发表机构
Max-Planck-Institut für Plasmaphysik; Technische Universität München; University of Twente（马克斯·普朗克等离子体物理研究所; 慕尼黑工业大学; 特文特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了GFHNNs、PGFHNNs等保结构神经网络架构，用于学习确定性与随机受迫哈密顿系统，其长期稳定性和准确性优于非几何残差神经网络，且所需训练数据更少。
AI 中文摘要
我们开发了一种基于神经网络学习确定性与随机受迫哈密顿系统的几何框架。受拉格朗日-达朗贝尔原理和变分积分器理论的启发，我们引入了拉格朗日-达朗贝尔映射的概念，并建立了一阶单步法的$C^r$收敛定理。基于这些结果，我们提出了广义受迫哈密顿神经网络（Generalized Forced Hamiltonian Neural Networks，GFHNNs），这是一类通过串联拉格朗日-达朗贝尔-欧拉映射得到的保结构神经网络，并证明了该架构的通用逼近定理。我们进一步将该框架扩展到参数依赖系统，得到参数化广义受迫哈密顿神经网络（Parametric Generalized Forced Hamiltonian Neural Networks，PGFHNNs）。当底层维纳过程的信息可用时，通过将斯特拉托诺维奇-泰勒展开中出现的多个斯特拉托诺维奇积分解释为参数，同一框架可应用于随机受迫哈密顿系统。我们的数值实验表明，与非几何残差神经网络相比，所提出的几何架构在长期稳定性和准确性上有显著提升，同时实现可比性能所需的训练数据量大幅减少。
英文摘要
We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.
