Model reduction from data

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.