Low-dimensional Invariant Coordinate Systems for Dynamic Modeling

9312092 Kirby这个建议提出了一个新的分析工具，用于提取一般动力系统的简化描述。 它详细说明了如何与低维不变流形或吸引子上的动态系统可以被重新表述为低维系统。 在研究这类系统时，人们通常会遇到偏微分方程形式的复杂模型，或常微分或差分方程的大型系统。 这种“高维性”与系统模型应该由少量方程控制的假设相矛盾。 该项目解决了如何客观地从基于建模考虑的过于详细和冗余的描述进行简化模型，准确地反映系统的真实维度的问题。 提出了一种技术，它允许重新制定的模型，其最佳的简化形式，它示出了如何可以用来促进研究的动力系统与混沌或湍流的解决方案。 该方法的基础是减少模型，一个偏微分方程或常微分方程组，以最简单的可能形式。 这是使用类似于瓶颈的神经网络架构进行的。 神经网络构造了一个多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尽可能低维度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。 这可以看作是瓶颈流形相空间的非线性压缩。 这可以被视为动力系统的相空间的非线性压缩，并且在数学上等价于流在非正交基上的投影。 ***

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. ***