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.

翻译：本文提出了一种基于倒向随机微分方程的新型近似贝叶斯滤波器。该方法利用滤波问题的非线性Feynman-Kac表示，并结合成熟的深度BSDE方法与神经网络来近似非归一化滤波密度。该算法通过离线训练实现，从而可在线应用于新观测数据。在抛物型Hörmander条件下，证明了混合先验-后验误差界。两个数值实例验证了理论收敛速率。