Dimensionless Learning Framework

• Dimensionless learning is a data-driven framework that discovers key scaling laws and dimensionless groups while ensuring unit consistency and transferable models.
• It reformulates physical problems in a unit-independent space using a two-level machine learning scheme to optimize predictive power and enforce dimensional invariance.
• It leverages techniques from information theory, latent-variable analysis, and symbolic regression to extract parsimonious, physically meaningful models from noisy experimental data.

Dimensionless learning is a data‑driven framework for discovering dimensionless groups, scaling laws, and related low‑dimensional representations directly from data while enforcing dimensional consistency. In one core formulation, it “automatically discover[s] dominant and unique dimensionless numbers and scaling laws from data” by embedding “the principle of dimensional invariance into a two‑level machine learning scheme” (Xie et al., 2021). Across the broader literature, the term also denotes learning in a dimensionless representation so that “the learned knowledge applies unchanged to any physically similar system,” exact “units equivariant machine learning” performed in a dimensionless space, and information‑theoretic or symbolic methods for selecting the dimensionless variables with the highest predictive power (Hromatko et al., 9 Dec 2025, Villar et al., 2022, Yuan et al., 4 Apr 2025). This suggests that dimensionless learning is not a single algorithm but a family of methods organized around the same principle: rewrite physical prediction, control, and inference problems in terms of dimensionless quantities, then use data to identify the combinations that are most informative, most parsimonious, or most transferable.

1. Conceptual basis and relation to classical dimensional analysis

Classical dimensional analysis guarantees that if variables are built from a set of independent base dimensions, physically meaningful relations can be written in terms of dimensionless groups. It also has a central limitation: “the Buckingham–$\pi$ theorem tells us the dimension of the space of possible dimensionless groups,” but “there are infinitely many valid choices of $\pi$ groups,” it “provides no guidance on which groups are most relevant to an actual dataset,” and “it does not supply the functional relationship $F$ among them” (Xie et al., 2021). Closely related statements recur throughout the literature: Buckingham–$\Pi$ gives a basis of admissible groups, but “this set is not unique” and classical analysis alone does not identify the “most physically meaningful dimensionless basis” for a given dataset (Bakarji et al., 2022).

In the data‑driven formulation, a generic physical relation

$y = f(p_1, p_2, \dots, p_n)$

is re‑expressed as

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$

where each $\Pi$ is a monomial

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$

The aim is to discover “dominant dimensionless numbers” and “scaling laws” directly from measurements, including “noisy experimental data,” while keeping the learned representation “scale‑free and unit‑independent” (Xie et al., 2021).

Several later formulations widen the scope of the term. “Dimensionless machine learning” is defined as imposing “exact units equivariance” by first constructing “a dimensionless version of its inputs using classic results from dimensional analysis, and then perform[ing] inference in the dimensionless space” (Villar et al., 2022). “Dimensionless learning based on information” treats the best dimensionless groups as those with “the highest predictive power by measuring their shared information content” (Yuan et al., 4 Apr 2025). “Hierarchical Dimensionless Learning (Hi‑$\pi$)” describes a “physics-data hybrid-driven approach for discovering dimensionless parameter combinations” that balances “accuracy and complexity” (Xia et al., 24 Jul 2025).

2. Mathematical structure and dimensionless parameterization

The common algebraic structure begins with the dimension matrix. If $p_i$ has dimension vector $\pi$0, then

$\pi$1

collects all dimensional information, and a candidate dimensionless monomial with exponent vector $\pi$2 must satisfy

$\pi$3

Hence all admissible exponent vectors lie in the null space of $\pi$4, with

$\pi$5

If $\pi$6 is a basis of that null space, then any admissible exponent vector can be written as

$\pi$7

where $\pi$8 and $\pi$9 are basis coefficients to be learned from data (Xie et al., 2021).

A closely related representation appears in “Dimensionally Consistent Learning with Buckingham Pi”: $F$0 with the values of the $F$1 groups over a dataset written as

$F$2

This makes the multiplicative structure linear in log space and turns dimensionless learning into a constrained search over the null space of the dimensional matrix (Bakarji et al., 2022).

