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Title: A data-driven approach to transfer operators in nonlinear dynamics using neural networks Authors:  Luigi Marangio - Universite Bourgogne Franche-Comte / Universita di Pisa (Italy) [presenting]
Christophe Guyeux - Universite de Franche Comte (France)
Abstract: Non linear dynamical systems arise almost everywhere in science: natural phenomenas, from biology to economics, can be described as a non linear dynamics in some suitable space. In several contexts finding a good dynamical model for the phenomena which is studied is a hard task. Recently, a new trend is taking hold: rather than observing a phenomena trying to model it with (partial differential) equations, this new approach aims to compute (usually) a matrix that should describe, up to some approximation error, the dynamic underling a big set of data (data-driven approach). Functional analysis provide a powerful tool to understand the statistical properties of dynamics, the so-called transfer operator: an infinite dimensional operator associated to a dynamical system, describing how the dynamics governs the evolution of initial probability densities instead of initial points. Under suitable assumptions, it is reasonable to approximate it with a matrix, which can be computed from the data. The goal is to shortly resume the data-driven techniques developed until now, clarify the mathematical theory involved in these approaches, to present a neural network based algorithm for the computation of a finite approximation of the transfer operator, and finally to apply this algorithm to a data set arising from fireman activity.