Dimensionally Consistent Change of Variables

• Dimensionally consistent change of variables is a transformation that preserves the intrinsic unit structure by ensuring derived variables are dimensionless through a nullspace condition.
• Buckingham’s Pi theorem and data–driven methods, such as constrained optimization and neural network architectures, facilitate the creation of reduced, invariant models.
• In operator theory and probabilistic modeling, these transformations maintain scaling properties and ensure that physical laws remain invariant during changes of variables.

A dimensionally consistent change of variables is a transformation within a mathematical or physical formulation that preserves the dimensional structure intrinsic to the problem—ensuring that all resulting expressions, models, or algorithms remain invariant under the change of physical units. Across applied mathematics, physics, statistical modeling, and operator theory, the principle of dimensional consistency constrains and guides the construction of reduced models, the manipulation of operators, and the design of learning algorithms, with rigorous methodologies grounded in symmetry, scaling, and invariance.

1. Mathematical Foundations of Dimensional Consistency

Let a set of $d = n + k$ measured quantities $\{x_j\}$ be associated with $p$ fundamental physical dimensions (such as $[\mathrm{M}]$, $[\mathrm{L}]$, $[\mathrm{T}]$). Each variable is assigned a $p \times 1$ integer “dimension vector” $\Omega(x_j) = [\alpha_1, \ldots, \alpha_p]^T$, representing its exponent in the fundamental units. The domains of these vectors are aggregated in the $p \times d$ “dimensional matrix” $$D = [\Omega(p_1)~\cdots~\Omega(p_n)~|~\Omega(q_1)~\cdots~\Omega(q_k)]$$.

A function or transformation is dimensionally consistent if the transformed variables and all functional relationships remain invariant under rescaling of the base units. In practice, this corresponds to seeking transformations such that the exponents $w \in \mathbb{R}^d$ of monomials $\pi = \prod_{j=1}^d x_j^{w_j}$ satisfy $D w = 0$. This nullspace criterion ensures the constructed variables (“$\Pi$-groups” in the terminology of Buckingham’s theorem) are dimensionless, as all derived exponents exactly cancel the units of the original variables (Bakarji et al., 2022).

In operator theory, dimensional consistency refers to transformations and pull-backs that preserve the “natural scaling” of functional spaces—e.g., Sobolev or Zygmund spaces—after a change of variables, paralleling the unit invariance required in classical applications (Said, 2020).

2. Buckingham $\Pi$ Theorem and Data-Driven Algorithms

The Buckingham $\Pi$ theorem provides a constructive procedure for generating all functionally independent dimensionless groups from a problem’s dimensional matrix $D$. Its role is central in formulating dimensionally consistent changes of variables, particularly when explicit governing equations are unavailable (Bakarji et al., 2022).

Advanced data-driven methodologies operationalize this principle in the following ways:

1. Constrained Optimization: Given data matrices of observed samples in physical space, one minimizes a loss function for a predictor $\psi$ in the space of dimensionless $\Pi$-groups over choices of exponents $\Phi_p$ (columns in null$(D_p)$), enforcing $D_p \Phi_p = 0$ as a hard constraint. Regularization terms ($\ell_1, \ell_2$) promote sparsity in the discovered exponents for interpretability and robustness.
2. BuckiNet Architecture: A feed-forward neural network is structured so that its first layer performs a linear transformation on the logarithm of physical variables, exponentiates, and thus outputs learnable $\Pi$-groups. Dimensional consistency is enforced either by freezing weights within null$(D_p)$ (hard constraint) or by penalizing deviation from nullity via a “null-space loss” in the training objective.
3. Dimensionless SINDy: Once $\Pi$-groups are computed, dynamics in these reduced variables are sought via sparse symbolic regression (e.g., SINDy). The dimensionless ODEs identified are parameterized by sparse coefficients, and time rescalings are likewise determined in a dimensionally consistent fashion.

These procedures enable automated discovery of invariant representations and reduced-order models that reflect the intrinsic symmetries and scales of the underlying physical system (Bakarji et al., 2022).

3. Dimensionally Consistent Change of Variables in Paradifferential Calculus

In the context of analysis, particularly for paradifferential and pseudodifferential operators, dimensionally consistent changes of variables are formalized as paracomposition or pull-back operations. The rigorous construction, as established by Alinhac and extended in (Said, 2020), ensures that the mapping $x^*: u \mapsto u \circ x$ (and its generalization for rough $x$) acts boundedly on $H^s$ and $C^s_*$ spaces.

The transformation of operators under pull-back is controlled so that the regularity and scaling (in terms of Sobolev or Zygmund indices) are preserved: for any $p > 0$, $x \in C^{1+p}$, and $u \in H^s$, one obtains

$$\|x^*u\|_{H^s} \leq C(\|\nabla x\|_{L^\infty}) \|u\|_{H^s},$$

and for invertible $x$, $x^*u - u \circ x \in H^{s+p}$. These sharp a priori bounds guarantee the absence of spurious losses or artificial gains in regularity under the change of variables. The calculus is thus dimensionally consistent in the analytic sense that all transformations respect the natural scaling structure of the theory.

