Sparse regression has emerged as a popular technique for learning dynamical systems from temporal data, beginning with the SINDy (Sparse Identification of Nonlinear Dynamics) framework proposed by arXiv:1509.03580. Quantifying the uncertainty inherent in differential equations learned from data remains an open problem, thus we propose leveraging recent advances in statistical inference for sparse regression to address this issue. Focusing on systems of ordinary differential equations (ODEs), SINDy assumes that each equation is a parsimonious linear combination of a few candidate functions, such as polynomials, and uses methods such as sequentially-thresholded least squares or the Lasso to identify a small subset of these functions that govern the system's dynamics. We instead employ bias-corrected versions of the Lasso and ridge regression estimators, as well as an empirical Bayes variable selection technique known as SEMMS, to estimate each ODE as a linear combination of terms that are statistically significant. We demonstrate through simulations that this approach allows us to recover the functional terms that correctly describe the dynamics more often than existing methods that do not account for uncertainty.

翻译：稀疏回归已成为从时间序列数据学习动力系统的流行技术，其开创性工作始于arXiv:1509.03580提出的SINDy（非线性动力系统稀疏识别）框架。量化从数据习得的微分方程中固有的不确定性仍是一个开放问题，因此我们提出利用稀疏回归统计推断的最新进展来解决该问题。针对常微分方程（ODE）系统，SINDy假设每个方程是少数候选函数（如多项式）的简约线性组合，并采用序贯阈值最小二乘法或Lasso等方法识别控制系统动力学的这些函数的子集。我们转而采用Lasso和岭回归估计量的偏差校正版本，以及称为SEMMS的经验贝叶斯变量选择技术，将每个ODE估计为统计显著项的线性组合。通过仿真实验证明，与未考虑不确定性的现有方法相比，该方法能够更准确地恢复正确描述动力学的函数项。