Dipartimento di Matematica

Prossimi Seminari

• Deep Learning meets Parametric Partial Differential Equations
  Gitta Kutyniok, Institute of Mathematics, Technische Universität Berlin (DE)
  giovedì 16 luglio 2020 alle ore 14:00, Online seminar: https://mox.polimi.it/elenco-seminari/?id_evento=1977&t=763724
  ABSTRACT
  ABSTRACT
  

  High-dimensional parametric partial differential equations (PDEs) appear in various contexts including control and optimization problems, inverse problems, risk assessment, and uncertainty quantification. In most such scenarios the set of all admissible solutions associated with the parameter space is inherently low dimensional. This fact forms the foundation for the reduced basis method.
  Recently, numerical experiments demonstrated the remarkable efficiency of using deep neural networks to solve parametric problems. In this talk, after an introduction into deep learning, we will present a theoretical justification for this class of approaches. More precisely, we will derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric PDEs. In fact, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical approximation results. We use this low-dimensionality to guarantee the existence of a reduced basis. Then, for a large variety of parametric PDEs, we construct neural networks that yield approximations of the parametric maps not suffering from a curse of dimensionality and essentially only depending on the size of the reduced basis.
  Finally, we present a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of parametric partial differential equations. These experiments strongly support the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.

Seminari Passati

• Physics-Informed Neural Networks (PINNs): An alphabet of Algorithms for Diverse Applications
  George Karniadakis, Brown University (USA)
  giovedì 25 giugno 2020 alle ore 14:00 precise, Online seminar: https://mox.polimi.it/elenco-seminari/?id_evento=1976&t=763724
  ABSTRACT
  

  We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical and biological systems and for discovering hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). We also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between Gauss Process Regression and NNs, and discuss the new powerful concept of meta-learning. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties. There are many versions of PINNs, e.g., variational (VPINNs), stochastic (sPINNs), conservative (cPINNs), nonlocal (nPINNs), generalized (xPINNs), etc, and we will provide some highlights. In addition, we will present our recent theoretical results on the convergence and generation of PINNs.
  ABSTRACT
• Design and Health
  Stefano Capolongo , Politecnico di Milano
  mercoledì 18 marzo 2020 alle ore 12:15, aula B21
• Comunicare la scienza: manuale di sopravvivenza
  Maurizio Melis , Giornalista scientifico
  mercoledì 11 marzo 2020 alle ore 12:15, aula B21
• Bergman projections on pseudoconvex domains containing complex manifolds in their boundary
  Gian Maria Dall'Ara, School of Mathematics, University of Birmingham
  mercoledì 4 marzo 2020 alle ore 10:45, aula seminari - 3° piano
• CANCELLED - this event has been cancelled
  Wolfgang Bangerth, Department of Mathematics, Campus Delivery, Fort Collins, CO, USA NUMETH-HPC
  venerdì 28 febbraio 2020 alle ore 14:00, Aula Saleri VI piano
  ABSTRACT
  

  CANCELLED
  
  ABSTRACT
• Modeling and simulation of thermo-poroelastic processes in fractured geothermal reservoirs
  Eirik Keilegavlen, Department of Mathematics, University of Bergen, Norway
  giovedì 20 febbraio 2020 alle ore 11:30, Aula Saleri - VI piano
  ABSTRACT
  

  We present a mathematical model for coupled thermo-hydro-mechanical processes in fractured porous media, motivated by applications to geothermal energy. The model is based on a mixed-dimensional representation of the host medium and fracture network. The dynamics in the host medium is governed by fully-coupled thermo-poroelasticity, while flow and energy transport can also take place in the fracture network. Fracture deformation, including opening and sliding, is modeled using techniques from contact mechanics. We present a finite volume approach for the mixed-dimensional model, combined with a primal-dual active set strategy for the contact problem.
  
  Contatto: luca.formaggia@polimi.it
  
  ABSTRACT
• Bail-in vs bail-out: Bank resolution and liability structure
  Alessandro Sbuelz, Università Cattolica (Milano)
  martedì 18 febbraio 2020 alle ore 12:00, Aula Consiglio
  ABSTRACT
  

  What is the joint impact of different resolution regimes and capital requirements on the optimal liability structure of a bank holding insured deposits and issuing non-bailinable debt and bail-inable Tier1-capital debt? We address this novel question and find that: 1) a credible bail-in resolution regime rules out extreme leverage and creates value by postponing default; 2) a positive probability of bail-out destroys credibility with dramatic effects on financial risk-taking, to the point of reversing the classical positive link between optimal leverage and growth prospects; and 3) a strict enforcement of the Basel III CET1 capital requirement strongly mitigates the impact of a non-credible resolution regime.
  ABSTRACT
• Kalman Filters for PDEs, a new hope
  Philppe Moireau, INRIA et Ecole Polytechnique, Paris, France
  lunedì 17 febbraio 2020 alle ore 16:00, Aula Saleri, VI piano
  ABSTRACT
  

  Kalman filters for PDEs is an old topic which is theoretically seducing but leads to a so-called curse of dimensionality in its numerical implementation. In order to circumvent this curse of dimensionality, it is now classical to base the numerical strategies on reduced order modeling. This raises the question of the accuracy of the resulting estimator.
  Alternatives approaches based on reduced-basis decomposition of the covariance were also developed in the past years, but here with an additional question of stability of the resulting estimator.
  Using H-matrix based discretization strategies, a recent numerical tool developed for integral equations discretization, we here show how a full Kalman estimator can now be envisioned PDE models and their associated large discretization, hence freshening up the interest for this old theory.
  
  
  Contatto: luca.dede@polimi.it
  
  ABSTRACT

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