发表机构
City St George’s, University of London（伦敦城市圣乔治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对混沌流降阶建模问题，提出通过聚类读取局部几何形状塑造径向基库的非侵入式降阶模型，经全局正则化最小二乘求解拟合降阶速度，在多个混沌系统上验证，能再现长期统计量，有效预测且匹配能量分布等，优于全局Galerkin投影。
AI 中文摘要
混沌系统通常在低维吸引子上演变，其几何形状因区域而异。我们提出一种非侵入式降阶模型，通过聚类读取局部几何形状，并用于塑造径向基库，其核适应每个区域。通过一次全局正则化最小二乘求解将降阶速度拟合到该库上，得到一个明确、可微的向量场，可再现长期统计量即不变测度，无需使用控制方程。由于径向基场远离数据衰减，自身无法返回逃逸状态，通过运动校正器稳定积分，其大小衡量每个结果依赖学习场而非校正器的程度。在Lorenz-63上，模型恢复吸引子、边缘密度以及正负Lyapunov指数，但对强横向收缩恢复不足。在Lorenz-96上，其有效预测时间与调谐神经网络和储层计算预测器竞争，且在全状态和降阶可观测量上都再现不变测度。在Kuramoto-Sivashinsky方程和准周期Kolmogorov流上，模型匹配侵入式量化局部Galerkin模型的能量分布和频谱，并优于相同维度的全局Galerkin投影，且无需投影控制方程。
英文摘要
Chaotic systems often evolve on a low-dimensional attractor whose geometry varies from one region to another. We propose a non-intrusive reduced-order model that reads this local geometry by clustering and uses it to shape a radial basis library whose kernels adapt to each region. Fitting the reduced velocity onto this library by one global least-squares solve gives an explicit, differentiable vector field that reproduces the long-term statistics without any use of the governing equations. A radial basis field decays away from the data and cannot by itself return an escaped state. The integration is therefore stabilised by a kinematic corrector, whose reported magnitude measures how far each result rests on the learned field. On Lorenz-63 the model recovers the attractor, its marginal densities and its Lyapunov spectrum. On Lorenz-96 its valid prediction time matches typical configurations of neural-network and reservoir-computing forecasters and trails their best-tuned ones, and the invariant measure is reproduced on both the full state and on a reduced observable. On the Kuramoto--Sivashinsky equation and the quasiperiodic Kolmogorov flow the model matches the energy distribution and spectrum of an intrusive quantised-local Galerkin model and improves on a global Galerkin projection of the same reduced dimension. The recovery is dictated by the distance from a state to its nearest kernels, not by the one-step regression error.
Comments29 pages, 24 Figures, Preprint
