Dipartimento di Matematica

Prossimi Seminari

• Nonintrusive reduced order models using physics informed neural networks
  Jan S. Hesthaven, Chair of Computational Mathematics and Simulation Science, EPFL, Lausanne, CH
  giovedì 29 ottobre 2020 alle ore 14:00 precise, Online seminar: https://mox.polimi.it/elenco-seminari/?id_evento=1979&t=763724
  ABSTRACT
  ABSTRACT
  

  The development of reduced order models for complex applications, offering the promise for rapid and accurate evaluation of the output of complex models under parameterized variation, remains a very active research area. Applications are found in problems which require many evaluations, sampled over a potentially large parameter space, such as in optimization, control, uncertainty quantification, and in applications where a near real-time response is needed. However, many challenges remain unresolved to secure the flexibility, robustness, and efficiency needed for general large-scale applications, in particular for nonlinear and/or time-dependent problems.
  
  After giving a brief general introduction to projection based reduced order models, we discuss the use of artificial feedforward neural networks to enable the development of fast and accurate nonintrusive models for complex problems. We demonstrate that this approach offers substantial flexibility and robustness for general nonlinear problems and enables the development of fast reduced order models for complex applications.
  
  In the second part of the talk, we discuss how to use residual based neural networks in which knowledge of the governing equations is built into the network and show that this has advantages both for training and for the overall accuracy of the model.
  
  Time permitting, we finally discuss the use of reduced order models in the context of prediction, i.e. to estimate solutions in regions of the parameter beyond that of the initial training. With an emphasis on the Mori-Zwansig formulation for time-dependent problems, we discuss how to accurately account for the effect of the unresolved and truncated scales on the long term dynamics and show that accounting for these through a memory term significantly improves the predictive accuracy of the reduced order model.
  
  Contatto: alfio.quarteroni@polimi.it

Seminari Passati

• Artificial Intelligence and Personalised Cardiac Modelling: Learning by Heart
  Maxime Sermesant, INRIA
  giovedì 17 settembre 2020 alle ore 14:00, Online seminar: mox.polimi.it/elenco-seminari/?id_evento=1978&t=763724
  ABSTRACT
  

  Machine learning and biophysical modelling are very complementary approaches. The recent progress in computing power and available data makes it possible to develop accurate data-driven approaches for healthcare, while biophysical models offer a principled way to represent anatomy and physiology. In this talk, I will present research where we combine both methodologies in order to leverage their strengths. Different clinical applications in computational cardiology will be presented.
  This seminar is organized within the ERC-2016-ADG Research project iHEART - An Integrated Heart Model forthe simulation of the cardiac function, that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 740132).
  Contact: alfio.quarteroni@polimi.it
  ABSTRACT
• 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
  

  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.
  ABSTRACT
• 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

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