非线性有限维非自治动力系统的精确且最优二次化
Exact and optimal quadratization of nonlinear finite-dimensional non-autonomous dynamical systems
AI总结：
本文研究非自治、任意维非线性 ODE 的精确最优二次化，提出存在性理论与维度无关算法，并扩展 QBee，以更低维提升和降阶模型学习应用验证优势。
摘要：
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都具有优势。二次化能够揭示模型的新变量和新结构，使其更易于分析、仿真和控制，并为学习提供便利的参数化。本文提出了用于非自治 ODE 二次化的新颖理论、算法和软件能力。针对可通过二次化获得双线性二次系统的情形，我们给出了取决于输入函数正则性的存在性结果。我们进一步发展了存在性结果和一种算法，将二次化过程推广到任意维度、且在维度增长时保留非线性结构的系统。对于此类系统，我们提供与维度无关的二次化。一个例子是半离散化 PDE：当离散化规模增大时，其非线性项在符号上保持相同。作为推动本研究实际应用的重要方面，我们将 QBee 软件的能力扩展到非自治 ODE 系统和任意维度 ODE。本文给出了若干先前文献中报道过的 ODE 示例，在这些示例中，我们的新算法找到了维度低于此前所报道提升变换的二次化 ODE 系统。我们还强调了二次化的一个重要应用领域：降阶模型学习。该领域可显著受益于在最优提升变量中工作，因为二次模型提供了模型的直接参数化，同时也避免了对非线性项进行额外的超缩减。一个太阳风示例展示了这些优势。
英文摘要：
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.
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