Young Researcher Seminars, Math, Applications and Models

Speaker:  Matteo Garbelli // Sara Galasso - Università di Trento/Università di Verona // Università degli Studi di Padova
  Wednesday, January 19, 2022 at 3:30 PM

First speaker: Matteo Garbelli (Università di Trento/Università di Verona) Title: Deep Learning and Mean Field Optimal Control from a Stochastic Optimal Control perspective Abstract:  In this talk, we illustrate an effective proposal for the mathematical formulation of Deep Learning (DL) based approaches within the contexts of Stochastic Optimal Control (SOC) and Mean-Field Control (MFC). Following a dynamical system perspective, we conduct an in-depth analysis of the Supervised Learning (SL) procedures characterizing Neural Network (NN) models and we show how to translate a SL problem into an optimal (stochastic) MFC one. We also derive two methods to solve this latter: the first is obtained considering the Hamilton–Jacobi–Bellman (HJB) approach in the Wasserstein space of measures, while the second is based on the MF-Pontryagin maximum principle, providing necessary, weaker, condition the solution must satisfy. Second speaker: Sara Galasso (Università degli Studi di Padova) Title: Synchronization in unforced mechanical systems with dissipation Abstract: Every mechanical system has intrinsically some sources of dissipation, with the consequence that the mechanical energy decreases along motions, at a certain rate in time. In fact, physical processes are typically affected by several such sources, each of which might possibly interest only some parts of the system. Mathematically, a first approach -- which allows the problem to be studied in the context of conservative systems -- is not to include in the model the dissipation. This is physically reasonable if the damping is negligible and if one is not interested in the dynamics for asymptotically long times. An alternative -- particularly important in the applications -- is to include in the model forcing terms that restore the energy lost. In between, the reality of dissipative unforced mechanical systems is variegated and challenging. In general, in the study of dissipative systems, questions regarding the asymptotic behaviour become relevant. For example: What can be said about the long-term dynamics when the damping is only partial, at least at first approximation? What about the mid-long-term dynamics? In this talk, we will try to better address these issues, and we shall see, in particular, the implications on the emergence of patterns and collective behaviours -- such as synchronization --, which actually motivated my research as a PhD student. Specifically, we will discuss models of coupled pendula, both finite- and infinite-dimensional, revisiting some aspects of the well-known Huygens's pendulum clocks.

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Publication date
January 14, 2022