Learning Stochastic Dynamical Systems with Structured Noise

TL;DR

The paper addresses learning stochastic dynamical systems with structured (singular) noise by formulating mixed stochastic differential equations ($\mathrm{mSDEs}$) and introducing a nonparametric, convex-loss framework to recover both drift and diffusion from trajectory data. It decomposes the state into low-dimensional components using feature maps $\boldsymbol{\xi}_{\mathbf{f}}$ and $\boldsymbol{\xi}_{\mathbf{g}}$, and estimates drift via convex losses $\mathcal{E}_f$ and $\mathcal{E}_g$, while diffusion is inferred from $\boldsymbol{\Sigma}^{\mathbf{y}} = (\boldsymbol{\sigma}^{\mathbf{y}})(\boldsymbol{\sigma}^{\mathbf{y}})^{\top}$ using quadratic variation. The authors validate the approach on a toy model and physics/biology-inspired systems (van der Pol, Vicsek, Henon-Heiles, stochastic Cucker-Smale), demonstrating accurate recovery of drift and diffusion and the ability to infer low-dimensional interaction structure from high-dimensional data. This framework enables data-driven, mechanistic modeling of complex stochastic systems with structured noise, with broad relevance to physics, biology, engineering, and finance; future work includes learning the optimal feature maps to enhance scalability and accuracy.

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.

Figures (10)

• Figure 1: Comparison of ${f}$ (left) and $\hat{{f}}$ (right) for the toy model equation \ref{['eq:toy model']}.
• Figure 2: Comparison of ${g}$ (left) and $\hat{{g}}$ (right) for toy model
• Figure 3: Van der Pol trajectory in the x-y plane. Left: Trajectory generated using the true drift function. Right: Trajectory generated using the estimated drift function.
• Figure 4: Comparison of ${f}$ (left) and $\hat{{f}}$ (right) for the Van der Pol oscillator equation \ref{['eq:Van der Pol']}.
• Figure 5: Comparison of ${g}$ (left) and $\hat{{g}}$ (right) for Van der Pol oscillator equation \ref{['eq:Van der Pol']}.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2