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

TL;DR

A general setup for deterministic system identification problems on graphs with Dirichlet and Neumann boundary conditions is introduced and a regularization technique based on graphical distance is introduced that helps with stabilizing the predictions at nodes far from the boundary.

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.

Figures (4)

• Figure 1: A visualization of boundary injected message passing as in Formula (\ref{['msg_passing']}) where node $u$ is boundary/control and gray nodes are interior. Plot (b) shows the flow of messages. Plot (c) shows the aggregation step. Plot (d) shows the update process.
• Figure 2: Graph $G_1$ used in SysID experiments
• Figure 3: Graph $G_2$ used in OC experiment
• Figure 4: Comparison of two boundary injected optimal controls against true diffusion and with $N=50$

Theorems & Definitions (6)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6