论文标题
中心流形近似值的内核方法和中心歧管定理的基于数据的版本
Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem
论文作者
论文摘要
对于具有非双曲平衡的动力学系统，可以通过中心歧管理论显着简化稳定性的研究。该理论允许将系统的复杂渐近行为隔离到均衡点附近，并通过分析所谓的中心歧管上的减少订单系统来获得对其行为的有意义的预测。 由于通常不知道中心歧管，因此良好的近似方法很重要，因为中心歧管定理指出，还原订单系统的原点的稳定性与完整阶系统的原点相同。 在这项工作中，我们建立了基于数据的中心歧管定理的基于数据的版本，该版本通过考虑近似代替精确的歧管来起作用。同样，量化了近似和原始还原动力学之间的误差。 然后，我们使用一种基于数据的核心方法来构建接近平衡的歧管的合适近似值，这与我们的一般误差理论兼容。通过高准确求解器对完整系统的重复数值模拟来收集数据，该求解器会生成一组离散轨迹，然后用作训练集。该方法在不同的示例上进行了测试，这些示例显示出有希望的性能和良好的准确性。
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.
