Low-dimensional Invariant Coordinate Systems for Dynamic Modeling

9312092 Kirby这一建议为提取一般动力系统的简化描述提供了一种新的分析工具。它详细说明了如何将位于低维不变流形或吸引子上的动力学系统重新表示为低维系统。在对这类系统的研究中，人们通常会遇到以偏微分方程组或常微分方程组或差分方程组的形式枯萎的复杂模型。这种“高维性”与系统模型应该由少数方程控制的猜想想法相矛盾。本项目涉及的问题是如何客观地从基于建模考虑的过于详细和冗余的描述转向准确反映系统真实维度的简化模型。提出了一种技术，允许将模型重新表述为其最优简化形式，并展示了如何利用该技术来促进具有混沌或湍流解的动力系统的研究。这种方法的基础是将模型简化为最简单的形式，即将偏微分方程组或常微分方程组简化为可能的形式。这是使用一种类似于瓶颈的神经网络结构来实现的。神经网络构造了多个变量的非线性向量函数逼近。瓶颈中节点的值对应于最优压缩到可能维度最低的流形上的动态信息。因此，模型方程可以用瓶颈流形上的变量来表示。这可以看作是对瓶颈流形相空间的非线性压缩。这可以被视为动力系统相空间的非线性压缩，并且在数学上等价于将流投影到非正交基上。***

9312092 Kirby This proposal presents a new analytic tool for extracting a reduced description of general dynamical systems. It details how systems with dynamics which lie on low-dimensional invariant manifolds or attractors can be reformulated into systems of low dimensionality. In the study of such systems one is typically confronted with complex models wither in the form of partial differential equations, or large systems of ordinary differential or difference equations. This "high- dimensionality" contradicts the conjectured idea that the model of the system should be governable by a small number of equations. This project addresses the issue of how to objectively proceed from an overly detailed and redundant description based on modeling considerations to a reduced model which accurately reflects the true dimensionality of the system. A technique is presented which allows the reformulation of the model to its optimally reduced form and it is shown how this can be used to facilitate the study of dynamical systems with chaotic or turbulent solutions. The basis of the approach is the reduction of the model, wither a partial differential equation or system of ordinary differential equations, to the simplest possible form. This is carried out using a neural network architecture which resembles and bottleneck. The neural network constructs a nonlinear vector function approximation in several variables. The values of the nodes in the bottleneck correspond to the dynamical information optimally compressed onto a manifold of lowest possible dimension. Consequently, the model equations can be formulated in terms of the variables on the bottleneck manifold. This can be viewed as a nonlinear compression of the phase space of the bottleneck manifold. This can be viewed as a nonlinear compression of the phase space of the dynamical system and is mathematically equivalent to a projection of the flow onto a non- orthogonal basis. ***