LAUSR

This article develops a new deep learning framework for general nonlinear filtering. Our main contribution is to present a computationally feasible procedure. The proposed algorithms have the capability of dealing with challenging (infinitely dimensional) filtering problems involving diffusions with randomly-varying switching. First, we convert it to a problem in a finite-dimensional setting by approximating the optimal weights of a neural network. Then, we construct a stochastic gradient-type procedure to approximate the neural network weight parameters, and develop another recursion for adaptively approximating the optimal learning rate. The convergence of the combined approximation algorithms is obtained using stochastic averaging and martingale methods under suitable conditions. Robustness analysis of the approximation to the network parameters with the adaptive learning rate is also dealt with. We demonstrate the efficiency of the algorithm using highly nonlinear dynamic system examples.