从轨迹数据中学习带结构性噪声的随机微分方程，可精准还原系统动力学。
Learning Stochastic Dynamical Systems with Structured Noise
- 基于经典物理中的二阶方程思路，建模噪声协方差矩阵奇异的系统。
- 在物理与生物多个实例中验证方法有效性，能准确估计漂移与扩散项。
- 适用于高维可降维系统，如群体行为模型，可反推低维相互作用核函数。
随机微分方程（SDEs）广泛应用于物理、生物、工程、社会科学和金融等领域。随着大规模数据集的出现，从观测数据中学习具有随机噪声的机制模型日益受到关注。本文提出一种非参数框架，用于学习具有奇异噪声的SDE系统中的漂移项与扩散项。受经典物理中二阶方程的启发，我们考虑噪声协方差矩阵奇异的系统，即具有结构性噪声的系统。我们给出基于轨迹数据构造估计器的算法，并通过物理与生物领域的多个例子验证了方法的有效性。由于该框架天然适用于具有高度降维特性的系统（即具有对称性），我们还将其应用于集体动力学中的高维Cucker-Smale flocking模型，成功从粒子数据中准确推断出低维相互作用核函数。
原文摘要 · Abstract (English)
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.
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