用神经网络保持能量守恒，高效模拟复杂系统演化
Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks
- 构建端到端对称神经网络，实现相空间降维与动力学学习统一
- 在典型哈密顿系统上实现高精度轨迹重建与长期稳定预测
- 适合需长期模拟的物理系统建模，如天体运动、分子动力学
我们提出一种新型数据驱动的辛诱导降阶建模（ROM）框架，用于高维哈密顿系统。该框架将隐空间发现与动力学学习统一于单一端到端神经架构中，编码器-解码器基于亨农神经网络（HenonNets），可添加线性SGS反射层，实现全空间与隐空间之间的精确辛映射。隐空间动力学由作为亨农网络实现的辛流映射推进。该统一神经架构确保在降阶层面精确保持底层辛结构，显著提升模型保真度与长期稳定性。我们在经典哈密顿系统上进行了全面数值实验，结果表明该方法能实现高精度轨迹重建、训练范围外的稳健预测以及准确的哈密顿量保持。这些成果凸显了本辛化ROM框架在复杂动力系统中的有效性与广泛应用潜力。
原文摘要 · Abstract (English)
We introduce a novel data-driven symplectic induced-order modeling (ROM) framework for high-dimensional Hamiltonian systems that unifies latent-space discovery and dynamics learning within a single, end-to-end neural architecture. The encoder-decoder is built from Henon neural networks (HenonNets) and may be augmented with linear SGS-reflector layers. This yields an exact symplectic map between full and latent phase spaces. Latent dynamics are advanced by a symplectic flow map implemented as a HenonNet. This unified neural architecture ensures exact preservation of the underlying symplectic structure at the reduced-order level, significantly enhancing the fidelity and long-term stability of the resulting ROM. We validate our method through comprehensive numerical experiments on canonical Hamiltonian systems. The results demonstrate the method's capability for accurate trajectory reconstruction, robust predictive performance beyond the training horizon, and accurate Hamiltonian preservation. These promising outcomes underscore the effectiveness and potential applicability of our symplectic ROM framework for complex dynamical systems across a broad range of scientific and engineering disciplines.
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