| Abstract: | 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. |
