Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Sofiia Huraka, Vakhtang Putkaradze

TL;DR

This work tackles predicting the collective motion of $N$ agents evolving on a Lie group $G$ from observations when the underlying control Hamiltonian is unknown. It introduces CO-LPNets, a framework that learns phase-space mappings by composing Poisson transformations derived from simple test Hamiltonians, ensuring exact preservation of the Lie–Poisson bracket and Casimir invariants. The method is demonstrated on groups ${\rm SO}(3)$ and ${\rm SE}(3)$ with two interaction topologies, ${\it Dictatorship}$ and ${\it Democracy}$, showing accurate trajectory reproduction and energy/Casimir conservation over long horizons with relatively little data and modest parameter counts. This structure-preserving, data-efficient approach promises robust, edge-deployable prediction for multi-vehicle control and lays groundwork for extensions to higher-dimensional groups and heterogeneous networks.

Abstract

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.

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

TL;DR

This work tackles predicting the collective motion of agents evolving on a Lie group from observations when the underlying control Hamiltonian is unknown. It introduces CO-LPNets, a framework that learns phase-space mappings by composing Poisson transformations derived from simple test Hamiltonians, ensuring exact preservation of the Lie–Poisson bracket and Casimir invariants. The method is demonstrated on groups and with two interaction topologies, and , showing accurate trajectory reproduction and energy/Casimir conservation over long horizons with relatively little data and modest parameter counts. This structure-preserving, data-efficient approach promises robust, edge-deployable prediction for multi-vehicle control and lays groundwork for extensions to higher-dimensional groups and heterogeneous networks.

Abstract

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 particles evolving on group, which describe coupled rigid bodies rotating about their center of mass, and 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 (/dimension) and parameters ( in our case), demonstrating potential for practical applications and edge deployment.
Paper Structure (57 sections, 3 theorems, 128 equations, 9 figures)

This paper contains 57 sections, 3 theorems, 128 equations, 9 figures.

Table of Contents

  1. Problem setup and general approach
  2. Background: controlled dynamics of interacting autonomous vehicles
  3. Data-driven discovery of differential equations with structure
  4. Discovering equations vs mappings in phase space.
  5. Data-based, structure-preserving computing of Poisson systems
  6. Data-based predictions for Lie-Poisson systems
  7. Contributions of this paper
  8. Novelty of results
  9. Plan of the paper
  10. Background work in the area
  11. The Lie-Poisson theory of optimal control of coupled interacting particles
  12. Pontryagin's maximum principle: Connection of optimal control problems with canonical Hamiltonian equations
  13. Symmetry reduction applied to optimal control
  14. The general theory of controlled dynamics of $N$ interacting particles
  15. Generating the ground truth data and predictions: the two cases
  16. ...and 42 more sections

Key Result

Lemma 3.2

Suppose $h(\mu) = f(\xi)$, where $\xi= \left\langle \alpha , \mu \right\rangle + \beta$, $\alpha$ is a vector belonging to the Lie algebra $\mathfrak{g}^N$, dual to $\breve{\mu} \in (\mathfrak{g}^*)^N$, and $\beta$ is a constant. Then the Lie--Poisson reduced dynamics LP_reduced_dynamics_equations

Figures (9)

  • Figure 1: A schematic of proposed neural network. The top of the Figure illustrates the learning procedure. The ground truth data involves sets of short pieces of trajectories with the beginning and end points. The neural network consists of a sequence of Poisson transformations $\mathbf{P}_1, \mathbf{P}_2, \ldots, \mathbf{P}_K$, depending on parameters that are applied in sequence to the original points. The Poisson transformations are obtained from the test Hamiltonian $h_k = w_k \mu_k$, $k = 1, \ldots K$, with the index $k$ running through all the indices of the vector $\breve{\mu}$. The parameters $w_k$ are defined by the neural network depending on the initial conditions for each time step $w_k = w_{k,NN}(\breve{\mu_0})$, indicated by the black arrows pointing to the network. The network parameters for $w_{k,NN}(\breve{\mu_0})$ defining each sequence of transformation are optimized to the output of the mapping is as close as possible to the ground truth data for the end of the interval. On the bottom of the Figure, procedure for reconstructing the next step starting from $\breve{\mu}_{NN}(t_\alpha)$. The reconstruction consists of $K$ intermediate Poisson transformations with the learned parameters.
  • Figure 2: Trajectory comparison in the case of three particles for ${\rm SO}(3)$ group, for the case of 'Democracy'. Ground truth case is denoted with blue colour, while CO-LPNets - with red colour, for one representative trajectory. All components $\mu_{k\alpha}$ are plotted, with particle index indicated by the row, and the component $\alpha$ by the column of the table.
  • Figure 3: Errors in the Casimir for each particle (left), Total energy (center) and Mean Absolute Error (MAE, right) for the trajectory presented on Figure \ref{['fig:SO3_test_trajectory_Democracy']}. MAE is computed for all ten reconstructed solutions, not just for the one shown on Figure \ref{['fig:SO3_test_trajectory_Democracy']}. Again, ground truth solution is denoted with blue colour, while the results of CO-LPNets are denoted with red colour.
  • Figure 4: The comparison of solutions for the case of three particles for ${\rm SO}(3)$ group, the 'Dictatorship' case, with all notations and color scheme as in Figure \ref{['fig:SO3_test_trajectory_Democracy']}.
  • Figure 5: Errors in the Casimir (left), Energy (center) and Mean Absolute Error (MAE, right) for the sample trajectory presented on Figure \ref{['fig:SO3_test_trajectory_dictatorship']}. Similar to Figure \ref{['fig:SO3_Casimir_Energy_error_Democracy']}, MAE is computed over all sample solutions. Color scheme and notations are the same as in Figure \ref{['fig:SO3_Casimir_Energy_error_Democracy']}.
  • ...and 4 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4: Geometric structure of equations
  • Remark 2.5
  • Remark 3.1: On data observability and applicability of our theory
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4: On the approximation of phase flow
  • Remark 3.5: On polynomial Hamiltonians
  • ...and 4 more