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The filtering equations govern the evolution of the conditional distributionof a signal process given partial, and possibly noisy, observations arrivingsequentially in time. Their numerical approximation plays a central role inmany real-life applications, including numerical weather prediction, financeand engineering. One of the classical approaches to approximate the solution ofthe filtering equations is to use a PDE inspired method, called thesplitting-up method, initiated by Gyongy, Krylov, LeGland, among othercontributors. This method, and other PDE based approaches, have particularapplicability for solving low-dimensional problems. In this work we combinethis method with a neural network representation. The new methodology is usedto produce an approximation of the unnormalised conditional distribution of thesignal process. We further develop a recursive normalisation procedure torecover the normalised conditional distribution of the signal process. The newscheme can be iterated over multiple time steps whilst keeping its asymptoticunbiasedness property intact.  We test the neural network approximations with numerical approximationresults for the Kalman and Benes filter.

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