Nondimensionalization is more science than art

Abstract: When faced with a mathematical model, often the first step is to reduce the complexity of the model by turning variables and parameters into dimensionless quantities. This process is often performed by hand, relying on a skill practiced over many years, and attempted for small models. Nondimensionalization is often considered an art, as there is no formal method accessible to applied scientists. Here we show how to systematically perform nondimensionalization for arbitrarily sized models described by rational first order ordinary differential equations. We translate and extend an existing approach for computing rational invariants of the maximal scaling symmetry, which combines ideas from differential algebra, invariant theory and linear algebra, to the setting arising in biological models. The modeler inputs the system of equations and our implemented algorithm outputs the nondimensional quantities for the corresponding nondimensionalized model. We extend the algorithm to include initial conditions, and the modeler's choice of invariants, thereby including a larger class of nondimensionalizations. We further prove that any dimensionally consistent change of variables preserves the dimension of the maximal scaling symmetry. We showcase the framework on various models, including the classical Michaelis-Menten equations, which serves as a benchmark for asking and answering specific modeling questions.

• The paper establishes an algorithmic approach that transforms nondimensionalization from an art into a robust scientific process by computing maximal scaling symmetries.
• It employs integer linear algebra and invariant theory, using Hermite and Smith normal forms to systematically extract rational invariants from large-scale ODE models.
• The method is validated through diverse case studies, including Michaelis-Menten kinetics and cell cycle control, demonstrating efficient, automated model reduction.

Systematic Nondimensionalization via Scaling Symmetries: An Algorithmic Approach

Introduction

The paper "Nondimensionalization is more science than art" (2512.13455) establishes a formal, algorithmic framework for nondimensionalization of mathematical models defined by rational first-order ordinary differential equations (ODEs). This is positioned against the longstanding view that effective nondimensionalization is an artisanal skill, rooted in hand-crafted selection and experience, and traditionally restricted to models small enough to allow manual analysis. The work demonstrates that leveraging concepts from differential algebra, linear algebra (particularly computation of Hermite normal forms), and invariant theory, it is possible to exhaustively and efficiently compute scaling symmetries and associated rational invariants for large-scale dynamical models. The approach generalizes and extends the computational symmetry-reduction algorithm of Hubert and Labahn [Hubert2013c], contextualizing the framework for biological and physical systems, and incorporates auxiliary model constraints, initial conditions, and user-selected invariants.

Framework and Mathematical Foundations

Scaling Symmetries and Dimensional Analysis

The formalism constructs invariance under independent scaling of each fundamental unit (e.g., time, concentration). For $n$ model variables/parameters and $m$ fundamental units, the classical Buckingham-$\pi$ theorem [buckingham1914physically] yields $n-m$ nondimensional quantities. The authors generalize this by seeking the maximal set of algebraic scaling symmetries, not limited to physical units, using so-called structural fundamental units, following the observation of Meinsma [meinsma2019dimensional].

Integer Linear Algebraic Algorithms

The core reduction algorithm is constructed over the exponent matrices of the model’s rational terms, representing scaling exponents of variables and parameters. Hermite normal forms (HNF) are computed to identify the kernel of the scaling symmetry action. The bottom $r$ rows of the unimodular transformation matrix (the Hermite multiplier) yield the maximal scaling action, and the associated rational invariants (maximal set of nondimensionalizations) correspond to generators of the kernel of the exponent matrix. This process is robust to variable order and monomial subtraction, and is provably complete (Proposition 5.1 from [Hubert2013c]).

Incorporation of Initial Conditions and Prescribed Invariants

The reduction scheme is systematically extended to represent initial conditions, conserved quantities, and arbitrary prescribed parameter combinations. This is achieved by supplementing the exponent matrix with additional columns encoding desired invariants, followed by checking compatibility (via kernel membership and Smith normal form). The authors prove, nontrivially, that any dimensionally consistent change of variables preserves the dimension of the maximal scaling symmetry (Theorem:ChangeOfVariables).

Algorithmic Nondimensionalization Pipeline

The framework proceeds with:

• Construction of the exponent matrix for all monomials appearing in model ODEs (including auxiliary constraints and invariants).
• Computation of the maximal scaling symmetry via Hermite normal form and extraction of the scaling matrix.
• Canonical selection of invariant generators via Hermite multiplier normalization.
• Explicit substitution to obtain reduced (nondimensionalized) ODEs, leveraging unimodular inverse matrices for variable transformation.
• Incorporation of prescribed invariants and completion via Smith normal form extensions.
• Automation: The approach is implemented in the Python package desr [desr], supporting applied mathematicians and engineers.

Theoretical Results

The paper demonstrates a number of mathematically rigorous, nontrivial results:

• Maximality of Scaling Symmetry Reduction: Any scaling symmetry acting compatibly with the model ODEs must be a right factor of the maximal scaling matrix. Thus, all possible nondimensionalizations are classified up to unimodular equivalence.
• Preservation of Symmetry under Sensible Change of Variables: Any invertible change of variables that respects dimensional consistency does not increase the rank of the scaling symmetry group, refuting naive beliefs.
• Flexibility in Choice of Invariants: Modelers may select any subset of compatible invariants and algorithmically extend this to a unimodular generating set for full reduction, facilitating integration of physically or biologically meaningful nondimensional parameterizations.

Empirical and Benchmark Applications

The paper provides detailed case studies across canonical biochemical kinetics and systems biology:

• Michaelis-Menten Kinetics: Recovery of classical nondimensionalization, including initial conditions and widely used constants such as the Michaelis constant $K_m$; alternative dimensional reductions for multiple timescale analysis (Segel-Slemrod).
• New Nondimensionalizations: Discovery of nonclassical nondimensional forms for extended reaction networks, finding reductions beyond the standard physical unit analysis.
• Cell Cycle Control Network: Full reduction of a multi-variable, multi-parameter ODE model to a minimized set of invariants, demonstrating scalability and automation potential.
• Compartment Models in Pharmacokinetics: Treatment of compartmental vaccine models, showing that maximal reduction and physical consistency are only achieved if invariance conditions on new parameters introduced by variable changes are explicitly enforced.

Strong numerical reductions are showcased, with dimension reduction matching classical results where appropriate, and exceeding it when additional (structural) symmetries are present.

Implications and Future Directions

This work fundamentally challenges the narrative that nondimensionalization is an art and establishes it as a principled science. Practical implications include:

• Automated Model Reduction: Symbolic preprocessing of ODE systems to canonical, minimal parameter sets prior to numerical simulation or inference.
• Integration in System Identification: Preprocessing for identifiability analysis, optimal experimental design, and statistical inference in systems biology and engineering [meshkat2025structural, feliu2022quasi].
• Broader Applications: Potential for extension to PDE models, spatial networks, and effective connection to machine learning techniques for structure discovery [desai2022symmetry, liu2022hierarchical].
• Algorithmic Discovery: The implemented algorithms systematically uncover all possible nondimensionalizations, providing transparency and interpretability lacking in hand-derived approaches.

Open questions include extension to Lie symmetries beyond scaling, automated detection of slow/fast time scales, and further integration in discovery pipelines for AI-driven dynamical system modeling.

Conclusion

The paper demonstrates that nondimensionalization, when formalized via scaling symmetry computation grounded in integer linear algebra and invariant theory, is a reproducible and automatable procedure. The provided algorithms and software can handle arbitrarily large, rational-ODE models, enforcing dimensional consistency and allowing flexible incorporation of model constraints and invariants. This advances both theoretical understanding and practical workflows in model reduction for mathematical biology, engineering, and applied sciences, with direct implications for symbolic and AI-based modeling systems.