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Structure-preserving learning and prediction in optimal control of collective motion

Wide-spread adoption of unmanned vehicle technologies requires the ability to predict the motion of the combined vehicle operation from observations. While the general prediction of such motion for an arbitrary control mechanism is difficult, for a particular choice of control, the dynamics reduces to the Lie-Poisson equations [33,34]. Our goal is to learn the phase-space dynamics and predict the motion solely from observations, without any knowledge of the control Hamiltonian or the nature of interaction between vehicles. To achieve that goal, we propose the Control Optimal Lie-Poisson Neural Networks (CO-LPNets) for learning and predicting the dynamics of the system from data. Our methods learn the mapping of the phase space through the composition of Poisson maps, which are obtained as flows from Hamiltonians that could be integrated explicitly. CO-LPNets preserve the Poisson bracket and thus preserve Casimirs to machine precision. We discuss the completeness of the derived neural networks and their efficiency in approximating the dynamics. To illustrate the power of the method, we apply these techniques to systems of $N=3$ particles evolving on ${\rm SO}(3)$ group, which describe coupled rigid bodies rotating about their center of mass, and ${\rm SE}(3)$ group, applicable to the movement of unmanned air and water vehicles. Numerical results demonstrate that CO-LPNets learn the dynamics in phase space from data points and reproduce trajectories, with good accuracy, over hundreds of time steps. The method uses a limited number of points ($\sim200$/dimension) and parameters ($\sim 1000$ in our case), demonstrating potential for practical applications and edge deployment.