Algorithmic Nondimensionalization Pipeline

• Algorithmic nondimensionalization is a computational method that automates the conversion of ODE systems into dimensionless form by exploiting maximal scaling symmetries.
• It employs integer linear algebra techniques such as Hermite and Smith normal forms to derive invariant quantities in a reproducible and efficient manner.
• The pipeline guarantees mathematical consistency by avoiding spurious symmetries and effectively handles complex, high-dimensional models like enzyme kinetics.

Algorithmic nondimensionalization, as formalized in Tanburn et al., constitutes a systematic, mathematically rigorous process to transform systems of rational first-order ordinary differential equations (ODEs) into dimensionless form by exploiting their maximal scaling symmetries using integer linear algebra. The pipeline replaces centuries of heuristic, manual nondimensionalization with a reproducible computational framework. This approach applies to high-dimensional models involving arbitrary combinations of parameters and variables, is grounded in the theory of scaling invariants, and extends to invariants chosen by the user—including initial data—while guaranteeing the absence of spurious symmetries under dimensionally consistent transformations (Tanburn et al., 15 Dec 2025).

1. Mathematical Foundation of Scaling Symmetries

A dynamical system with variables $z = (z_1,\ldots,z_n)$ and constant parameters $c = (c_1,\ldots,c_p)$ described by rational ODEs is canonically formulated in a "homogeneous" form:

$\frac{dz_i}{dt} = \frac{z_i}{t} F_i(t, z, c)$

for each state variable $z_i$, with $F_i$ a rational function. Introducing the augmented variable $z_0 = t$ and collecting all dynamical quantities in $\bar{z} = (z_0, z_1, ..., z_n)$, yields

$$\frac{d\bar{z}}{dt} = \frac{\bar{z}}{t} \bar{F}(\bar{z})$$

with $\bar{F} = (1, F_1, ..., F_n)$. The maximal scaling symmetry is characterized by a one-parameter scaling ansatz

$$\bar{z}_j \mapsto \lambda^{a_j} \bar{z}_j, \quad j=0,\ldots,n$$

with exponent vector $a = (a_0, a_1, ..., a_n) \in \mathbb{Z}^{n+1}$. The infinitesimal generator, $$V = \sum_{j=0}^n a_j \bar{z}_j \frac{\partial}{\partial \bar{z}_j}$$, imposes the invariance condition $V[F_i] = 0$ for all $i$, which can be recast as a first-order linear PDE satisfied by invariants $$I(x): \sum_j a_j x_j \frac{\partial I}{\partial x_j} = 0$$.

2. Algorithmic Computation of Scaling Lattice

Each rational $F_i$ is decomposed into monomials; for each, an exponent vector $m = (m_0,\ldots,m_{n+p})$ records the power of each variable and parameter. To encode scaling invariance, one selects a reference monomial, subtracts its exponents from those of all monomials, and assembles the resulting difference vectors as columns in the integer exponent matrix $K_{F_i}$. The full matrix $K = [K_{F_1} | K_{F_2} | ... | K_{F_n}]$ collects these across ODE components.

The scaling lattice corresponds to all integer solutions $a \in \mathbb{Z}^{n+p+1}$ with $a K = 0$. The set of independent scaling exponents is obtained by computing the row Hermite normal form $U K = H$, with rank $r$ determined by the number of nonzero rows. The bottom $r$ rows of unimodular $U$ yield the scaling action matrix $A \in \mathbb{Z}^{r \times (n+p+1)}$:

Step	Object	Description
Exponent Extraction	$m$-vectors	For each monomial in $F_i$
Exponent Matrices	$K_{F_i}$, $K$	Difference matrices, full concatenated matrix
Symmetry Action	$A$	Basis for maximal scaling action (bottom $r$ rows of $U$)

3. Algorithmic Construction of Dimensionless Invariants

Given $r$ scaling generators, the dimension of the invariant space is $(n+p+1) - r$. To explicitly obtain algebraically independent invariants, one computes a column Hermite multiplier $V$ such that $A V = [I_r | 0]$. Partitioning $V = [V_a | V_b]$ yields $V_b$ of size $(n+p+1) \times ((n+p+1)-r)$. The dimensionless variables are:

$y_j = \prod_{i=0}^{n+p} z_i^{(V_b)_{i,j}}$

Naming the resulting invariants as $$y_0 = \tau, y_1 = I_1, \ldots, y_{n+p-r} = I_{n+p-r}$$, the system is thus reduced to algebraically independent, dimensionless quantities.

