发表机构
Mathematical Institute, Utrecht University; School of Mathematics and Statistics, The University of Sydney（乌得勒支大学; 悉尼大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种从向量场样本联合推断哈密顿量及泊松结构的通用方法，覆盖已知、常数及线性结构情形，通过特征值问题和约束拟合恢复系统，并在Lorenz-86与Kirchhoff模型上验证高精度与正确拓扑。
AI 中文摘要
我们提出了一种从向量场样本数据驱动推断哈密顿与Lie-Poisson动力系统的方法，其中哈密顿量与底层泊松张量被联合估计，而非假设结构已知。哈密顿量被表示为具有固定随机内部权重的随机特征映射的线性组合，因此对于任何给定的泊松结构，其估计都是一个线性回归问题。我们处理三种日益一般的情形：已知的泊松张量、未知的常数可逆（辛）结构矩阵，以及结构矩阵随状态线性变化的Lie-Poisson与仿射泊松结构。在常数情形下，哈密顿量与结构矩阵通过一个特征值问题联合恢复，该问题避免了因哈密顿系统的缩放对称性而产生的平凡零解。对于Lie-Poisson与仿射泊松系统，我们交替进行：先通过广义特征值问题从近似首次积分中初步估计哈密顿量，再进行结构常数的约束非线性最小二乘拟合，该拟合通过惩罚参数上的延拓强制满足Jacobi恒等式，对于仿射结构还需满足余循环条件。正交匹配追踪选择一组紧凑且条件良好的随机特征子集。由于学习到的哈密顿量是单特征项之和，该模型允许显式的泊松/辛分裂积分器，我们用它通过长时间庞加莱截面来验证学习到的系统。我们在四维和五维Lorenz-86模型以及六维Kirchhoff刚体系统上展示了该方法，高精度地恢复了哈密顿量、结构常数和Casimir不变量，并表明随着随机特征数量的增加，推断出的庞加莱截面收敛到正确的不变环面拓扑。
英文摘要
We present a method for the data-driven inference of Hamiltonian and Lie-Poisson dynamical systems from vector-field samples, in which the Hamiltonian and the underlying Poisson tensor are estimated jointly rather than assuming the structure is known. The Hamiltonian is represented as a linear combination of random feature maps with fixed random internal weights, so its estimation is a linear regression problem for any given Poisson structure. We treat three cases of increasing generality: a known Poisson tensor, an unknown constant invertible (symplectic) structure matrix, and Lie-Poisson and affine Poisson structures, with structure matrix linear in the state. In the constant case, the Hamiltonian and structure matrix are recovered jointly from an eigenvalue problem that avoids the trivial zero solution due to the scaling symmetry of Hamiltonian systems. For Lie-Poisson and affine Poisson systems we alternate between an initial Hamiltonian estimate from a generalized eigenvalue problem for an approximate first integral, and a constrained nonlinear least-squares fit of the structure constants that enforces the Jacobi identity and, for affine structures, the cocycle condition, via continuation in a penalty parameter. Orthogonal matching pursuit selects a compact, well-conditioned subset of random features. Because the learned Hamiltonian is a sum of single-feature terms, the model admits an explicit Poisson/symplectic splitting integrator, which we use to validate the learned systems via long-time Poincare sections. We demonstrate the approach on the four- and five-dimensional Lorenz-86 model and a six-dimensional Kirchhoff rigid-body system, recovering Hamiltonians, structure constants and Casimir invariants to high accuracy, and showing the inferred Poincare sections converge to the correct invariant-tori topology as the number of random features increases.
