Quadratization of polynomial and nonpolynomial systems of ordinary differential equations 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, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous 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 semi-discretized 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 non-autonomous 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.
翻译:将多项式和非多项式的常微分方程组进行二次化在各个学科中都很有优势,如系统理论、流体力学、化学反应建模和数学分析。通过二次化可以揭示出模型的新变量和结构,这些结构可能更容易分析、模拟、控制,并可提供方便的学习参数化。本文提出了针对非自治常微分方程的二次化的新理论、算法和软件功能。我们提供存在结果,该结果取决于输入函数的正则性,当通过二次化得到二次二元系统时,我们可以存在。当维度增加时仍保留非线性结构时,我们进一步开发了存在结果和算法来推广二次化过程。对于这种系统,我们提供了不依赖于维度的二次化过程。一个例子是半离散的偏微分方程,其中当离散化大小增加时,非线性项仍保持符号相同。作为该研究实用采用的一个重要方面,我们将QBee软件的能力扩展到包括非自治的ODE系统和任意维度的ODE。我们提供了几个ODE的示例,这些示例先前在文献中报道,我们的新算法发现了比先前报告的提升变换低维度的二次化ODE系统。我们还强调了二次化的一个重要领域:降阶模型学习。这个领域可以从在最优提升变量中工作中受益,其中二次模型提供了直接的模型参数化,也避免了非线性项的额外超降。一个太阳风的例子突出了这些优点。