Multi-Agent System Identification with Nonlinear Sheaf Diffusion
Authors: Nivar Anwer, Hans Riess, Matthew Hale
Organizations: Department of Computer Science, Georgia Tech · Department of Electrical & Computer Engineering, Georgia Tech
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.
We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.
Recovering latent interaction structures from multi-agent dynamics is important for understanding and predicting interacting systems. Trajectory-based structural inference has achieved promising performance, but conventional formulations assume that the trajectories of all modeled agents are available. In practice, agents may become unobserved at deployment because of limited sensing, occlusion, or communication failure. Existing studies have considered unseen-node estimation, structural inference under partial observations, and missing-value imputation, yet the joint recovery of hidden-agent trajectories and their interactions remains underexplored. We formulate this problem as structural inference under hidden agents. Its key difficulty is a circular dependency: recovering interactions involving a hidden agent requires an estimate of its trajectory, while trajectory reconstruction can itself benefit from structural information. To address this challenge, we propose Structural Inference under Hidden Agents (SIHA), which combines structure-agnostic initialization with structure-guided iterative refinement. SIHA reconstructs hidden trajectories from visible observations, infers interactions using Neural Relational Inference, and feeds the estimated structure back into hidden-state reconstruction through multi-strength structural attention and iterative state--structure updates. Experiments on three benchmark dynamical systems demonstrate consistent improvements in visible-to-visible structural inference, while also showing benefits in hidden-state reconstruction and future prediction. Motion-capture experiments with simulated whole-limb occlusion further demonstrate its effectiveness in realistic hidden-agent settings.
We present a differentiable optimization framework for multi-agent coordination. An input is decomposed into overlapping local views, each processed by an agent that solves a convex subproblem parameterized by a neural encoder. Agents coordinate through the Alternating Direction Method of Multipliers (ADMM) with inter-agent constraints specified by a cellular sheaf. The sheaf specifies which aspects of neighboring solutions must agree, allowing for heterogeneous notions of global consensus. Backpropagating through the unrolled optimization jointly trains all components of the multi-agent system. We evaluate on maze pathfinding, image classification, and Sudoku, where agents with individually insufficient local views learn to coordinate to produce correct global outputs. On MNIST, the local-view decomposition yields improved robustness to distribution shifts relative to a standard CNN. On Sudoku, the optimization-derived structure yields markedly higher solve rates than parameter-matched MPNN baselines. Finally, the ADMM structure exposes distinct primal, consensus, and dual state variables, opening the coordination dynamics to direct analysis and intervention -- a property unavailable in standard message-passing architectures.