An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions (Communications in Computational Physics, 2022)
This paper is concerned with a novel deep learning method for variationalproblems with essential boundary conditions. To this end, we first reformulate theoriginal problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinitedimensional setting. Based on this, by expressing the primal and dual variables withtwo individual deep neural network functions, we present an augmented Lagrangiandeep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penaltymethod, the new method admits two main advantages: i) the choice of the penaltyparameter is flexible and robust, and ii) the numerical solution is more accurate in thesame magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essentialboundary conditions, and numerical experiments are presented to show the effectiveness of the new method.
Keywords: The augmented Lagrangian method; deep learning; variational problems; saddle point problems; essential boundary conditions.