Multi-Agent System Identification with Nonlinear Sheaf Diffusion

TL;DR

This paper investigates the challenges of identifying local interaction laws in multi-agent systems governed by nonlinear sheaf Laplacians, revealing topological obstructions and conditions for successful recovery from trajectory data.

Contribution

It introduces a topological framework using sheaf cohomology to characterize when the interaction laws can be uniquely recovered from trajectory observations.

Findings

Topological obstructions to recovery are measured by sheaf cohomology.

Unique recovery is possible if and only if the sheaf cohomology vanishes.

Finite-dimensional parameterized recovery is feasible when a data-dependent matrix is positive definite.

Abstract

Local interaction laws governing multi-agent systems can be difficult to recover from trajectory data, even when the dynamics are observed faithfully. In systems governed by a nonlinear sheaf Laplacian -- a generalization of the graph Laplacian accommodating heterogeneous state spaces and asymmetric communication channels -- the coordination law is encoded by edge potential functions whose gradients produce the inter-agent forces. Because trajectory observations record node-state evolution, they expose only the aggregate effect of the edge forces at each node: distinct interaction laws that agree at the node level are indistinguishable from trajectory data alone. We show that the fundamental obstruction to recovery is topological, measured by sheaf cohomology, and that unique recovery from an unconstrained function class is possible if and only if this cohomology vanishes. When the obstruction is nontrivial, we show that recovery within a finite-dimensional parameterized class is possible precisely when a data-dependent information matrix is positive definite. Experiments validate the theory and illustrate that accurate trajectory reproduction need not certify recovery of the underlying interaction law.