Euler–Poincaré Neural Dynamics: A Geometric-Mechanics Framework for Scientific Simulation

We introduce Euler--Poincaré Neural Dynamics (EPND), a geometric-mechanics framework that casts evolution-operator learning as Lie-group flows for long-horizon dynamical modeling. Unlike conventional operator-learning approaches that treat temporal propagation as an unconstrained black-box map, EPND places geometric mechanics at the core of its architecture, playing a role of the mathematical engine. This foundation enables a principled treatment of curvature, symmetry, and conservation, with the learned evolution expressed in geometric terms. Building on this foundation, we develop the Euler--Poincaré Parallel Scan, a parallel algorithm that leverages the associative algebra of Lie-group compositions to overcome the inefficiencies of sequential computation. By unifying geometric structure with scalable computation, EPND achieves high accuracy, strong stability, and significant parallel acceleration in modeling long-horizon dynamics in versatile scientific simulations.