A more geometric version is given in “The Algebra of Units: From Buckingham’s $F$3 Theorem to Latent‑Variable Learning.” If $F$4 and $F$5, then for an exponent vector $F$6,

$F$7

The paper states that, after logarithmic transformation, measurements “lie on a low-dimensional manifold whose geometry is determined by the underlying dimensionless groups,” and that singular value decomposition identifies this manifold directly from data (Valorani, 15 Jun 2026). This suggests a direct connection between dimensional analysis and latent‑variable methods such as PCA, SVD, and POD.

3. Learning formulations and algorithmic families

A central formulation is the “two‑level optimization” of dimensionless learning. Level 1 searches over admissible dimensionless groups, i.e. over $F$8, and Level 2 fits a scaling relation and evaluates predictive quality. For each candidate $F$9, the method computes the corresponding $\Pi$0-groups, fits a regressor such as a polynomial or XGBoost model, and evaluates a score such as $\Pi$1 on held‑out data (Xie et al., 2021). Practical search strategies include grid search, pattern search, and “gradient‑based schemes,” while parsimony is encouraged by limiting the number of groups $\Pi$2, constraining coefficients to small ranges such as $\Pi$3, and favoring “small rational numbers” for the exponents (Xie et al., 2021).

Other papers instantiate the same principle with different learners. “Dimensionally Consistent Learning with Buckingham Pi” develops three techniques: a constrained optimization with kernel ridge regression, BuckiNet, and a dimensionless SINDy procedure. BuckiNet constrains its first layer to implement

$\Pi$4

and adds the soft dimensional-consistency penalty

$\Pi$5

The SINDy-based variant searches over candidate $\Pi$6-groups and time scales to discover sparse, dimensionless dynamical equations (Bakarji et al., 2022).

Two later developments emphasize model selection rather than only prediction. IT‑$\Pi$7 defines an information‑theoretic irreducible error bound

$\Pi$8

and chooses the dimensionless variables that minimize that lower bound, or equivalently maximize the shared information between $\Pi$9 and $y = f(p_1, p_2, \dots, p_n)$0 (Yuan et al., 4 Apr 2025). Hi‑$y = f(p_1, p_2, \dots, p_n)$1 uses dimensional analysis to generate initial dimensionless variables, symbolic regression to search for nonlinear recombinations, and a polynomial map

$y = f(p_1, p_2, \dots, p_n)$2

to score “accuracy and complexity” (Xia et al., 24 Jul 2025).

A different branch of the literature formulates the problem as exact symmetry enforcement. “Dimensionless machine learning: Imposing exact units equivariance” represents changes of units as a group action and defines a model to be units equivariant if

$y = f(p_1, p_2, \dots, p_n)$3

Its prescription is to construct dimensionless invariants first, learn only in that invariant space, and then decode back to dimensional outputs (Villar et al., 2022).

4. Empirical discoveries and representative physical applications

The original “dimensionless learning” paper demonstrates the framework on “noisy experimental measurements (not synthetic data)” from several engineering problems (Xie et al., 2021). In turbulent Rayleigh–Bénard convection, the method discovers the unique optimum at $y = f(p_1, p_2, \dots, p_n)$4, yielding

$y = f(p_1, p_2, \dots, p_n)$5

and learns

$y = f(p_1, p_2, \dots, p_n)$6

with a 5th‑order polynomial, collapsing the data in log–log space with $y = f(p_1, p_2, \dots, p_n)$7 (Xie et al., 2021). In laser–metal interaction, it recovers the “keyhole number”

$y = f(p_1, p_2, \dots, p_n)$8

and a nearly linear law

$y = f(p_1, p_2, \dots, p_n)$9

with $\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$0 for keyhole aspect ratio (Xie et al., 2021). In metal 3D printing porosity, the method shows that a naive energy density

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$1

gives only $\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$2, whereas two learned dimensionless groups, including

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$3

and

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$4

allow XGBoost models to reach $\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$5 up to about $\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$6 on test data (Xie et al., 2021).

“Dimensionally Consistent Learning with Buckingham Pi” recovers classical variables in several canonical examples. For the bead on a rotating hoop, BuckiNet identifies

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$7

and dimensionless SINDy recovers the associated dimensionless ODE structure (Bakarji et al., 2022). For the Blasius boundary layer, constrained optimization discovers the similarity variable

$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$8

to numerical precision from sparse dimensional data (Bakarji et al., 2022). For Rayleigh–Bénard convection, the same paper identifies the inverse Rayleigh number as the best control parameter for a Landau‑type normal form near onset (Bakarji et al., 2022).

