2023/08/30 by Miguel Vaquero, Jorge Cortés, Vaquero, Miguel +3 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #37C79 #37J06 #65L05 #70H15 #70H30 #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #I.2.6 #I.6.1 #Machine Learning (cs.LG) #Mathematical Physics (math-ph) #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Optimization and Control (math.OC) #Symplectic Geometry (math.SG) #Time Series Analysis and Forecasting
paper · pdf · doi:10.48550/arxiv.2308.16331
openalex publication_date 2023/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether's Theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether's Theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry towards simulating and learning problems.