A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks
This work addresses computational efficiency in solving partial differential equations using PINNs, but it appears incremental as it builds on existing multi-level methods with a new algorithmic perspective.
The authors tackled the challenge of solving large-scale nonlinear optimization problems by proposing a block-coordinate multi-level algorithm, which they applied to physics-informed neural networks (PINNs) for solving partial differential equations, resulting in better solutions and significant computational savings.
Multi-level methods are widely used for the solution of large-scale problems, because of their computational advantages and exploitation of the complementarity between the involved sub-problems. After a re-interpretation of multi-level methods from a block-coordinate point of view, we propose a multi-level algorithm for the solution of nonlinear optimization problems and analyze its evaluation complexity. We apply it to the solution of partial differential equations using physics-informed neural networks (PINNs) and show on a few test problems that the approach results in better solutions and significant computational savings