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.

翻译：随机微分方程（SDEs）是一种普遍存在的建模框架，广泛应用于物理学、生物学、工程学、社会科学和金融学领域。由于大规模数据集的可用性，从带有随机噪声的观测数据中学习机理模型引起了日益增长的兴趣。在本工作中，我们提出了一种非参数框架，用于学习随机噪声为奇异情况下的SDE系统中的漂移项和扩散项。具体而言，受经典物理学中二阶方程的启发，我们考虑具有结构化噪声的系统，即具有奇异协方差矩阵的噪声。我们提供了一种基于轨迹数据构建估计量的算法，并通过物理学和生物学中的多个示例展示了我们方法的有效性。由于所开发的框架最自然地适用于具有高度降维（即对称性）的系统，我们还将其应用于集体动力学中研究的高维Cucker-Smale集群模型，并表明它能够从粒子数据中准确推断出低维相互作用核。