This paper is concerned with online filtering of discretely observed nonlinear diffusion processes. Our approach is based on the fully adapted auxiliary particle filter, which involves Doob's $h$-transforms that are typically intractable. We propose a computational framework to approximate these $h$-transforms by solving the underlying backward Kolmogorov equations using nonlinear Feynman-Kac formulas and neural networks. The methodology allows one to train a locally optimal particle filter prior to the data-assimilation procedure. Numerical experiments illustrate that the proposed approach can be orders of magnitude more efficient than state-of-the-art particle filters in the regime of highly informative observations, when the observations are extreme under the model, or if the state dimension is large.

翻译：本文关注离散观测非线性扩散过程的在线滤波问题。我们基于完全自适应的辅助粒子滤波方法，该方法涉及通常难以计算的Doob的$h$变换。我们提出一个计算框架，通过使用非线性Feynman-Kac公式和神经网络求解背后的后向Kolmogorov方程来近似这些$h$变换。该方法允许在数据同化过程之前训练一个局部最优的粒子滤波。数值实验表明，在观测信息量丰富、观测值在模型下极端或状态维数较大的情况下，该方法相对于最先进的粒子滤波算法可实现数量级的效率提升。