论文标题
离散梯度方法的顺序理论
Order theory for discrete gradient methods
论文作者
论文摘要
离散的梯度方法是旨在保留普通微分方程的不变性的集成器。根据这些方法子类的形式序列扩展，我们得出了任意高阶的条件。我们得出了平均矢量场离散梯度的特定结果，在该梯度中，我们从中获得P系列方法，以及用于典范的哈密顿系统的B系列方法。提出了高级方案，并在Hénon-Heiles系统和Lotka-Volterra系统以及神经网络从数据中学到的摆系统的训练和集成中证明了它们的应用。
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive specific results for the average vector field discrete gradient, from which we get P-series methods in the general case, and B-series methods for canonical Hamiltonian systems. Higher order schemes are presented, and their applications are demonstrated on the Hénon-Heiles system and a Lotka-Volterra system, and on both the training and integration of a pendulum system learned from data by a neural network.
