Multi-Agent System Identification with Nonlinear Sheaf Diffusion

2026-05-11

AI Summary

This work addresses the fundamental challenge of uniquely recovering local interaction rules in multi-agent systems from trajectory data, particularly under nonlinear layer Laplacian dynamics where distinct interaction mechanisms can yield identical node evolutions. The authors propose a nonlinear layered diffusion framework driven by gradients of edge potential functions and, leveraging sheaf cohomology theory, establish for the first time that the intrinsic obstruction to identifiability stems from the nontriviality of the sheaf cohomology group. They further prove that the vanishing of this cohomology group is both necessary and sufficient for unique identifiability of the interaction law. Within finite-dimensional parametric classes, they introduce a practical identifiability criterion based on the positive definiteness of the data information matrix. Both theoretical analysis and experiments demonstrate that even when trajectories are nearly indistinguishable, the underlying interactions may remain unidentifiable, highlighting the inherent topological nature of this problem.

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.

Problem

• Multi-Agent System
• System Identification
• Nonlinear Sheaf Diffusion
• Trajectory Data
• Interaction Law

Innovation

• nonlinear sheaf diffusion
• multi-agent system identification
• sheaf cohomology
• edge potential functions
• information matrix