Structure-Preserving Learning and Prediction in Optimal Control of Collective Motion

The widespread adoption of autonomous vehicle technologies requires accurate predictions of coordinated multi-agent motion. While predicting such motion under arbitrary control mechanisms is generally intractable, this paper focuses on certain classes of optimal control where the system dynamics reduce to Lie-Poisson equations. In this context, the goal of the paper is to learn the dynamics solely from data, without prior knowledge of the control Hamiltonian or the inter-agent interaction laws. The main achievement of this paper is the introduction of Control Optimal Lie-Poisson Neural Networks (CO-LPNets), which are built from a composition of Poisson maps. By design, CO-LPNets preserve the system’s Casimir invariants to machine precision. The paper also demonstrates the completeness of these neural networks and highlights their representational efficiency. CO-LPNets are applied to systems of interacting particles on the SO(3) and SE(3) Lie groups, modeling coupled rigid body rotations and the spatial navigation of unmanned vehicles, respectively. Numerical evaluations confirm that CO-LPNets accurately learn the global phase-space dynamics from sparse data, faithfully reproducing trajectories over hundreds of time steps. Furthermore, the paper demonstrates the robustness of the architecture against observational noise. Requiring minimal training data (∼200 points per dimension) and highly compact architectures (∼1000 parameters), CO-LPNets offer a highly efficient, structure-preserving solution well-suited for practical edge deployment in autonomous systems.