论文标题

学习Koopman操作员的不变子空间 - 部分:异质词典混合以近似子空间不变性

Learning Invariant Subspaces of Koopman Operators--Part 2: Heterogeneous Dictionary Mixing to Approximate Subspace Invariance

论文作者

Johnson, Charles A., Balakrishnan, Shara, Yeung, Enoch

论文摘要

这项工作建立在第1部分中提出的模型和概念上,以从数据中学习Koopman运营商的近似词典表示。本文的第一部分提出了一种论证Koopman词典的子空间不变性的方法。这种方法是根据国家包容的逻辑提升(SILL)的。这是具有连接逻辑功能的仿射基础。 SILL字典的非线性功能是同质的,这是数据驱动的Koopman操作员词典学习的规范。在本文中,我们发现,从不同类别的非线性函数绘制的异质词典函数的结构化混合具有与基于深度学习的DEEPDMD算法相同的精度和尺寸缩放。我们通过构建一个由基尔函数和结合性径向基函数(RBFS)组成的异质词典来特别表明这一点。该混合词典与DEEPDMD具有相同的精度和尺寸缩放,同时保持了几何可解释性,同时降低了参数的数量级。这些结果增强了基于字典的Koopman模型来解决高维非线性学习问题的生存能力。

This work builds on the models and concepts presented in part 1 to learn approximate dictionary representations of Koopman operators from data. Part I of this paper presented a methodology for arguing the subspace invariance of a Koopman dictionary. This methodology was demonstrated on the state-inclusive logistic lifting (SILL) basis. This is an affine basis augmented with conjunctive logistic functions. The SILL dictionary's nonlinear functions are homogeneous, a norm in data-driven dictionary learning of Koopman operators. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm. We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves the same accuracy and dimensional scaling as deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

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