The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard approaches typically require high-fidelity training data. In many practical settings, the data are indirectly observed through noisy and nonlinear measurement. The challenge lies not only in approximating the coefficients of the SDEs, but in simultaneously inferring the posterior updates given the observations. In this work, we present a neural path estimation approach to solve stochastic dynamical systems based on variational inference. We first derive a stochastic control problem that solve filtering posterior path measure corresponding to a pathwise Zakai equation. We then construct a generative model that maps the prior path measure to posterior measure through the controlled diffusion and the associated Randon-Nykodym derivative. Through an amortization of sample paths of the observation process, the control is learned by an embedding of the noisy observation paths. Thus, we learn the unknown prior SDE and the control can recover the conditional path measure given the observation sample paths and we learn an associated SDE which induces the same path measure. In the end, we perform experiments on nonlinear dynamical systems, demonstrating the model's ability to learn multimodal, chaotic, or high dimensional systems.

翻译：从数据中重构和推断随机动力系统是反问题与统计学习中的基础任务。虽然代理建模推进了近似这些动力系统的计算方法，但标准方法通常需要高保真度的训练数据。在许多实际场景中，数据是通过含噪声的非线性测量间接观测得到的。挑战不仅在于近似随机微分方程的系数，更在于同时根据观测数据推断后验更新。本文提出一种基于变分推断的神经路径估计方法来解决随机动力系统问题。我们首先推导了一个随机控制问题，该问题求解对应于路径wise Zakai方程的滤波后验路径测度。随后，我们构建了一个生成模型，该模型通过受控扩散过程及相应的Radon-Nikodym导数将先验路径测度映射为后验测度。通过对观测过程样本路径的摊销化处理，控制函数通过嵌入含噪声的观测路径进行学习。由此，我们能够学习未知的先验随机微分方程，且该控制函数能够恢复给定观测样本路径条件下的条件路径测度，同时我们学习到一个能诱导相同路径测度的关联随机微分方程。最后，我们在非线性动力系统上进行了实验，证明了该模型学习多模态、混沌或高维系统的能力。