Multi-Agent System Identification with Nonlinear Sheaf Diffusion
This work provides a theoretical framework for understanding identifiability in multi-agent system identification, which is crucial for applications in robotics, biology, and network science.
The paper identifies a topological obstruction, measured by sheaf cohomology, to uniquely recovering local interaction laws from trajectory data in multi-agent systems governed by nonlinear sheaf Laplacians, and provides conditions for unique recovery within parameterized classes. Experiments show that accurate trajectory reproduction does not guarantee correct law recovery.
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.