中心流形逼近的核方法和基于数据的中心流形定理
Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem
摘要：
对于具有非双曲平衡点的动力系统，利用中心流形理论可以大大简化稳定性的研究。该理论允许隔离系统接近平衡点的复杂渐近行为，并通过分析所谓的中心流形上的降阶系统来获得其行为的有意义的预测。由于中心流形通常是未知的，良好的近似方法是重要的，因为中心流形定理指出，降阶系统的原点的稳定性性质与全阶系统的原点的稳定性性质相同。在这项工作中，我们建立了一个基于数据的版本的中心流形定理，其工作原理是通过考虑近似代替精确流形。此外，近似和原始的减少动态之间的误差进行量化。然后，我们使用适当的基于数据的核方法来构建一个合适的近似流形接近平衡，这是与我们的一般误差理论兼容。通过高精度求解器对整个系统进行重复的数值模拟来收集数据，该求解器生成离散轨迹集，然后将其用作训练集。该方法进行了测试，不同的例子，表现出良好的性能和精度。
英文摘要：
For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
