Inverse Boundary Value and Optimal Control Problems on Graphs: A Neural and Numerical Synthesis

TL;DR

This paper introduces a neural and numerical framework for inverse boundary value and optimal control problems on graphs, utilizing boundary-injected message passing neural networks and regularization for improved stability and accuracy.

Contribution

It presents a novel boundary-injected message passing neural network architecture and a regularization technique based on graphical distance for system identification on graphs.

Findings

Enhanced prediction accuracy near boundaries.

Improved stability of predictions at distant nodes.

Effective regularization method for graph-based control problems.

Abstract

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.