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

TL;DR

This paper introduces a novel neural network-based approach combined with the splitting-up method to approximate solutions to filtering equations, enabling efficient recursive estimation of conditional distributions in signal processing applications.

Contribution

It develops a new methodology integrating neural networks with the splitting-up PDE method for filtering equations, maintaining unbiasedness over multiple time steps.

Findings

Successfully approximates Kalman and Benes filters

Demonstrates recursive normalization preserves distribution accuracy

Applicable to low-dimensional filtering problems

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, finance and engineering. 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. 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. 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.