Nonlinear filtering based on density approximation and deep BSDE prediction

2025-08-14

AI Summary

This work addresses the challenge of efficiently approximating unnormalized filtering densities in nonlinear Bayesian filtering. Methodologically, it introduces a novel framework integrating deep learning with backward stochastic differential equations (BSDEs): leveraging the nonlinear Feynman–Kac formula, the filtering density is formulated as the solution to a BSDE; its unknown drift and diffusion components are parameterized by deep neural networks, enabling end-to-end density approximation under an offline-training/online-inference paradigm. Theoretically, under ellipticity assumptions, we derive a prior-posterior mixed error bound and establish an explicit convergence rate for the algorithm. Empirically, the method achieves significantly higher accuracy than conventional particle filters and the extended Kalman filter (EKF) on two benchmark nonlinear filtering tasks, while maintaining computational scalability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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 mixed a priori-a posteriori error bound is proved under an elliptic condition. The theoretical convergence rate is confirmed in two numerical examples.

Problem

• Develops a novel Bayesian filter using deep BSDE methods
• Approximates unnormalized filtering density via neural networks
• Enables offline training for online filtering applications

Innovation

• Deep BSDE method for Bayesian filtering
• Neural networks approximate unnormalized density
• Offline training enables online application