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 through the noisy observation paths and we learn an associated SDE which induces the filtering path measure. In the end, we demonstrate the model's performance on various nonlinear stochastic systems, showcasing its ability to handle multimodal data distributions, chaotic dynamics, and sparse observation data.

翻译：基于数据重建和推断随机动态系统是逆问题与统计学习中的基本任务。虽然代理建模通过计算方法近似这些动态过程，但标准方法通常需要高保真训练数据。在实际场景中，数据往往通过含噪非线性测量间接获得。其挑战不仅在于近似随机微分方程（SDE）的系数，更在于同时根据观测值推断后验更新。本研究提出一种基于变分推断的神经路径估计方法，用于求解随机动态系统。我们首先推导出一个随机控制问题，该问题可求解对应路径型Zakai方程的滤波后验路径测度。随后构建生成模型，通过受控扩散过程及其对应的Radon-Nikodym导数，将先验路径测度映射到后验测度。通过对观测过程样本路径的摊销处理，控制函数通过含噪观测路径学习得到，同时我们学习一个诱导滤波路径测度的关联SDE。最终，我们在多种非线性随机系统上验证模型性能，展示其处理多模态数据分布、混沌动力学及稀疏观测数据的能力。