This paper is concerned with the problem of nonlinear filtering, i.e., computing the conditional distribution of the state of a stochastic dynamical system given a history of noisy partial observations. Conventional sequential importance resampling (SIR) particle filters suffer from fundamental limitations, in scenarios involving degenerate likelihoods or high-dimensional states, due to the weight degeneracy issue. In this paper, we explore an alternative method, which is based on estimating the Brenier optimal transport (OT) map from the current prior distribution of the state to the posterior distribution at the next time step. Unlike SIR particle filters, the OT formulation does not require the analytical form of the likelihood. Moreover, it allows us to harness the approximation power of neural networks to model complex and multi-modal distributions and employ stochastic optimization algorithms to enhance scalability. Extensive numerical experiments are presented that compare the OT method to the SIR particle filter and the ensemble Kalman filter, evaluating the performance in terms of sample efficiency, high-dimensional scalability, and the ability to capture complex and multi-modal distributions.

翻译：本文研究非线性滤波问题，即根据含噪部分观测历史计算随机动力系统状态的条件分布。传统的序贯重要性重采样（SIR）粒子滤波在面临退化似然函数或高维状态场景时，受权重退化问题的制约存在根本性局限。本文探索了一种替代方法，该方法基于从当前状态先验分布到下一时刻后验分布的Brenier最优传输（OT）映射估计。与SIR粒子滤波不同，OT公式无需似然函数的解析形式，且能够利用神经网络的逼近能力对复杂多峰分布进行建模，并采用随机优化算法提升可扩展性。本文通过大量数值实验将OT方法与SIR粒子滤波及集合卡尔曼滤波进行对比，从样本效率、高维可扩展性及复杂多峰分布捕获能力三个维度评估其性能。