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：多项式和非多项式常微分方程系统的二次化在系统理论、流体力学、化学反应建模及数学分析等多个学科领域具有显著优势。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。本文提出了用于非自治常微分方程二次化的新理论、算法及软件能力。我们根据输入函数的正则性，给出了可通过二次化获得二次-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及一种算法，该算法将二次化过程推广至任意维度且随维度增长保持非线性结构的系统。针对此类系统，我们提供了维度无关的二次化方法，例如半离散化偏微分方程——其非线性项在离散化尺度增加时仍保持符号一致性。作为本研究实际应用的重要方面，我们将QBee软件的功能扩展至非自治常微分方程组及任意维度的常微分方程。通过文献中多组示例验证，我们的新算法能够找到比先前提升变换方法维度更低的二次化常微分方程组。此外，我们重点阐述了二次化的一个重要应用领域：降阶模型学习。该领域可通过使用最优提升变量显著受益，其中二次模型不仅提供模型的直接参数化，还避免了非线性项的超降阶处理。太阳风案例充分展示了这些优势。