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

We introduce Euler–Poincaré Neural Dynamics (EPND), a geometric-mechanics–driven framework that redefines how Koopman-type neural models learn dynamical systems. Unlike conventional operator-learning methods that rely on function-space linearization, EPND places geometric mechanics at the core of its architecture the mathematical engine governing evolution through Lie-group flows. This foundation enables a principled treatment of curvature, symmetry, and conservation, that ensures both interpretability and physical consistency. 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 of versatile scientific simulation.