Model reduction from data

动力学模型的历史可以追溯到拉普拉斯的工作：他的恶魔可以根据数学模型(和初始条件)确定宇宙中每个原子的位置。动态模型也是现代科学、工程和技术的核心：准确描述行为、系统和过程以进行分析和设计，需要使用动态模型。这一观点突显了动力学模型在现代科学技术中的重要性。模型要么是通过利用物理基本定律获得的，要么是越来越多地在系统与人类和环境交互的情况下利用机器学习和人工智能技术获得的。最后，最近开发了一种纯数据驱动的方法，因此是在假设基础系统具有特定属性(例如线性)的情况下开发的。无论采用哪种方法，所得到的模型的复杂性都随着建模的系统、过程、现象的复杂性而增加。我们的结论是，当代自然和人造系统的复杂性使得精确的模型是不切实际的或不可能的，人们必须求助于近似。本研究计划的目标是为一种新的、数据驱动的建模范例奠定基础，该范例提供的近似模型保留了底层系统的特定属性，并且对于用户选择的一组操作条件是任意准确的。这是通过定义所谓的洛夫纳算符、它们的移位形式以及发展洛夫纳演算来实现的。“洛夫纳观点”允许用函数(洛纳算子，它是有限维运算符)、“移位”(它扮演时间导数或离散微分器的角色)和洛夫纳演算(它从洛夫纳算符的操作生成准确的、低复杂性的模型)来描述复杂的动力学行为。方法论结果将得到数值方法的发展的支持，以实时实施建议的算法，并生成数据驱动的控制策略，以保证底层系统的安全、可靠和最佳运行。研究计划还旨在确定可以从建议的建模方法中受益的应用领域，初步考虑波能转换器的最优功率提取问题，PI已经在该问题上做出了开创性的成果，这是增加这种可再生能源渗透率的关键。

The history of dynamical models traces back to the work of Laplace: his demon could determine the location of every atom in the universe on the basis of a mathematical model (and initial conditions). Dynamic models are also at the core of modern science, engineering and technology: the accurate description of behaviours, systems and processes for analysis and design, requires the use of dynamical models. This perspective highlights the importance of dynamical models for in modern science and technology.Models are either obtained by exploiting the fundamental laws of physics or, more and more often in the case in which systems interact with humans and the environment, exploiting machine learning and AI techniques. Finally, a purely data-driven, hence model-free, approach has been recently developed under the assumption that the underlying system possesses specific properties, such as linearity. Regardless of the approach, the complexity of the obtained model grows with the complexity of the system, process, phenomenon, to model. We conclude that the complexity of contemporary natural and man-made systems is such that accurate models are impractical, or impossible, to derive and one must resort to approximations.The objective of this research programme is to lay the foundations of a new, data-driven, modelling paradigm which provides approximate models which retain specific properties of the underlying systems and are arbitrarily accurate for a set of user-selected operating conditions. This is accomplished by defining the so-called Loewner operators, their shifted versions, and by developing the Loewner calculus. The "Loewner perspective" allows describing complex dynamical behaviours in terms of functions (the Lowner operators, which are finite dimensional operators), "shifts" (which plays the role of the time-derivative or of a discrete-differentiator) and the Loewner calculus (which generates accurate, low-complexity models, from manipulations of the Loewner operators). The methodological results will be supported by the development of numerical methods for real-time implementation of the proposed algorithms and for the generation of data-driven control strategies guaranteeing safe, reliable, and optimal operation of the underlying system.The research programme aims also at identifying application domains which could benefit from the proposed modelling methodology, with initial considerations to the problem of optimal power extraction for wave energy converters, problems for which the PI has already contributed ground-breaking results and which is key for increasing the penetration of this renewable energy source.