  This paper is concerned with the problem of nonlinear filtering, i.e.,
computing the conditional distribution of the state of a stochastic dynamical
system given a history of noisy partial observations. Conventional sequential
importance resampling (SIR) particle filters suffer from fundamental
limitations, in scenarios involving degenerate likelihoods or high-dimensional
states, due to the weight degeneracy issue. In this paper, we explore an
alternative method, which is based on estimating the Brenier optimal transport
(OT) map from the current prior distribution of the state to the posterior
distribution at the next time step. Unlike SIR particle filters, the OT
formulation does not require the analytical form of the likelihood. Moreover,
it allows us to harness the approximation power of neural networks to model
complex and multi-modal distributions and employ stochastic optimization
algorithms to enhance scalability. Extensive numerical experiments are
presented that compare the OT method to the SIR particle filter and the
ensemble Kalman filter, evaluating the performance in terms of sample
efficiency, high-dimensional scalability, and the ability to capture complex
and multi-modal distributions.
