Revealing hidden dynamics from the stochastic data is a challenging problem as randomness takes part in the evolution of the data. The problem becomes exceedingly complex when the trajectories of the stochastic data are absent in many scenarios. Here we present an approach to effectively modeling the dynamics of the stochastic data without trajectories based on the weak form of the Fokker-Planck (FP) equation, which governs the evolution of the density function in the Brownian process. Taking the collocations of Gaussian functions as the test functions in the weak form of the FP equation, we transfer the derivatives to the Gaussian functions and thus approximate the weak form by the expectational sum of the data. With a dictionary representation of the unknown terms, a linear system is built and then solved by the regression, revealing the unknown dynamics of the data. Hence, we name the method with the Weak Collocation Regression (WCR) method for its three key components: weak form, collocation of Gaussian kernels, and regression. The numerical experiments show that our method is flexible and fast, which reveals the dynamics within seconds in multi-dimensional problems and can be easily extended to high-dimensional data such as 20 dimensions. WCR can also correctly identify the hidden dynamics of the complex tasks with variable-dependent diffusion and coupled drift, and the performance is robust, achieving high accuracy in the case with noise added.

翻译：从随机数据中揭示隐藏动力学是一个具有挑战性的问题，因为随机性参与了数据的演化过程。当随机数据的轨迹在许多场景中缺失时，该问题变得尤为复杂。本文基于福克-普朗克（FP）方程的弱形式，提出了一种无需轨迹即可有效建模随机数据动力学的方法。通过将高斯函数配点作为FP方程弱形式的测试函数，我们将导数转移至高斯函数，从而将弱形式近似为数据的期望和。利用未知项的字典表示，构建线性系统并通过回归求解，进而揭示数据的未知动力学。基于其三个关键组成部分——弱形式、高斯核配点与回归，我们将该方法命名为弱配点回归（Weak Collocation Regression, WCR）方法。数值实验表明，该方法灵活快速，可在数秒内揭示多维问题的动力学，并能轻松扩展至高维数据（如20维）。WCR还能正确识别具有变量依赖扩散和耦合漂移的复杂任务中的隐藏动力学，且性能稳健，在添加噪声的情况下仍能实现高精度。