A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic Hörmander condition, and empirically in numerical examples. In a prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, followed by an exact update through Bayes' formula. This results in a classical prediction-update filtering algorithm that operates online for new observation sequences post-training. The algorithm employs a sampling-based Feynman--Kac approach, designed to mitigate the curse of dimensionality. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem. The convergence analysis is complemented by a nonlinear $10$-dimensional numerical example demonstrating the robustness of the method.

翻译：本文提出一种用于逼近非线性滤波密度的数值格式，并在抛物型赫曼德条件下从理论上建立其收敛速率，同时通过数值算例进行经验验证。在预测步骤中，该格式利用深度分裂方案逼近离散时刻含噪声部分观测量之间的福克-普朗克方程，随后通过贝叶斯公式进行精确更新。这构成了经典的预测-更新滤波算法，可在训练后对新观测序列进行在线处理。该算法采用基于采样的费曼-卡克方法，旨在缓解维度灾难。作为推论，我们独立于滤波问题获得了福克-普朗克方程逼近的收敛速率。收敛性分析通过一个$10$维非线性数值算例加以补充，证明了该方法的鲁棒性。