A numerical scheme for approximating the nonlinear filtering density is introduced and its convergence rate is established, theoretically under a parabolic H\"{o}rmander condition, and empirically for two examples. For the prediction step, between the noisy and partial measurements at discrete times, the scheme approximates the Fokker--Planck equation with a deep splitting scheme, and performs 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. Our convergence proof relies on the Malliavin integration-by-parts formula. As a corollary we obtain the convergence rate for the approximation of the Fokker--Planck equation alone, disconnected from the filtering problem.

翻译：本文提出了一种近似非线性滤波密度的数值格式，并在理论（满足抛物型Hörmander条件）与两个算例上验证了其收敛速率。在离散时刻含噪声与部分观测之间的预测步骤中，该格式采用深度分裂方法近似Fokker–Planck方程，并通过贝叶斯公式执行精确更新，从而构建了一种经典的预测-更新滤波算法，可在训练后在线处理新观测序列。算法采用基于采样的Feynman–Kac方法，旨在缓解维度灾难。收敛性证明依赖于Malliavin分部积分公式。作为推论，我们得到了脱离滤波问题、单独近似Fokker–Planck方程的收敛速率。