An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations

Abstract

The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction [13,33], finance [7, 10][9, Part IX] and engineering [35]. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors, see e.g., [17,31]. This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation inspired by [21]. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.

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

“…Deep learning for the filtering problem is not an extensively studied topic. To the best of our knowledge, there are only two papers, [3,10], that are based on partial differential equations. In [3] a deep splitting method is used to solve the Zakai equation, while in [10] the Fokker-Planck equation is solved by deep splitting.…”

“…An approach based on the Fokker-Planck approach. In [10] the authors present a similar approach as in [3]. It also involves a deep splitting scheme on an underlying equation.…”

“…It also involves a deep splitting scheme on an underlying equation. But in [10] this is done directly on the Fokker-Planck equation which gives the unconditional density of X instead. To obtain a corresponding filter estimate, a likelihood normalization is applied after each update of the Fokker-Planck approximation.…”

“…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.…”

“…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.…”