Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms

2026/02/25 by Albani Olivieri, Gleb Pogudin, Boris Kramer · 1 voice
Computer Science · Mathematics · Physics and Astronomy · #Computation #Model Reduction and Neural Networks #Nonlinear system #Numerical methods for differential equations #Partial differential equation #Polynomial #Polynomial and algebraic computation #Process (computing) #Quadratic equation #Symbolic computation #Transformation (genetics) #cs.MS #cs.SC #math.DS #math.NA

paper · pdf · doi:10.48550/arxiv.2602.22371

openalex publication_date 2026/02/25 · arxiv published 2026/02/25 · arxiv updated 2026/02/25 · openalex created_date 2026/02/28 · openalex updated_date 2026/08/06

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.

Citations

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Discussions

• Albani Olivieri, Gleb Pogudin, Boris Kramer: Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms https://arxiv.org/abs/2602.22371 https://arxiv.org/pdf/2602.22371 https: [bsky, 0 points, 0 comments]

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