Mathematics and Numerics of Deep Neural Networks for Physical Simulations

Christian Lessig
christian.lessig@ovgu.de

Thomas Richter
thomas.richter@ovgu.de



Date and time

The seminar will take place Wednesdays 09:15-10:45 in G29-E037. The first introductory session starts on October 28. Here we discuss possible topics and start the process of allocating these projects and topics to you. Starting mid December we plan to have first presentations of projects and talks.

Due to Covid-19 we will have to be flexible and will organize the seminar in a hybrid fashion. All online parts as well as broadcasts of sessions at the University will happen in the seminar's Zoom channel.

Corona-Rules

We kindly ask you to read and acknowledge the Corona guidelines of the University

Topics

This seminar is a joint event of the faculties for computer science and mathematics. Students will give talks or present projects on different mathematical and numerical aspects of deep neural networks used for physical simulations. The focus can either be on the presenation of the underlying mathematical theory or on the implementation of approaches and a demonstration and ciritical discussion of the results.

  1. Jonathan Irmscher
    A. R. Barron. Approximation and Estimation Bounds for Artificial Neural Networks. Machine Learning, 14(1):115-133, jan 1994.
  2. Henry von Wahl
    H. Bölcskei, P. Grohs, G. Kutyniok, and P. Petersen. Optimal Approximation with Sparsely Connected Deep Neural Networks. SIAM Journal on Mathematics of Data Science, 1(1):8-45, jan 2019.
  3. Harshit Kapadia
  4. Robert Piel, Robert Brockhoff
    E. Haber, L. Ruthotto Stable architectures for deep neural networks. Inverse Problems 34(1):014004, 2017.
  5. Leopold Latsch, Martyna Minakowska, Martyna Soszynska
    W. E, B. Yu. The deep ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12, 2018.
    L. Lu, X. Meng, Z. Mao, G.E. Karniadakis DeepXDE: A deep learning library for solving differential equations. https://arxiv.org/abs/1907.04502
  6. Piotr Minakowski
    Approximation of Elasticity Problems with Deep Neural Networks
Further Topics:
  1. C. Gin, B. Lusch, S. L. Brunton, and J. N. Kutz.
    Deep Learning Models for Global Coordinate Transformations that Linearize PDEs. arXiv e-prints, page arXiv:1911.02710, Nov. 2019.
  2. P. Grohs, F. Hornung, A. Jentzen, and P. Zimmermann.
    Space-time error estimates for deep neural network approximations for differential equations. arXiv e-prints, page arXiv:1908.03833, Aug. 2019.
  3. G. Kutyniok, P. Petersen, M. Raslan, and R. Schneider.
    A Theoretical Analysis of Deep Neural Networks and Parametric PDEs. mar 2019.
  4. W. E, B. Yu.
    The deep ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12, 2018.
  5. E. Haber, L. Ruthotto.
    Stable architectures for deep neural networks. Inverse Problems, 34(1):014004, dec 2017.

Audience

This seminar is intended for students in Master programs of mathematics and computer science.

Participation and registration

If you are interested in participating in the seminar please let us know by mail, thomas.richter@ovgu.de or christian.lessig@ovgu.de and register online LSF.