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Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms
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This work addresses the challenges in analyzing and controlling non-quadratic partial differential equations (PDEs) by introducing a rigorous definition of PDE quadratization and establishing the first theoretical framework for this transformation. Leveraging symbolic computation and discrete optimization, the authors develop an automated algorithm, QuPDE, which introduces auxiliary variables to convert one-dimensional polynomial or rational PDEs into lower-order quadratic forms. This approach overcomes the limitations of traditional methods that rely on manual construction. The method successfully achieves quadratization for 14 representative non-quadratic PDEs from fluid dynamics, space physics, chemical engineering, and biology, with some results outperforming existing techniques and several systems—previously lacking any quadratic representation—now admitting such formulations for the first time.
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📝 Abstract
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.
Problem
Research questions and friction points this paper is trying to address.
quadratization
partial differential equations
nonquadratic PDEs
autonomous PDEs
symbolic transformation
Innovation
Methods, ideas, or system contributions that make the work stand out.
quadratization
partial differential equations
symbolic computation
discrete optimization
nonlinear PDEs
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