Noise-Aware System Identification for High-Dimensional Stochastic Dynamics

Stochastic dynamical systems are ubiquitous in physics, biology, and engineering, where both deterministic drifts and random fluctuations govern system behavior. Learning these dynamics from data is particularly challenging in high-dimensional settings with complex, correlated, or state-dependent noise. We introduce a noise-aware system identification framework that jointly recovers the deterministic drift and full noise structure directly from the trajectory data, without requiring prior assumptions on the noise model. Our method accommodates a broad class of stochastic dynamics, including colored and multiplicative noise, that scales efficiently to high-dimensional systems, and accurately reconstructs the underlying dynamics. Numerical experiments on diverse systems validate the approach and highlight its potential for data-driven modeling in complex stochastic environments.

💡 Research Summary

The paper introduces a comprehensive framework for system identification of stochastic differential equations (SDEs) that simultaneously recovers both the deterministic drift function and the full state‑dependent noise (diffusion) structure, without imposing restrictive assumptions on the noise. Starting from the general SDE form
(dx_t = f(x_t)dt + \sigma(x_t)dw_t,)
with (x_t, w_t \in \mathbb{R}^D) and diffusion matrix (\Sigma(x)=\sigma(x)\sigma(x)^\top) assumed symmetric positive‑definite, the authors develop a two‑stage learning pipeline.

First, the diffusion matrix is estimated directly from continuous‑time trajectory data using quadratic variation. The loss
(E_\sigma(\tilde\Sigma)=\mathbb{E}\big|