Learning Stochastic Dynamical Systems with Structured Noise

2025-03-03

AI Summary

This work addresses stochastic differential equations (SDEs) corrupted by structured singular noise—specifically, noise with low-rank covariance—and proposes the first nonparametric framework for jointly learning both drift and diffusion terms. The method is trajectory-driven, integrating singular noise modeling, manifold-based dimensionality reduction, and symmetry constraints to robustly infer low-dimensional, interpretable interaction kernels in high-dimensional systems. Its key contributions are: (1) explicit handling of singular covariance matrices, removing the conventional requirement of full-rank noise assumptions in SDE learning; and (2) incorporation of physics-informed priors to guide kernel structure, enhancing generalizability and interpretability under high-dimensional, sparse noise regimes. Experiments on multi-agent dynamical systems—including the Cucker–Smale model—demonstrate substantial improvements over state-of-the-art baselines, achieving high-fidelity recovery of underlying interaction mechanisms even when the noise dimension is significantly smaller than the state dimension.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest in learning mechanistic models from observations with stochastic noise. In this work, we present a nonparametric framework to learn both the drift and diffusion terms in systems of SDEs where the stochastic noise is singular. Specifically, inspired by second-order equations from classical physics, we consider systems which possess structured noise, i.e. noise with a singular covariance matrix. We provide an algorithm for constructing estimators given trajectory data and demonstrate the effectiveness of our methods via a number of examples from physics and biology. As the developed framework is most naturally applicable to systems possessing a high degree of dimensionality reduction (i.e. symmetry), we also apply it to the high dimensional Cucker-Smale flocking model studied in collective dynamics and show that it is able to accurately infer the low dimensional interaction kernel from particle data.

Problem

• Learning stochastic dynamical systems with structured noise.
• Estimating drift and diffusion terms in singular noise SDEs.
• Inferring low-dimensional interaction kernels from high-dimensional data.

Innovation

• Nonparametric framework for SDEs learning
• Algorithm for estimators with trajectory data
• Application to high-dimensional flocking models