具有离散时滞的时滞微分方程的数据驱动发现
Data-driven Discovery of Delay Differential Equations with Discrete Delays
AI总结：
本研究扩展了SINDy框架，通过增广库和贝叶斯优化识别具有离散时滞的微分方程，显著减少了计算调用次数并支持多变量多时滞参数识别。
摘要：
非线性动力学稀疏识别（SINDy）框架是一种识别控制方程的稳健方法，已成功应用于常微分方程、偏微分方程和随机微分方程。在本研究中，我们通过使用包含延迟样本的增广库和贝叶斯优化，扩展了SINDy以识别时滞微分方程。为了识别可能未知的时滞，我们在一组候选值上最小化重构误差。由此产生的方法通过显著减少相对于暴力方法的SINDy调用次数，提高了整体性能。我们还处理了多元设置，以识别多个未知时滞和（非乘法性）参数。对具有不同长期行为、变量数、时滞和参数的时滞微分方程进行的几项数值测试支持了贝叶斯优化的使用，突出了所提出方法的有效性及其计算优势。因此，可发现的模型类别得到了显著扩展。
英文摘要：
The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a robust method for identifying governing equations, successfully applied to ordinary, partial, and stochastic differential equations. In this work we extend SINDy to identify delay differential equations by using an augmented library that includes delayed samples and Bayesian optimization. To identify a possibly unknown delay we minimize the reconstruction error over a set of candidates. The resulting methodology improves the overall performance by remarkably reducing the number of calls to SINDy with respect to a brute force approach. We also address a multivariate setting to identify multiple unknown delays and (non-multiplicative) parameters. Several numerical tests on delay differential equations with different long-term behavior, number of variables, delays, and parameters support the use of Bayesian optimization highlighting both the efficacy of the proposed methodology and its computational advantages. As a consequence, the class of discoverable models is significantly expanded.
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