Faster Gröbner bases for Lie derivatives of ODE systems via monomial orderings
Mariya Bessonov, Ilia Ilmer, Tatiana Konstantinova, Alexey Ovchinnikov, Gleb Pogudin, Pedro Soto
TL;DR
The paper addresses the computational bottleneck of deriving Gröbner bases for polynomial systems formed by Lie derivatives of ODE outputs with parameters. It introduces a weighted monomial ordering, computed from the ODE structure via a Level function and corresponding Weight assignments, to accelerate Gröbner-basis computations; the ordering compares by weights and breaks ties with a derivative-aware reverse-lex rule, implemented within the F4 framework. The authors provide a detailed weight-generation algorithm, analyze the non-homogeneous nature of these systems, and present extensive experimental results across Maple and Magma showing significant runtime and memory improvements for many models, thereby enabling faster global identifiability analyses via SIAN. They also explore inverted weight schemes and report occasional slowdowns, underscoring the method's empirical character and the need for model-specific tuning and further theoretical grounding.
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 Gröbner 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.