用核方法学习耗散哈密顿系统，提升噪声数据下的预测精度。
Learning dissipative Hamiltonian dynamics with reproducing kernel Hilbert spaces and random Fourier features
- 通过赫尔姆霍兹分解分离保守与耗散项，分别用对称核和无旋核建模。
- 在两个耗散哈密顿系统上验证，相比高斯可分核显著提升预测准确率。
- 适合对物理规律建模、需保持能量结构的动态系统研究者。
本文提出一种新方法，从有限且含噪数据中学习耗散哈密顿动力学。该方法利用赫尔姆霍兹分解将向量场分解为辛分量与耗散分量，分别采用定义于辛核与无旋核的再生核希尔伯特空间进行建模，核函数设计为满足奇对称性以强化物理一致性。通过随机傅里叶特征近似核函数，有效降低优化问题维度。在两个耗散哈密顿系统的仿真中验证了方法性能，结果表明其预测精度显著优于使用高斯可分核的方法。
原文摘要 · Abstract (English)
This paper presents a new method for learning dissipative Hamiltonian dynamics from a limited and noisy dataset. The method uses the Helmholtz decomposition to learn a vector field as the sum of a symplectic and a dissipative vector field. The two vector fields are learned using two reproducing kernel Hilbert spaces, defined by a symplectic and a curl-free kernel, where the kernels are specialized to enforce odd symmetry. Random Fourier features are used to approximate the kernels to reduce the dimension of the optimization problem. The performance of the method is validated in simulations for two dissipative Hamiltonian systems, and it is shown that the method improves predictive accuracy significantly compared to a method where a Gaussian separable kernel is used.
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