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Evolutions, their Mean-Field Approximation, and Learning (Fornasier, TUM)

Zeit: 12.05.2020, 14-15 Uhr
Ort: Online (contact marius.yamakou@fau to get the data for the VC) Lehrstuhl für Angewandte Analysis (Alexander von Humboldt-Professur)

 

Evolutions, their Mean-Field Approximation, and Learning
Prof. Massimo Fornasier, TUM

We present three related models of time evolving phenomena: gradient flows, quasi-static evolutions,
and spatially inhomogenous evolutionary games. All of them are driven by suitable energy or payoff function
minimization/maximization. Gradient flows are motivated by physics for which conservative forces are the derivatives of the potential energies. Given the expected frequency for steady states to occur, quasi-static evolutions focus on the dynamics of critical points.
Spatially inhomogenous evolutionary games describe the motion of players whose strategies evolve simultaneously by a replicator dynamics influenced by players’ position and return a feedback on the velocity field guiding their motion.
For all of these types of evolutions we present a theory of mean-field approximation, which describes the transport of
the probability measure of the initial conditions along trajectories. While for gradient flows mean-field equations are
well-established, a corresponding theory for quasi-static evolutions and evolutionary games constitutes a novel contribution.
For these evolutive models to be applied in practices, we present reliable methods to solve the inverse problem of
inferring/learning the governing energies or payoff functions from observation of realized evolutions.

References:

S. Almi, M. Fornasier, R. Huber, Data-driven evolutions of critical points, arXiv:1911.00298

L. Ambrosio, M. Fornasier, M. Morandotti, and S. Savaré. Spatially Inhomogeneous Evolutionary Games <https://arxiv.org/abs/1805.04027>, May 2018.
https://arxiv.org/pdf/1805.04027.pdf <https://arxiv.org/pdf/1805.04027.pdf>

M. Bongini, M. Fornasier, M. Hansen and M. Maggioni. Inferring Interaction Rules from Observations of Evolutive Systems I: The Variational Approach <http://doi.org/10.1142/S0218202517500208>, Math. Models Methods Appl. Sci., 27(05):909-951, 2017.