Pathwise Learning of Stochastic Dynamical Systems with Partial Observations

2026-01-29

AI Summary

This work proposes a variational inference–based neural path estimation method for stochastic dynamical systems under partial observability, noisy measurements, and nonlinear dynamics. By constructing a controlled diffusion process, the approach maps a prior path measure to the posterior measure conditioned on observations, leveraging an observation-embedded representation to learn an optimal control policy. This enables joint learning of the underlying unknown stochastic differential equation (SDE) and its conditional path distribution. The method innovatively integrates the Zakai equation in path space with neural control, achieving, for the first time, end-to-end inference of posterior path measures. Experiments demonstrate that the framework accurately reconstructs system dynamics and efficiently infers posterior path distributions across multimodal, chaotic, and high-dimensional settings.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• stochastic dynamical systems
• partial observations
• pathwise inference
• posterior path measure
• noisy measurements

Innovation

• pathwise inference
• stochastic control
• variational inference
• generative modeling
• partial observations