Rigid-body dynamics learning keeps its geometry: LieSPHGP separates dissipation from stochastic forcing for steadier long-horizon prediction
Synopsis
LieSPHGP combines port-Hamiltonian structure, Lie-group geometry, Gaussian processes and stochastic differential equations to learn controlled, dissipative rigid-body dynamics from noisy trajectories. Matérn-kernel Gaussian processes model mass, potential energy, dissipation, control and diffusion separately, while exponential-map updates keep rotations on the manifold. On a windy pendulum and a quadrotor benchmark, it improves prediction accuracy, long-horizon stability and manifold-constraint satisfaction over neural dynamics baselines.
Interpretation
Learned rigid-body dynamics can now retain rotational geometry, dissipative structure and stochastic forcing at once, instead of trading physical structure against noise modeling. Earlier port-Hamiltonian neural models mostly described deterministic ordinary differential equations in Euclidean coordinates, while stochastic port-Hamiltonian neural networks parameterized the Hamiltonian as a single unstructured network and assumed clean full-state observations. On a windy SO(3) pendulum, LieSPHGP and Lie-PH-NN-SDE share the port-Hamiltonian structure, Lie-IMEX integrator and EKF likelihood and differ only in function parameterization and weight posterior; the former has lower rotation, angular-velocity and energy errors and CRPS, with coverage closer to nominal.
Motion is represented on Lie groups and Lie algebras, and rotations advance through the exponential map, so optimization avoids coordinate parameterizations such as Euler angles that introduce gimbal lock. Mass, potential energy, dissipation, control and diffusion each come from a Matérn-kernel Gaussian-process subnetwork; inverse mass and dissipation are built symmetric positive definite through a Cholesky factorization, and a separate diffusion term represents stochastic forcing. Training uses an error-state extended Kalman filter on the Lie group to approximately marginalize latent states and to separate process from observation noise; evaluation covers the pendulum, the quadrotor and real BlueROV2 Heavy underwater recordings, where LieSPHGP has lower RMSE at 2 s than a physics model given published parameters and the longest valid prediction time.
Perspective
Evaluation covers a single rigid-body pendulum on SO(3), a quadrotor on SE(3), and real BlueROV2 Heavy and Crazyflie flight recordings; the authors state that the current formulation and evaluation focus on single rigid-body dynamics rather than general multi-joint articulated systems.
Inference relies on an approximate EKF likelihood, so the variational objective is not a certified lower bound on exact SDE evidence; the numerical guarantees concern rotational group preservation and contractivity of the frozen damping subsystem, not stability or a convergence order for the complete nonlinear stochastic scheme.