Lie-Algebraic Acceleration of Neural Koopman Dynamics

We present a Lie-algebraic approach to model Koopman dynamics that integrates algebraic structure with computational scalability. The proposed formulation constrains the neural generators to evolve within prescribed Lie subalgebras and constructs finite-time flows through a neural Magnus expansion construction, thereby maintaining consistency with the associated Lie-group composition over each time segment. To address the computational burden inherent in sequential propagation, we exploit the associativity of Lie-group compositions and construct segmentwise propagators via a prefix-scan algorithm, which reduces the depth of temporal composition from linear to logarithmic. Consequently, the framework enables accurate long-horizon prediction while improving computational efficiency, and provides a principled foundation for scalable Koopman operator learning for nonlinear systems.

Accurately and efficiently predicting the long-term evolution of complex dynamical systems is an important challenge in scientific simulation. We introduce a Neural Koopman dynamics framework built on a mathematical structure called Lie algebra, allowing the learned dynamics to respect the algebraic structure of the system. We further develop a prefix-scan scheme that combines short-time evolutions in parallel, enabling efficient prediction over long time horizons. Together, these components provide a scalable approach to accurate and computationally efficient modeling of nonlinear dynamics.