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：稀疏非线性动力学识别（SINDy）算法可应用于随机微分方程，通过随机微分方程轨迹的观测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，而样本数据通常通过系统状态的数值估计得到。我们分析了先前提出的漂移与扩散估计量在有限数据条件下的误差界性能。然而，由于该算法仅在采样频率与轨迹长度均趋于无穷时收敛，在特定容差内获得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量。在给定采样频率下，这些估计量能够更精确地逼近漂移函数与扩散函数，使SINDy成为更实用的系统辨识方法。