The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm can be applied to stochastic differential equations to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.

翻译：随机非线性动力学稀疏识别（SINDy）算法可应用于随机微分方程，通过随机微分方程实现过程中的数据估计漂移函数与扩散函数。该算法需要从这些函数中获取样本数据，这些数据通常通过状态数据的数值方法进行估计。我们分析了现有漂移及扩散函数估计的性能，给出了有限数据条件下的误差界。然而，由于该算法仅当采样频率与轨迹长度同时趋于无穷时才收敛，因此获得满足特定容差标准的近似解可能难以实现。为解决此问题，我们开发了适用于SINDy框架的高阶精度估计方法。在给定采样频率下，这些估计能更精确地近似漂移函数与扩散函数，从而显著提升SINDy作为系统辨识方法的可行性。