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 with 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：本文提出的算法旨在通过将弱形式应用于随机SINDy方法，为随机微分方程（SDEs）的发现提供一种新颖的数据驱动框架。该算法的弱形式提供了一种具有噪声鲁棒性的方法论，避免了传统上使用有限差分计算含噪导数的弊端。另一创新点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$K_j(X_{t_n})$的使用保证了期望意义上的无偏性，并防止了时间测试函数中存在的结构性回归偏差。该框架将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基准上验证了该算法，成功恢复了所有活跃非线性项，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均忠实复现了真实弛豫时间尺度。