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

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