Faster Gr\"obner bases for Lie derivatives of ODE systems via monomial orderings

TL;DR

This paper introduces a method to accelerate Gröbner basis computations for polynomial systems derived from Lie derivatives of ODE models by employing specialized monomial orderings based on the model's structure.

Contribution

The paper presents a novel approach using structure-informed monomial orderings to improve the efficiency of Gröbner basis calculations for differential equation models.

Findings

Significant speed-up in Gröbner basis computation observed

Method improves structural identifiability analysis efficiency

Empirical results show compatibility across frameworks

Abstract

Symbolic computation for systems of differential equations is often computationally expensive. Many practical differential models have a form of polynomial or rational ODE system with specified outputs. A basic symbolic approach to analyze these models is to compute and then symbolically process the polynomial system obtained by sufficiently many Lie derivatives of the output functions with respect to the vector field given by the ODE system. In this paper, we present a method for speeding up Gr\"obner basis computation for such a class of polynomial systems by using specific monomial ordering, including weights for the variables, coming from the structure of the ODE model. We provide empirical results that show improvement across different symbolic computing frameworks and apply the method to speed up structural identifiability analysis of ODE models.