DGLG: A Novel Deep Generalized Legendre–Galerkin Approach to Optimal Filtering Problem

The optimal filtering problem for general nonlinear and continuous state-observation systems attracts lots of attention in the control theory. The essence of optimal filtering requires solving the Duncan–Mortensen–Zakai (DMZ) equation in a computationally feasible way. Under the pioneering work of Yau-Yau filtering, the DMZ equation is reduced to a pathwise computation of a forward Kolmogorov equation with time-varying initial conditions, which is very challenging. To overcome the computational difficulty, in this article, we proposed a new efficient filtering algorithm consisting of a forward Kolmogorov equation solver based on a physics-informed neural network and a probability density approximator based on generalized Legendre polynomials. By utilizing the advanced deep learning method and classical Galerkin approximation, our developed algorithm not only maintains the high accuracy of the spectral method but also removes massive computational loads in the offline part. Furthermore, the convergence of our method is proved. Numerical experiments have been carried out to verify the feasibility of the new method. Regarding accuracy and efficacy, the newly proposed deep generalized Legendre–Galerkin algorithm outperforms other popular suboptimal methods including the extended Kalman filter and particle filter.