Inverse Boundary Value and Optimal Control Problems on Graphs: A Neural and Numerical Synthesis
This work addresses inverse boundary value and optimal control problems on graphs, which is incremental as it builds on existing graph neural network methods with specific enhancements for boundary conditions.
The paper tackles the problem of system identification and optimal control on graphs with boundary conditions, introducing a boundary injected message passing neural network and a graphical distance regularization technique to achieve more accurate and stable predictions, especially near and far from the boundary.
A general setup for deterministic system identification problems on graphs with Dirichlet and Neumann boundary conditions is introduced. When control nodes are available along the boundary, we apply a discretize-then-optimize method to estimate an optimal control. A key piece in the present architecture is our boundary injected message passing neural network. This will produce more accurate predictions that are considerably more stable in proximity of the boundary. Also, a regularization technique based on graphical distance is introduced that helps with stabilizing the predictions at nodes far from the boundary.