Exact and Optimal Quadratization of Nonlinear Finite-Dimensional Nonautonomous Dynamical Systems

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-03-26 DOI:10.1137/23m1561129

Andrey Bychkov, Opal Issan, Gleb Pogudin, Boris Kramer

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引用次数: 0

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 982-1016, March 2024.
Abstract. Quadratization of polynomial and nonpolynomial systems of ordinary differential equations (ODEs) is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling, and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, and control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms, and software capabilities for quadratization of nonautonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semidiscretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both nonautonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.

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非线性有限维非自治动力系统的精确和最优四分化

SIAM 应用动力系统期刊》，第 23 卷第 1 期，第 982-1016 页，2024 年 3 月。摘要多项式和非多项式常微分方程（ODE）系统的四元化在系统理论、流体力学、化学反应建模和数学分析等多个学科中都很有优势。四元数化揭示了模型的新变量和新结构，可能更易于分析、模拟和控制，并为学习提供了方便的参数化。本文介绍了非自治 ODE 四元化的新理论、算法和软件功能。根据输入函数的正则性，我们提供了通过四元化获得二次线性系统的存在性结果。我们进一步开发了存在性结果和一种算法，将四分法过程推广到具有任意维度的系统，当维度增加时，该系统仍保留非线性结构。对于此类系统，我们提供了与维度无关的四分法。半离散 PDEs 就是一个例子，当离散尺寸增大时，其非线性项在符号上保持一致。作为本研究实用化的一个重要方面，我们将 QBee 软件的功能扩展到了非自治的 ODEs 系统和任意维度的 ODEs 系统。我们举了几个以前在文献中报道过的 ODEs 例子，在这些例子中，我们的新算法找到了比以前报道的提升变换维度更低的四元化 ODE 系统。我们进一步强调了四元化的一个重要领域：降阶模型学习。这一领域可以从最优提升变量中大大受益，二次模型提供了模型的直接参数化，同时也避免了非线性项的额外超还原。一个太阳风的例子凸显了这些优势。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.