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.

翻译：无人机技术的广泛应用要求能够从观测数据中预测多机协同运行的运动轨迹。尽管针对任意控制机制的运动轨迹通用预测较为困难，但对于特定控制选择，其动力学可简化为李-泊松方程[33,34]。我们的目标是在完全未知控制哈密顿量或飞行器间相互作用机制的情况下，仅通过观测数据学习相空间动力学并预测运动轨迹。为实现该目标，我们提出了控制最优李-泊松神经网络（CO-LPNets），用于从数据中学习和预测系统动力学。该方法通过泊松映射的复合学习相空间映射，这些映射作为可显式积分的哈密顿量流获得。CO-LPNets 保持泊松括号结构，从而以机器精度保持卡西米尔不变量。我们讨论了所推导神经网络的完备性及其在动力学近似中的效率。为展示该方法的能力，我们将其应用于 $N=3$ 的粒子系统：在 ${\rm SO}(3)$ 群上演化的系统（描述绕质心旋转的耦合刚体），以及适用于无人空中/水下航行器运动的 ${\rm SO}(3)$ 群系统。数值结果表明，CO-LPNets 能够从数据点学习相空间动力学，并以较高精度在数百个时间步长上复现运动轨迹。该方法仅需有限数据点（约每维度 $\sim200$ 个）和参数（本文案例约 $\sim 1000$），展现出实际应用与边缘部署的潜力。