Learning Stochastic Dynamical Systems with Structured Noise (2503.01077v1)

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.

Learning Stochastic Dynamical Systems with Structured Noise: An Expert Overview

This paper introduces a novel nonparametric framework for learning stochastic dynamical systems modeled by Stochastic Differential Equations (SDEs) with structured noise. The essence of this framework lies in simultaneously estimating both the drift and diffusion components of SDEs, enabling robust modeling of systems where stochastic noise exhibits a singular covariance structure. Such systems often reflect scenarios with inherent dimensionality reductions, which are common in physics-inspired settings.

Key Contributions and Methodology

The paper's primary contribution is the formulation of a learning framework that transparently handles singular noise structures—a form of colored noise with noise terms only active along certain directions. This is a prevalent feature in dynamical systems inspired by classical physics that conform to second-order dynamics. The authors employ a scheme that simultaneously learns the drift function and diffusion matrix by leveraging trajectory data with structured stochasticity.

In modeling terms, the research capitalizes on the mixed SDE (mSDE) representation:

$\dif_t = (_t) + (_t)\dif_t, \quad _t, _t \in R^{D}.$

Here, their approach ensures the mSDE's noise vector has deficient rank, implying certain noise-free evolution paths. Unlike existing methodologies that often rely on parametric assumptions, this method adopts a nonparametric approach. The paper describes a well-formulated numerical algorithm to implement this, designed to operate on trajectory data efficiently.

Numerical Simulations and Outcomes

The paper provides an empirical validation of its methods through various simulation setups across different domains:

• Toy Model: The authors demonstrate accurate recovery of drift and noise parameters within an artificially constructed SDE, yielding minimal relative errors across multiple metrics.
• Van der Pol Oscillator: For this system, prone to nonlinear oscillations, their approach effectively captures the dynamic behavior, closely mirroring the true trajectories under systematic noise conditions.
• Stochastic Cucker-Smale Flocking Model: As a representative high-dimensional system, this offers insights into scaling the learning algorithms to complex systems. The interaction kernels and noise terms were estimated with high fidelity, showcasing the framework's viability for modeling collective dynamics.

The quantitative analysis reveals the effectiveness of their estimator, as reflected by relative $L^2(\rho)$ errors and Wasserstein distances between observed and learned trajectory distributions. Notably, the framework seems robust across systems with varying dimensionality and structural complexity.

Implications and Future Directions

This work has substantial implications for both theoretical research and practical applications. Theoretically, it advances the field by showing how structured noise situations in high-dimensional SDEs can be understood and modeled comprehensively without restrictive assumptions. Practically, it opens avenues for more accurate modeling of real-world phenomena in physics, biology, and engineering, where noise often follows structured, singular patterns.

The authors note ongoing work in extending their approach to automatically learn feature maps alongside drift and diffusion terms—an endeavor expected to mitigate dimensionality issues further and improve estimation fidelity. This is particularly salient when applied to systems such as multi-agent robotics, complex biological systems, and financial markets, where the interplay of numerous interacting agents and noise is crucial.

In conclusion, this paper provides a significant stepping stone towards more flexible and accurate modeling of complex systems governed by stochastic dynamics. The methodology proposed holds promise for various scientific fields, particularly those grappling with high-dimensional data and structured randomness.