Quadratization for partial differential equations (PDEs) is a process that transforms a nonquadratic PDE into a quadratic form by introducing auxiliary variables. This symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of nonlinear and nonquadratic PDE models. This paper presents a rigorous definition of PDE quadratization, theoretical results for the PDE quadratization problem of spatially one-dimensional PDEs-including results on existence and complexity-and introduces QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.

翻译：偏微分方程（PDE）的二次化是一种通过引入辅助变量将非二次PDE转化为二次形式的符号变换过程。这一符号变换已被广泛应用于简化非线性及非二次PDE模型的分析、模拟与控制。本文提出了PDE二次化的严格定义，给出了空间一维PDE二次化问题的理论结果——包括存在性与复杂性结论，并介绍了QuPDE算法。该算法基于符号计算与离散优化，可为任意空间一维多项式或有理PDE输出一个二次化形式，是迄今为止首个能够为PDE寻找二次化的计算工具。我们通过将QuPDE应用于流体力学、空间物理、化学工程及生物过程等领域的十四个非二次PDE来展示其性能。QuPDE在每种情况下均能给出低阶二次化：对于部分算例，所揭示的二次变换比文献中先前发现的辅助变量更少；对于某些此前从未被转化为二次形式的系统，该算法也成功找到了其二次化形式。