Quadratization of Autonomous Partial Differential Equations: Theory and Algorithms

TL;DR

This paper introduces QuPDE, a novel algorithm that symbolically transforms one-dimensional nonlinear PDEs into quadratic form, simplifying analysis and control, and demonstrates its effectiveness across various scientific fields.

Contribution

The paper provides the first computational tool for PDE quadratization, offering a rigorous definition, theoretical insights, and an algorithm that finds low-order quadratic transformations.

Findings

QuPDE successfully quadratizes 14 diverse PDEs.

It produces simpler, low-order quadratic forms with fewer auxiliary variables.

Some PDEs were quadratized for the first time using this method.

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.