  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.
