Learning solutions to some toy constrained optimization problems in infinite dimensional Hilbert spaces
This is an incremental application of existing methods to new data, with potential benefits for computational efficiency in specific theoretical or physics-related optimization tasks.
The authors tackled the problem of solving constrained optimization in infinite-dimensional Hilbert spaces by implementing deep learning versions of penalty and augmented Lagrangian methods on toy problems from calculus of variations and physics, achieving decent approximations and significant speedups in cases where constraint function outputs are functions.
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.