4. Incorporating Initial Conditions and User-Selected Invariants

To enforce invariance of specific ratios—e.g., $z_i/z_{i0}$ or parameter functions—one appends the corresponding exponent vectors to $K$ and repeats the calculation. For a chosen set of $s$ invariants encoded in $P \in \mathbb{Z}^{(n+p+1)\times s}$, compatibility (i.e., $A P = 0$) is required. A Smith normal form-based extension algorithm extends $P$ to a full basis, using the Smith normal form of $W_b P$ (where $W = V^{-1}$), verifying a full-rank diagonal, and extending to a unimodular matrix $C$ such that $V_b \leftarrow V_b C$. The first $s$ columns of the resulting $V_b$ coincide with $P$.

5. Algorithmic Pipeline and Complexity

The algorithm admits a stepwise, tractable implementation. Letting $M$ denote the total number of monomials, construction of $K$ requires $O(M(n+p+1))$ time. Hermite and Smith normal forms dominate computational cost, typically requiring polynomial time in the size and bit lengths of matrix entries. The pipeline is summarized as follows:

1. Rewrite $dz_i/dt = (z_i/t) F_i(t, z, c)$.
2. Build exponent matrices $K_{F_i}$ for all $F_i$.
3. Concatenate $K = [K_{F_1}|\ldots|K_{F_n}]$ and append additional columns for any prescribed invariants.
4. Compute row Hermite normal form $U K = H$, determine rank $r$.
5. Extract scaling action $A$ (bottom $r$ rows of $U$).
6. Calculate Hermite multiplier $V$ of $A$ and identify $V_b$.
7. If invariants are user-supplied, perform Smith normal form extension as described.
8. Define invariants $y = z^{V_b}$ and substitute into the ODE system.
9. Integrate auxiliary variables as needed.

The computational complexity is dominated by integer linear algebra operations, scaling polynomially in the number of variables, parameters, and bit-lengths of integer entries.

6. Illustrative Example: Michaelis-Menten System

Consider the classical Michaelis–Menten enzyme kinetics with post-conservation variables $(t, s, c, k_{-1}, k_2, k_1, e_0)$:

$$\begin{align*} \frac{ds}{dt} &= -k_1 (e_0 - c) s + k_{-1} c \ \frac{dc}{dt} &= k_1 (e_0 - c) s - (k_{-1}+k_2) c \end{align*}$$

Rewriting each equation in homogeneous form yields the rational representation required for matrix construction. The scaling action matrix $A$ obtained through the Hermite decomposition corresponds to two symmetries:

• $\lambda:$ dilates $$(t, s, c, k_{-1}, k_2, k_1, e_0) \to (\lambda t, s, c, \lambda^{-1}k_{-1}, \lambda^{-1}k_2, \lambda^{-1}k_1, e_0)$$
• $\mu:$ dilates $$(t, s, c, k_{-1}, k_2, k_1, e_0) \to (t, \mu s, \mu c, ..., \mu^{-1}k_1, \mu e_0)$$

The invariants computed via $V_b$ are:

• $\tau = k_1 e_0 t$
• $u = s/e_0$
• $v = c/e_0$
• $c_0 = k_{-1}/(k_1 e_0)$
• $c_1 = k_2/(k_1 e_0)$

The system, upon substitution, reduces to:

$$\begin{align*} \frac{du}{d\tau} &= c_0 v + uv - u \ \frac{dv}{d\tau} &= -(c_0 + c_1)v - uv + u \end{align*}$$

Inclusion of constraints representing the initial condition and alternative parameter groupings, as in the Murray nondimensionalization form, can be accommodated by appending invariants and recomputing the pipeline, yielding standard nondimensional parameters:

• $\tau = e_0 k_1 t$
• $u = s/s_0$
• $v = c/e_0$
• $\epsilon = e_0/s_0$
• $K_m/s_0$
• $\lambda = k_2/(k_1 s_0)$

7. Theoretical Guarantees and Consistency

The pipeline is underpinned by a structural theorem (Theorem 4.10 of Tanburn et al.) asserting that any invertible, dimensionally consistent change of variables (i.e., one preserving scaling exponents) cannot increase the dimension of the scaling symmetry. Thus, the maximal scaling action is intrinsic to the underlying ODE, independent of the model's specific realization, and the resulting invariants are canonical. No extraneous symmetries can be created through well-posed reparametrizations, guaranteeing the optimality and consistency of computed nondimensionalizations (Tanburn et al., 15 Dec 2025).