  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.
