[Paper Review] Polynomial Time Nondimensionalisation of Ordinary Differential Equations via their Lie Point Symmetries

Évelyne Hubert, Alexandre Sedoglavic|arXiv (Cornell University)|Apr 13, 2006

Nonlinear Waves and Solitons11 references5 citations

TL;DR

This paper presents a polynomial-time algorithm for nondimensionalising ordinary differential equations (ODEs) using Lie point symmetries, specifically scaling and translation symmetries, to reduce the number of parameters. By computing rational invariants from these symmetries, the method rewrites ODE systems in dimensionless coordinates with fewer parameters, achieving computational complexity bounded by O(ℓ³), where ℓ is the number of parameters.

ABSTRACT

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and translations that are Lie point symmetries of considered ordinary differential system. By rewriting original problem in an invariant coordinates set for these symmetries, one can reduce the involved number of parameters. This process is classically call nondimensionalisation in dimensional analysis. We present an algorithm based on this standpoint and show that its arithmetic complexity is polynomial in input's size.

Motivation & Objective

• To develop an algorithm that systematically reduces the number of parameters in ODE systems arising from physical and biological models.
• To formalize the process of nondimensionalisation as a symmetry-based transformation using Lie point symmetries.
• To prove that the computation of rational invariants and system rewriting is polynomial-time in the input size.
• To provide a systematic, algorithmic framework for parameter reduction that avoids ad hoc manual manipulations.
• To extend dimensional analysis to a computational algebraic framework using Lie group theory and invariant theory.

Proposed method

• Identify one-parameter groups of scaling and translation symmetries that leave the ODE system invariant.
• Construct the infinitesimal generators of these symmetry groups using the determining system of PDEs.
• Compute rational invariants of the symmetry groups via linear algebra, specifically Gaussian elimination on the coefficient matrix of the symmetry action.
• Rewrite the original ODE system in terms of these rational invariants, resulting in a dimensionless system with fewer parameters.
• Use the solvability of the Lie algebra generated by symmetries to prioritize translation symmetries before scaling symmetries in the algorithm.
• Ensure the algorithm’s arithmetic complexity remains polynomial, bounded by O(ℓ³), by leveraging linear algebraic techniques on the symmetry exponents.

Experimental results

Research questions

• RQ1 Can the process of nondimensionalisation for ODE systems be formalized as a symmetry-based algorithm with guaranteed polynomial-time complexity?
• RQ2 How can rational invariants derived from Lie point symmetries be systematically computed to reduce the number of parameters in ODEs?
• RQ3 What is the computational complexity of computing these invariants and rewriting the system in invariant coordinates?
• RQ4 In what order should different types of symmetries (translation vs. scaling) be applied to maximize parameter reduction?
• RQ5 Can this method be extended to handle more general symmetry types beyond scaling and translation, such as non-linear symmetries?

Key findings

• The algorithm computes rational invariants from Lie point symmetries using Gaussian elimination, ensuring polynomial-time complexity bounded by O(ℓ³).
• The method successfully reduces the number of parameters in ODE systems by exploiting symmetry groups, as demonstrated on Verhulst’s logistic model and a prey-predator system.
• For the Verhulst model with linear predation, the system is reduced from three parameters to one via invariants (t̃ = (a−c)t, x̃ = b/(a−c) x), yielding the dimensionless form dx̃/dt̃ = x̃(1−x̃).
• In the prey-predator model, three parameters (r, k₁, k₂) are eliminated by choosing a scaling transformation, resulting in a system with only three remaining parameters: s, e, h.
• The method is robust to the choice of symmetry basis, and user control over which parameters to eliminate is preserved by proper variable ordering during Gaussian elimination.
• The framework extends to non-standard symmetries, such as the one-parameter group ∂/∂y + b∂/∂a + ∂/∂d in the FitzHugh-Nagumo model, enabling parameter reduction via invariants like ỹ = y−d and â = a+bd.