University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > A Deep Ritz method with r-adaptivity for solving Partial Differential Equations

A Deep Ritz method with r-adaptivity for solving Partial Differential Equations

Add to your list(s) Download to your calendar using vCal

  • UserDavid Pardo (University of the Basque Country, BCAM - Basque Center for Applied Mathematics)
  • ClockThursday 18 November 2021, 14:30-15:00
  • HouseSeminar Room 1, Newton Institute.

If you have a question about this talk, please contact nobody.

MDLW03 - Deep learning and partial differential equations

(joint work with Javier Omella, Jon A. Rivera, and Jamie M. Taylor) The Ritz method is a traditional method for solving symmetric and positive definite problems governed by Partial Differential Equations (PDEs). This method minimizes the energy functional, and a first Neural Network (NN) formulation using this method was proposed in [1]. In this talk, we first illustrate how traditional methods for solving PDEs using NNs (like Deep-Ritz, Deep Least-Squares, and other Deep Galerkin methods) may suffer from strong quadrature problems, leading to poor approximate solutions. We envision four alternatives to overcome this challenge: a) Monte Carlo methods, b) adaptive integration, c) piecewise-polynomial approximations of the NN solution, and d) the inclusion of regularization terms in the loss following the ideas of [2]. From all these methods, we develop an r-adaptive method, which falls under the category of piecewise-polynomials approximations of the NN. We consider a piecewise-linear solution over a grid—allowing for exact integration—and simultaneously optimize the node positions (r-adaptivity) and the solution values. We show promising numerical results of the r-adaptive Deep Ritz method in one- and two-dimensional domains.

Weinan E and Bing Yu, The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems. Commun. Math. Stat., vol. 6, no. 1, pp. 1–12 (2018). https://doi.org/10.1007/s40304-018-0127-z

Siddhartha Mishra and Roberto Molinaro, Estimates on the generalization error of physics informed neural networks (PINNs) for approximating PDEs. arXiv preprint arXiv:2006.16144  (2020).

This talk is part of the Isaac Newton Institute Seminar Series series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.

 

© 2006-2022 Talks.cam, University of Cambridge. Contact Us | Help and Documentation | Privacy and Publicity