动力系统控制方程的稀疏与非参数估计及其在生物学中的应用
Sparse and nonparametric estimation of equations governing dynamical systems with applications to biology
AI总结：
提出一种结合稀疏参数估计与非参数技术的新框架，用于克服Sindy等参数化模型在捕捉复杂系统非线性时的局限，并在复杂生物现象估计的示例中验证了其有效性。
摘要：
数据驱动的模型方程发现是理解许多科学领域中动力系统行为的强大方法。特别是，从数据中学习数学模型的能力将有利于系统生物学，因为这些系统的复杂性通常使得自下而上的建模方法不可行。近年来，稀疏估计技术在系统辨识中受到重视，主要采用参数化范式以最小的模型复杂度高效捕捉系统动力学。特别是，Sindy算法成功利用稀疏性，通过从函数库中提取捕捉这些系统动力学所需的少数关键项来估计非线性系统。然而，参数化模型往往无法准确表示复杂系统固有的某些非线性。为了解决这一限制，我们引入了一个将稀疏参数估计与非参数技术相结合的新框架。它能够捕捉Sindy无法描述的非线性，而无需关于其函数形式的先验信息。也就是说，无需扩展函数库以包含试图发现的函数。我们在与复杂生物现象估计相关的几个示例上展示了我们的方法。
英文摘要：
Data-driven discovery of model equations is a powerful approach for understanding the behavior of dynamical systems in many scientific fields. In particular, the ability to learn mathematical models from data would benefit systems biology, where the complex nature of these systems often makes a bottom up approach to modeling unfeasible. In recent years, sparse estimation techniques have gained prominence in system identification, primarily using parametric paradigms to efficiently capture system dynamics with minimal model complexity. In particular, the Sindy algorithm has successfully used sparsity to estimate nonlinear systems by extracting from a library of functions only a few key terms needed to capture the dynamics of these systems. However, parametric models often fall short in accurately representing certain nonlinearities inherent in complex systems. To address this limitation, we introduce a novel framework that integrates sparse parametric estimation with nonparametric techniques. It captures nonlinearities that Sindy cannot describe without requiring a priori information about their functional form. That is, without expanding the library of functions to include the one that is trying to be discovered. We illustrate our approach on several examples related to estimation of complex biological phenomena.
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