  In this paper, we present a novel approach to approximate the gain function
of the feedback particle filter (FPF). The exact gain function is the solution
of a Poisson equation involving a probability-weighted Laplacian. The numerical
problem is to approximate the exact gain function using only finitely many
particles sampled from the probability distribution.
  Inspired by the recent success of the deep learning methods, we represent the
gain function as a gradient of the output of a neural network. Thereupon
considering a certain variational formulation of the Poisson equation, an
optimization problem is posed for learning the weights of the neural network. A
stochastic gradient algorithm is described for this purpose.
  The proposed approach has two significant properties/advantages: (i) The
stochastic optimization algorithm allows one to process, in parallel, only a
batch of samples (particles) ensuring good scaling properties with the number
of particles; (ii) The remarkable representation power of neural networks means
that the algorithm is potentially applicable and useful to solve
high-dimensional problems. We numerically establish these two properties and
provide extensive comparison to the existing approaches.
