  In this work we present deep learning implementations of two popular
theoretical constrained optimization algorithms in infinite dimensional Hilbert
spaces, namely, the penalty and the augmented Lagrangian methods. We test these
algorithms on some toy problems originating in either calculus of variations or
physics. We demonstrate that both methods are able to produce decent
approximations for the test problems and are comparable in terms of different
errors produced. Leveraging the common occurrence of the Lagrange multiplier
update rule being computationally less expensive than solving subproblems in
the penalty method, we achieve significant speedups in cases when the output of
the constraint function is itself a function.
