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

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  • 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.

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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).

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.

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