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：多项式与非多项式常微分方程系统的二次化在系统理论、流体力学、化学反应建模和数学分析等多个学科中具有优势。二次化能够揭示模型的新变量和结构，使其更易于分析、仿真和控制，并为学习过程提供便捷的参数化方法。本文提出了关于非自治常微分方程二次化的新理论、算法及软件功能。我们根据输入函数的正则性，给出了可通过二次化得到二次-双线性系统的存在性结果。进一步地，我们发展了存在性结果和一种算法，该算法将二次化过程推广到具有任意维度且维度增长时保持非线性结构的系统。对于此类系统，我们提供了与维度无关的二次化方法。例如在半离散化偏微分方程中，当离散化尺寸增大时，非线性项在符号上保持不变。作为本研究实践应用的重要方面，我们将QBee软件的功能扩展至非自治常微分方程组以及具有任意维度的常微分方程。我们展示了文献中先前报道的多个常微分方程实例，通过这些实例，我们的新算法找到了维度低于先前提升变换方法的二次化常微分方程组。此外，我们重点阐述了一个重要的二次化应用领域：降阶模型学习。该领域可显著受益于最优提升变量的应用，其中二次模型不仅为模型提供直接参数化，还避免了非线性项额外的超约化处理。一个太阳风实例充分体现了这些优势。