带推断的常微分方程稀疏重构
Sparse reconstruction of ordinary differential equations with inference
AI总结：
针对 SINDy 从数据学习 ODE 时缺乏不确定性量化的问题，本文结合偏差校正 Lasso、岭回归及 SEMMS 经验贝叶斯变量选择进行稀疏重构，模拟显示其更准确恢复动力学项。
摘要：
稀疏回归已成为从时间数据中学习动力系统的一种流行技术，其开端是 arXiv:1509.03580 提出的 SINDy（Sparse Identification of Nonlinear Dynamics，非线性动力学稀疏识别）框架。量化从数据中学习到的微分方程所固有的不确定性仍是一个开放问题，因此我们提议利用稀疏回归统计推断的最新进展来解决这一问题。聚焦于常微分方程（ODE）系统，SINDy 假设每个方程都是少数候选函数（例如多项式）的简约线性组合，并使用序贯阈值最小二乘或 Lasso 等方法来识别这些函数中支配系统动力学的一小部分子集。我们则采用 Lasso 和岭回归估计量的偏差校正版本，以及一种称为 SEMMS 的经验贝叶斯变量选择技术，将每个 ODE 估计为具有统计显著性的项的线性组合。我们通过模拟证明，与现有未考虑不确定性的方法相比，该方法使我们能够更频繁地恢复正确描述动力学的函数项。
英文摘要：
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.
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