Moreover, under such transformations, the operators themselves are conjugated in a dimensionally consistent way, with their symbol classes and expansion properties preserved, and with full control over remainder terms (Said, 2020).

4. Score Change of Variables and Probabilistic Modeling

For mappings between probability spaces, dimensional consistency is manifested in the transformation rules for differential quantities such as score functions. Given a smooth, invertible map $y = \phi(x)$ between random vectors, the score (gradient of log-density) transforms as

$$\nabla_y \log q(y) = J_{\phi^{-1}(y)}^\top \nabla_x \log p(x)|_{x=\phi^{-1}(y)} + \nabla_{x\cdot}(J_{\phi^{-1}}^\top)|_{x=\phi^{-1}(y)},$$

where $J_{\phi^{-1}}$ is the Jacobian of $\phi^{-1}$ (Robbins, 10 Dec 2024).

Both terms are vector-valued in the $y$-coordinate system, and the formula is manifestly dimensionally consistent—mapping covectors and correcting for local volume distortion. This exact invariance is critical in machine learning applications, such as diffusion modeling and generalized score matching, where model performance and interpretability depend on correct treatment of variable transformations. The methodology enables, for example, model training in one space and exact sampling in another.

When specialized to the case of smooth projections (e.g., as in generalized sliced score matching), the change-of-variables retains dimensional consistency by ensuring every projected estimate remains compatible with the original high-dimensional geometry (Robbins, 10 Dec 2024).

5. Field Theoretic and Geometric Perspectives

In field theory, particularly in the context of general relativity, dimensional consistency under arbitrary changes of variables (unit transformations) requires invariance not just under coordinates, but under pointwise rescalings of the metric (Weyl or conformal transformations). Conventional unit systems do not guarantee invariance of the dimensionless action under such local rescalings, exposing ambiguities in the physical interpretation of dimensional observables (Shimon, 2013).

Shimon’s formalism (Shimon, 2013) achieves invariance by promoting all “constants” ($c$, $G$, $\hbar$, $m$, $e$, …) to spacetime-dependent fields with assigned conformal weights, and by augmenting the Lagrangian with compensating geometric scalar terms constructed to precisely cancel the non-invariant contributions. The resulting dimensionless action,

$$\phi = \frac{1}{\hbar\,c}\,\frac{1}{2\,\ell_P^2} \int d^4x\,\sqrt{-g}\,\Big[R-2\Lambda +6\,\frac{\Box f}{f} - 12\,g^{\mu\nu}\partial_\mu \ln f\,\partial_\nu \ln f\Big],$$

is manifestly scale-invariant and physically unambiguous. This paradigm ensures all measurable predictions are entirely basis-independent and shifts the conceptual locus of “physicality” from unitful constructs to dimensionless observables.

6. Computational Complexity and Robustness Considerations

The computational complexity of dimensionally consistent change-of-variable frameworks is determined primarily by operations such as nullspace search and operator evaluation. For the Buckingham $\Pi$ approach, the construction and nullspace computation for the dimensional matrix $D$ scales as $\mathcal{O}(d^3)$, but physical problems rarely entail large $d$. The constrained optimization formulations and BuckiNet architectures benefit from reduced parameter count and robust regularization, which mitigate overfitting even with moderate sample sizes ($m\sim 100\text{--}10^3$) (Bakarji et al., 2022). The SINDy regression further exploits reduced dimensionality for efficient sparse regression.

In operator-theoretic applications, refined control on spectral cut-offs, as developed in paradifferential calculus, ensures that repeated compositions of transformations remain well-posed and do not introduce runaway growth in computational overhead or loss of analytic precision (Said, 2020).

7. Illustrative Examples and Significance

The utility of dimensionally consistent change of variables is demonstrated in several canonical physical and computational settings:

• Bead on a Rotating Hoop: Recovery of classical dimensionless groups and derivation of reduced, non-dimensional ODEs for dynamical modeling via BuckiNet and SINDy (Bakarji et al., 2022).
• Laminar Boundary Layer (Blasius Profile): Automated discovery of similarity variables matching traditional analysis through constrained optimization in $\Pi$-space (Bakarji et al., 2022).
• Probability Simplex and Diffusion Models: Precise transformation of stochastic differential equations and score estimators under nonlinear mappings, enabling reliable modeling of diffusion on constrained domains such as simplices (Robbins, 10 Dec 2024).
• Field-Theoretic Invariance: Theoretical establishment of a fully scale-invariant gravitational action, exemplifying the importance of such changes in quantum gravity and cosmology (Shimon, 2013).

These examples underscore the centrality of dimensional consistency as an organizing structural principle across mathematical, computational, and physical sciences.