An augmented Lagrangian deep learning method for variational problems with essential boundary conditions

by   Jianguo Huang, et al.

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.



There are no comments yet.


page 1

page 2

page 3

page 4


Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

We propose a method due to Nitsche (Deep Nitsche Method) from 1970s to d...

Extended full waveform inversion in the time domain by the augmented Lagrangian method

Extended full-waveform inversion (FWI) has shown promising results for a...

Unification of variational multiscale analysis and Nitsche's method, and a resulting boundary layer fine-scale model

We show that in the variational multiscale framework, the weak enforceme...

Augmented Lagrangian method for a TV-based model for demodulating phase discontinuities

In this work we reformulate the method presented in App. Opt. 53:2297 (2...

Computational homogenization of time-harmonic Maxwell's equations

In this paper we consider a numerical homogenization technique for curl-...

Adaptive Learning on the Grids for Elliptic Hemivariational Inequalities

This paper introduces a deep learning method for solving an elliptic hem...

Uniform Convergence Guarantees for the Deep Ritz Method for Nonlinear Problems

We provide convergence guarantees for the Deep Ritz Method for abstract ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.