Nonlinear filtering based on density approximation and deep BSDE prediction
It provides a new theoretical framework for nonlinear filtering with rigorous error bounds, but the method is domain-specific to filtering problems.
The paper introduces a novel approximate Bayesian filter using deep BSDEs, achieving offline training and online application with a proven convergence rate confirmed in numerical examples.
A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using the well-known deep BSDE method and neural networks. The method is trained offline, which means that it can be applied online with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic Hörmander condition. The theoretical convergence rate is confirmed in two numerical examples.