Hi‑$\Pi_{\text{out}} = F(\Pi_1,\dots,\Pi_k),$9 extends this program to multi‑parameter discovery. In Rayleigh–Bénard convection it “accurately extracted two intrinsic dimensionless parameters: the Rayleigh number and the Prandtl number,” and on a dataset spanning small and large $\Pi$0 it reports interpolation relative error $\Pi$1 with $\Pi$2 and extrapolation relative error $\Pi$3 with $\Pi$4 (Xia et al., 24 Jul 2025). In viscous pipe flow it “automatically discovers two optimal dimensionless parameters: the Reynolds number and relative roughness,” and with a 4th‑order polynomial reports relative error $\Pi$5 and $\Pi$6 (Xia et al., 24 Jul 2025). In compressibility correction for subsonic flow, it extracts the Prandtl–Glauert‑like combination

$\Pi$7

as a useful intermediate variable for recovering the Karman–Tsien structure (Xia et al., 24 Jul 2025).

The LPBF literature uses a related workflow to learn reduced process descriptions from high‑fidelity simulation. A thermo‑fluid finite‑element model is non‑dimensionalized using quantities such as

$\Pi$8

and a learned heat‑absorption index

$\Pi$9

is then correlated with meltpool morphology, temperature gradients, and cooling rate (Bhagat et al., 2022).

5. Control, units equivariance, and transfer across scales

In control, dimensionless learning is used in a different but related sense: “learning control laws, policies, and tuning parameters in a dimensionless space, so that the learned knowledge applies unchanged to any physically similar system” (Hromatko et al., 9 Dec 2025). The core construction starts from

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$0

and introduces scalings

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$1

yielding the dimensionless dynamics

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$2

After discretization,

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$3

and, for similar systems, the dimensionless model is identical (Hromatko et al., 9 Dec 2025).

This structure is extended to dimensionless MDPs and a dimensionless MPC problem in which costs and constraints are also non‑dimensionalized: $\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$4 The key claim is that if two systems are dynamically similar and share the same dimensionless parameters and sampling time, then “their dimensionless MPC problems are identical,” so the tuned controller transfers directly (Hromatko et al., 9 Dec 2025).

The paper demonstrates this on cartpole swing‑up and race‑car lap time minimization. For the cartpole, it uses

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$5

and defines the state and input scalings

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$6

The result is that with “dimensionless MPC,” reinforcement learning continues smoothly after switching to a dynamically similar system of different physical scale, whereas with a “purely dimensional MPC,” performance “nearly resets when the scale changes” (Hromatko et al., 9 Dec 2025). For the race car, Bayesian optimization on a small‑scale car transfers to a full‑size car under preserved $\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$7-groups, without re‑tuning (Hromatko et al., 9 Dec 2025).

A broader machine‑learning treatment of the same idea appears in “Dimensionless machine learning: Imposing exact units equivariance.” There, the units rescaling group is $\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$8, a units‑equivariant model satisfies

$\Pi = \prod_{i=1}^n p_i^{\alpha_i}.$9

and dimensionless invariants are constructed by solving

$\pi$0

for the units vectors $\pi$1 (Villar et al., 2022). In the springy double pendulum benchmark, the dimensionless Hamiltonian neural network has in‑distribution relative error $\pi$2, compared with $\pi$3 for a dimensional baseline, but under units‑rescaled OOD testing the dimensionless model remains at $\pi$4 while the baseline rises to $\pi$5 (Villar et al., 2022). This suggests that exact units equivariance and scale transfer are practically significant consequences of learning in a dimensionless space.

6. Information-theoretic, geometric, and domain-specific extensions

Several papers generalize dimensionless learning beyond regression over hand‑selected features. IT‑$\pi$6 defines optimal dimensionless variables as those minimizing a model‑free lower bound on the prediction error based on Rényi mutual information (Yuan et al., 4 Apr 2025). It states that the method can “rank variables by predictability, identify distinct physical regimes, uncover self-similar variables, determine the characteristic scales of the problem, and extract its dimensionless parameters” (Yuan et al., 4 Apr 2025). It also defines “model efficiency”

$\pi$7

thereby comparing any fitted model with the information‑theoretic lower bound (Yuan et al., 4 Apr 2025).

