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.

翻译：本文提出了一个通用的几何框架，用于模拟和学习在李群变换下保持不变的哈密顿系统的动力学。这意味着已知一组对称性作用于系统并尊重其动力学，因此根据诺特定理，可以观察到守恒量。我们提出通过构建$G$-不变拉格朗日子流形来模拟和学习感兴趣映射，这些子流形是辛几何中的关键对象。我们构造的一个显著特性是：模拟/学习的动力学也保留了与原系统相同的守恒量，从而比非对称感知方法更忠实地替代原始动力学，并能更准确地预测未观测轨迹。此外，我们的框架不仅能模拟/学习哈密顿流，还能模拟/学习任何李群等变辛变换。我们的设计利用了辛几何和几何力学中的关键技术及概念：约化理论、诺特定理、拉格朗日子流形、动量映射以及余迷向约化等。我们还提出了在保持底层几何结构的同时学习泊松变换的方法，以及如何为非几何积分器赋予几何性质。因此，本文旨在探索利用辛几何与泊松几何的力量来解决模拟与学习问题的新途径。