Deep learning based numerical approximation algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems

Abstract

In this article we introduce and study a deep learning based approximation algorithm for solutions of stochastic partial differential equations (SPDEs). In the proposed approximation algorithm we employ a deep neural network for every realization of the driving noise process of the SPDE to approximate the solution process of the SPDE under consideration. We test the performance of the proposed approximation algorithm in the case of stochastic heat equations with additive noise, stochastic heat equations with multiplicative noise, stochastic Black-Scholes equations with multiplicative noise, and Zakai equations from nonlinear filtering. In each of these SPDEs the proposed approximation algorithm produces accurate results with short run times in up to 50 space dimensions.

An energy-based deep splitting method for the nonlinear filtering problem

“…See [2] for details on the derivation of the equation. Following [3], we consider a more general equation of the form…”

“…These theoretically appealing facts have not been extensively used to derive practical algorithms, mostly because of the computational load associated with approximating SPDE with classical methods such as finite element or finite difference methods. Following the recent years' extensive developments in solving PDE and SPDE with deep neural networks [3,4,10,39], new opportunities arise. In [4] the authors present a deep splitting method for high-dimensional PDE.…”

“…This grid-free method was demonstrated by approximating solutions to PDE in up to 10 000 dimensions. In the follow up paper [3] the method was applied successfully to SPDE in up to 50 dimensions. In particular, it was applied to the Zakai equation for nonlinear filtering with a fixed observation sequence and a quite peculiar dynamics, chosen to admit an analytic benchmark solution.…”

“…The energy-based approach and the full algorithm is presented in Section 4. It also contains a discussion of two previous approaches to solving the optimal filtering problem with deep splitting methods [3,10]. In Section 5 we present our numerical results.…”

Computation of conditional expectations with guarantees

“…, X d (see, e.g., Chatterjee and Hadi, 2015;Draper and Smith, 1998;Hastie et al, 2009;Ryan, 2009). But it also appears in different computational problems, such as the numerical approximation of partial differential equations and backward stochastic differential equations (see, e.g., Bally, 1997;Beck et al, 2021;Bouchard and Touzi, 2004;Chevance, 1997;Fahim et al, 2011;Gobet et al, 2005;Gobet and Turkedjiev, 2006), stochastic partial differential equations and (see, e.g., Beck et al, 2020), stochastic control problems (see, e.g., Åström, 1970;Bain and Crisan, 2008), stochastic filtering (see, e.g., Jazwinski, 2007), the approximation of posterior distributions in Bayesian statistics (see, e.g. Gelman et al, 2013), complex valuation problems (see, e.g.…”