Learning Stochastic Dynamical Systems with Structured Noise
This work addresses the challenge of modeling complex systems with structured noise, which is incremental as it builds on existing SDE learning methods by incorporating singular covariance structures.
The authors tackled the problem of learning stochastic differential equations (SDEs) with singular noise by developing a nonparametric framework to estimate drift and diffusion terms, demonstrating its effectiveness on examples from physics and biology, including accurately inferring low-dimensional interaction kernels in high-dimensional flocking models.
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.