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，并为模型学习提供便捷的参数化方法。本文针对非自治常微分方程（ODEs）的二次化问题，提出了新颖的理论、算法和软件功能。我们根据输入函数的正则性条件，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，我们发展了适用于任意维数系统的存在性结果和算法，这些系统在维数增长时仍保持非线性结构。针对此类系统，我们提出了维度无关的二次化方法——以半离散化偏微分方程为例，当离散化尺度增加时，其非线性项在符号形式上保持不变。作为该研究实用化的重要方面，我们扩展了QBee软件的功能，使其能够处理非自治ODEs系统以及任意维数的ODEs。通过多个文献报道的ODEs示例，我们的新算法能获得比现有提升变换方法维度更低的二次化ODE系统。我们进一步强调了二次化的重要应用领域：降阶模型学习。该领域可显著受益于最优提升变量的选择，其中二次模型不仅直接参数化模型，还避免了非线性项的额外超约化处理。太阳风实例充分展现了这些优势。