A geometric extension appears in “The Algebra of Units: From Buckingham’s $\pi$8-grec Theorem to Latent-Variable Learning,” where gauge variation across scaled realizations leads to a within‑cluster deviation matrix

$\pi$9

and the last $p_i$0 right singular vectors span $p_i$1 exactly under the stated assumptions (Valorani, 15 Jun 2026). On a synthetic compressor dataset of 16,000 measurements, the method recovers the flow coefficient, head coefficient, and Mach number to numerical precision, with “error below $p_i$2” on the performance map (Valorani, 15 Jun 2026).

Other domain‑specific examples show that the same dimensionless perspective can organize inference in very different fields. “Dimensionless cosmology” argues that “only dimensionless combinations of ‘fundamental constants’ can be operationally defined and compared,” and recasts Big Bang nucleosynthesis, recombination, and CMB inference in terms of

$p_i$3

rather than varying dimensional constants such as $p_i$4 directly (Narimani et al., 2011). In neutron‑star structure, the TOV equations are recast using

$p_i$5

leading to EOS‑independent mass–radius–compactness scalings and an “Intrinsic and Perturbative Analyses of the Dimensionless (IPAD) TOV equations” program for inferring core EOS structure from observables (Cai et al., 30 Jan 2025). In reacting porous‑flow systems, a “dimensionless framework” maps a particle size distribution into distributions of Damköhler numbers such as

$p_i$6

and uses these distributions to compare column and heap leaching across scale (Segura, 20 Jan 2026). These cases suggest that dimensionless learning can denote both explicit learning algorithms and a broader strategy of expressing governing relations, control laws, and inference problems in a reduced dimensionless representation.

7. Limitations, misconceptions, and open directions

A persistent misconception is that dimensional analysis alone identifies the physically relevant variables. The literature consistently states the opposite: Buckingham–$p_i$7 provides a null space, not a unique answer, and does not identify the functional relationship or the most predictive groups (Xie et al., 2021, Bakarji et al., 2022). Dimensionless learning addresses that ambiguity, but it does not remove all assumptions.

Most formulations require that “relevant variables and their dimensions are correctly specified,” and that “the true physics admits a description by dimensionless monomials” or by a dimensionless representation of the chosen form (Xie et al., 2021). If important variables are missing, “no high‑quality scaling can be recovered,” although the resulting low maximal $p_i$8 can reveal that omission (Xie et al., 2021). The black‑body example in units‑equivariant machine learning shows the same issue differently: without including Planck’s constant $p_i$9, dimensional analysis yields only the Rayleigh–Jeans form and cannot recover Planck’s law (Villar et al., 2022).

Another limitation is computational. Search over $\pi$00-space or over candidate exponent vectors can be expensive in high‑dimensional null spaces (Xie et al., 2021, Bakarji et al., 2022). Multiple‑group extraction “needs more data” and is more sensitive to sample quality than one‑group discovery (Xia et al., 24 Jul 2025). In the tutorial literature, key open problems include “managing the computational cost of identifying multiple dimensionless groups,” “automating the selection of relevant input variables,” understanding the role of sample size and data distribution, and developing “user-friendly tools for experimentalists” (Gan et al., 12 Dec 2025).

The effect of noise is a recurrent concern. The tutorial on dimensionless learning studies “measurement noise and discrete sampling,” and reports that a “quantization regularizer” pushing learned coefficients toward integers or half‑integers improves robustness and interpretability (Gan et al., 12 Dec 2025). IT‑$\pi$01 adds a different perspective by replacing model‑specific error with a bound determined by shared information; this suggests a route for deciding whether poor predictive performance is due to the model class or to information missing from the inputs (Yuan et al., 4 Apr 2025).

Taken together, these developments suggest several research directions already present in the literature: integrating symbolic regression, sparse regression, and deeper nonlinear learners while preserving dimensional consistency; building dimensionless surrogates for PDEs and control; exploiting SVD and latent‑variable geometry in log space; using information‑theoretic objectives to rank or select $\pi$02-groups; and extending dimensionless learning from single numbers to distributions of dimensionless quantities in heterogeneous systems (Bakarji et al., 2022, Valorani, 15 Jun 2026, Yuan et al., 4 Apr 2025, Segura, 20 Jan 2026). In essence, the field treats dimensionless representation not merely as a preprocessing step, but as a principled way to constrain learning, organize physical interpretation, and improve extrapolation across scales.