用单变量函数替代MLP，提升哈密顿神经网络的稳定性与精度
Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks
- 以单变量变换替代MLP，更好捕捉高频多尺度动态
- 在四类经典问题上能量漂移降低，长期预测更稳定
- 适合高维复杂物理系统建模，尤其参数稀缺场景
我们提出基于柯尔莫哥洛夫-阿诺德表示的哈密顿神经网络（KAR-HNN），用一元变换取代多层感知机（MLP）。尽管哈密顿神经网络（HNN）通过直接从数据学习哈密顿函数实现能量守恒，但现有方法多依赖MLP，对超参数敏感且难以探索复杂能量景观。本方法利用局部函数逼近能力，更有效地捕捉高频与多尺度动力学，显著减少能量漂移，提升长期预测稳定性。网络保持哈密顿系统的辛结构，确保可解释性与物理一致性。在弹簧质量、单摆、二体与三体问题四个基准测试中验证了其有效性，预示其在高维、参数稀疏的真实物理过程建模中具有广泛适用性。
原文摘要 · Abstract (English)
We propose a Kolmogorov-Arnold Representation-based Hamiltonian Neural Network (KAR-HNN) that replaces the Multilayer Perceptrons (MLPs) with univariate transformations. While Hamiltonian Neural Networks (HNNs) ensure energy conservation by learning Hamiltonian functions directly from data, existing implementations, often relying on MLPs, cause hypersensitivity to the hyperparameters while exploring complex energy landscapes. Our approach exploits the localized function approximations to better capture high-frequency and multi-scale dynamics, reducing energy drift and improving long-term predictive stability. The networks preserve the symplectic form of Hamiltonian systems, and thus maintain interpretability and physical consistency. After assessing KAR-HNN on four benchmark problems including spring-mass, simple pendulum, two- and three-body problem, we foresee its effectiveness for accurate and stable modeling of realistic physical processes often at high dimensions and with few known parameters.
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