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：多项式及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与结构，从而更便于分析、仿真与控制，并为模型学习提供便捷的参数化形式。本文针对非自治常微分方程系统的二次化问题，提出了新颖的理论、算法及软件功能。我们给出了依赖于输入函数正则性的存在性结果，阐明了通过二次化可生成二次-双线性系统的条件。进一步地，我们发展了适用于任意维度系统的存在性结果与算法，此类系统在维度增长时仍保留非线性结构。针对这类系统，我们实现了与维度无关的二次化方法——例如半离散化偏微分方程，其非线性项在离散规模增加时保持符号一致性。作为推动本研究实际应用的关键环节，我们将QBee软件的功能扩展至非自治常微分方程系统及任意维度的常微分方程系统。通过多个文献中报道的经典算例，验证了新算法能够获得维度低于此前提升变换方法的二次化系统。此外，我们重点强调了二次化在降阶模型学习领域的重要应用：在最优提升变量框架下，二次模型既可直接参数化模型本身，又可避免对非线性项进行额外超约化，从而显著提升该领域的研究效能。太阳风实例充分展示了这些优势。