MOX-Mathematics Department, Politecnico di Milano, Milano, Italy.
M3DISIM, Institut National de Recherche en Informatique et en Automatique, Palaiseau, France.
Int J Numer Method Biomed Eng. 2021 Jul;37(7):e3471. doi: 10.1002/cnm.3471. Epub 2021 Jun 6.
We propose a method to discover differential equations describing the long-term dynamics of phenomena featuring a multiscale behavior in time, starting from measurements taken at the fast-scale. Our methodology is based on a synergetic combination of data assimilation (DA), used to estimate the parameters associated with the known fast-scale dynamics, and machine learning (ML), used to infer the laws underlying the slow-scale dynamics. Specifically, by exploiting the scale separation between the fast and the slow dynamics, we propose a decoupling of time scales that allows to drastically lower the computational burden. Then, we propose a ML algorithm that learns a parametric mathematical model from a collection of time series coming from the phenomenon to be modeled. Moreover, we study the interpretability of the data-driven models obtained within the black-box learning framework proposed in this paper. In particular, we show that every model can be rewritten in infinitely many different equivalent ways, thus making intrinsically ill-posed the problem of learning a parametric differential equation starting from time series. Hence, we propose a strategy that allows to select a unique representative model in each equivalence class, thus enhancing the interpretability of the results. We demonstrate the effectiveness and noise-robustness of the proposed methods through several test cases, in which we reconstruct several differential models starting from time series generated through the models themselves. Finally, we show the results obtained for a test case in the cardiovascular modeling context, which sheds light on a promising field of application of the proposed methods.
我们提出了一种从快速尺度的测量数据中发现描述具有多尺度时间行为现象的长期动力学的微分方程的方法。我们的方法基于数据同化（DA）和机器学习（ML）的协同组合，用于估计与已知快速尺度动力学相关的参数，并用于推断慢尺度动力学的基本规律。具体来说，通过利用快速和慢动力学之间的尺度分离，我们提出了一种时间尺度的解耦方法，大大降低了计算负担。然后，我们提出了一种从要建模的现象的时间序列中学习参数数学模型的 ML 算法。此外，我们研究了在本文提出的黑盒学习框架内获得的数据驱动模型的可解释性。特别是，我们表明每个模型都可以用无限多种不同的等价方式重写，从而使从时间序列学习参数微分方程的问题本质上是不适定的。因此，我们提出了一种策略，允许在每个等价类中选择一个唯一的代表模型，从而提高结果的可解释性。我们通过几个测试案例证明了所提出方法的有效性和抗噪性，在这些测试案例中，我们从通过模型本身生成的时间序列中重建了几个微分模型。最后，我们展示了在心血管建模背景下的一个测试案例的结果，这为所提出方法的一个有前途的应用领域提供了启示。
