The proposed algorithm seeks to provide a novel data-driven framework for the discovery of stochastic differential equations (SDEs) by application of the Weak-formulation to stochastic SINDy. This Weak formulation of the algorithm provides a noise-robust methodology that avoids traditional noisy derivative computation using finite differences. An additional novelty is the adoption of spatial Gaussian test functions in place of temporal test functions, wherein, the use of the kernel weight $K_j(X_{t_n})$ guarantees unbiasedness in expectation and prevents the structural regression bias that is otherwise pertinent temporal test functions. The proposed framework converts the SDE identification problem into two SINDy based linear sparse identification problems. We validate the algorithm on three SDEs, for which we recover all active non-linear terms with coefficient errors below 4\%, stationary-density total-variation distances below 0.01, and autocorrelation functions that reproduce true relaxation timescales across all three benchmarks faithfully.

翻译：该算法旨在通过将弱形式应用于随机SINDy，为随机微分方程（SDEs）的发现提供一种新颖的数据驱动框架。算法的弱形式公式提供了一种抗噪声的方法，避免了使用有限差分法进行传统含噪导数计算。另一创新之处在于采用空间高斯测试函数替代时间测试函数，其中核权重$K_j(X_{t_n})$的使用保证了期望无偏性，并防止了时间测试函数中固有的结构回归偏差。所提出的框架将SDE识别问题转化为两个基于SINDy的线性稀疏识别问题。我们在三个SDE上验证了该算法，成功恢复了所有活跃的非线性项，系数误差低于4%，平稳密度总变差距离低于0.01，且自相关函数在所有三个基准测试中忠实再现了真实弛豫时间尺度。