A Novel Logarithmic Transformed Deep Galerkin Approach To Optimal Filtering Problem

Ji Shi, Xiaopei Jiao, Stephen S.-T. Yau

Abstract: The optimal filtering problem for general nonlinear state-observation systems has garnered significant attention in control theory. At its core, optimal filtering involves determining the probability density function of the system state conditioned on historical observations. The Yau-Yau method [25], a pioneering framework, offers a viable approach with comprehensive theoretical guarantees and practical numerical implementation. Specifically, the Yau-Yau framework comprises two key components: offline solution of the forward Kolmogorov equation (FKE) and online data assimilation updates. The primary challenge lies in efficiently and accurately solving the FKE, as it directly impacts the real-time filtering process. To address this fundamental obstacle, we propose a highly efficient filtering algorithm that combines a FKE solver based on deep neural networks and a PDF approximator using generalized Legendre polynomials. By integrating advanced deep learning techniques with Galerkin approximation, we introduce the logarithmic transformed deep Galerkin approach (LTDG). The numerical simulations showcase the effectiveness and accuracy of our newly proposed algorithm. LTDG demonstrates superior performance compared to other methods, such as the extended Kalman filter and particle filter, and it successfully maintains the high accuracy of the Galerkin spectral method while having fewer online computational burdens.