilizing DeepOnets neural networks.

Findings

Successful extension of Euler schemes to infinite dimensions

Implementation of DeepOnets for infinite-dimensional PDE approximation

Framework adaptable to stochastic wave models and Levy processes

Abstract

We consider the nonlinear Kolmogorov equation posed in a Hilbert space $H$, not necessarily of finite dimension. This model was recently studied by Cox et al. [24] in the framework of weak convergence rates of stochastic wave models. Here, we propose a complementary approach by providing an infinite-dimensional Deep Learning method to approximate suitable solutions of this model. Based in the work by Hure, Pham and Warin [45] concerning the finite dimensional case, and our previous work [20] dealing with L\'evy based processes, we generalize an Euler scheme and consistency results for the Forward Backward Stochastic Differential Equations to the infinite dimensional Hilbert valued case. Since our framework is general, we require the recently developed DeepOnets neural networks [21, 51] to describe in detail the approximation procedure. Also, the framework developed by Fuhrman and Tessitore [35] to fully describe the stochastic approximations will be adapted to our setting