Machine Learning-Based Dimensional Analysis

• Machine learning-based DA is a method that fuses classical dimensional invariance with data-driven techniques to discover and validate dimensionless groups.
• It employs null-space methodologies from Buckingham’s π theorem alongside neural networks and optimization to resolve the non-uniqueness of scaling laws.
• Its applications span turbulent convection, laser-metal interaction, and computer experiments, achieving high predictive accuracy even under noisy conditions.

Machine learning-based dimensional analysis (DA) denotes a family of methods that combine the invariance structure of classical dimensional analysis with statistical learning, surrogate modeling, or neural networks in order to discover dimensionless groups, identify scaling laws, and construct predictive models on physically valid feature spaces. Across the recent literature, the central idea is to encode unit structure through a dimension matrix, restrict learned representations to the null space implied by Buckingham’s $\pi$ theorem, and then use data to resolve the non-uniqueness, relevance, and predictive utility of candidate dimensionless groups. In this sense, machine learning-based DA extends classical similitude theory from a purely structural statement—existence of dimensionless groups—to an empirical program for selecting dominant groups, learning $q_0=f(\pi_1,\ldots,\pi_k)$ relations, and, in some formulations, discovering dimensionally homogeneous differential equations (Xie et al., 2021, Bakarji et al., 2022, Rodriguez-Arelis et al., 2023).

1. Classical dimensional analysis and the motivation for learning-based extensions

Classical DA begins with a set of dimensional variables and a choice of base dimensions such as mass $M$, length $L$, time $T$, and temperature $\Theta$. Buckingham’s $\pi$ theorem states that if $n$ variables are expressed in terms of $r$ independent base dimensions, then the physical law can be rewritten using $k=n-r$ independent dimensionless groups. If the dimension matrix is denoted by $q_0=f(\pi_1,\ldots,\pi_k)$0, admissible exponent vectors lie in the null space $q_0=f(\pi_1,\ldots,\pi_k)$1, and each dimensionless group has monomial form

$q_0=f(\pi_1,\ldots,\pi_k)$2

This construction guarantees unit invariance and underpins similitude arguments, reduced-order descriptions, and scale-transfer reasoning (Xie et al., 2021).

The limitation, emphasized repeatedly in the literature, is that the null space basis is not unique. Any invertible linear recombination of a basis yields another valid set of $q_0=f(\pi_1,\ldots,\pi_k)$3-groups, so classical DA does not by itself determine which groups are most informative, which are dominant in a particular regime, or what the functional dependence among them should be. For highly multivariable systems, incomplete governing equations, and noisy data, this non-uniqueness becomes a practical obstacle: arbitrary response-surface modeling on ad hoc $q_0=f(\pi_1,\ldots,\pi_k)$4 choices can be inefficient or misleading, and the theorem alone does not supply a scaling law such as $q_0=f(\pi_1,\ldots,\pi_k)$5 or indicate whether a single dominant group exists (Xie et al., 2021, Constantine et al., 2017).

Learning-based DA addresses precisely these deficiencies. The various formulations differ in their optimization strategy and predictive model class, but they share a common objective: search within the dimensionally admissible manifold for groups that maximize data collapse, predictive accuracy, extrapolation performance, sparsity, or interpretability. This has led to several distinct but related frameworks, including active-subspace-based semi-empirical DA, constrained optimization over null-space coefficients, neural architectures with $q_0=f(\pi_1,\ldots,\pi_k)$6-layers, data-adaptive Gaussian-process surrogates in DA coordinates, and ensemble gradient-boosted regressors trained on handcrafted $q_0=f(\pi_1,\ldots,\pi_k)$7 sets (Constantine et al., 2017, Bakarji et al., 2022, Rodriguez-Arelis et al., 2023, Pierzyna et al., 2023).

2. Mathematical structure: dimension matrices, null spaces, and learned invariants

The common formalism starts from a variable vector $q_0=f(\pi_1,\ldots,\pi_k)$8 together with a dimension matrix $q_0=f(\pi_1,\ldots,\pi_k)$9 or $M$0, whose columns encode the exponents of the base units for each variable. If $M$1, then $M$2, and any exponent vector defining a dimensionless monomial must satisfy $M$3 (Xie et al., 2021, Gan et al., 12 Dec 2025).

A standard parameterization writes

$M$4

where $M$5 spans $M$6 and $M$7 is learned from data. In dimensionless learning, the discovered groups are inserted into a predictive model,

$M$8

subject to $M$9 for each group. The regularizer $L$0 is used to promote parsimony, group sparsity, or small integer and rational exponents; log-linear relations are handled through forms such as

$L$1

In this formulation, the null-space parameterization guarantees dimensional consistency by construction, while the loss function supplies a criterion for selecting one distinguished representation from the classically non-unique family (Xie et al., 2021).

The neural-network tutorial formulation makes the same structure explicit in log space. Basis $L$2-groups are computed from null-space vectors $L$3, and a linear combination layer learns coefficients $L$4:

$L$5

The total loss combines prediction error with a quantization regularizer,

$L$6

where $L$7 attracts the learned coefficients toward a discrete set $L$8, such as half-integers, and always includes $L$9 to promote sparsity. This soft rounding mechanism is explicitly intended to stabilize discovery and yield interpretable exponents (Gan et al., 12 Dec 2025).

A separate but mathematically connected line of work interprets DA models as ridge functions of log-transformed inputs. If $T$0, then the empirical law induced by DA can be written as a ridge function in a reduced coordinate system. Active subspaces are then obtained from the gradient covariance

$T$1

whose eigendecomposition yields orthogonal directions ordered by derivative-based global sensitivity. Rotating the original DA coordinates by these eigenvectors produces unique, relevance-ordered dimensionless groups, conditional on the chosen input density and distinct eigenvalues (Constantine et al., 2017).

3. Principal algorithmic families

The current literature does not present a single canonical algorithm. Instead, machine learning-based DA has developed as a set of related design patterns that impose Buckingham constraints at different stages of the learning pipeline.

Framework	Core mechanism	Representative paper
Active-subspace DA	Response surfaces or finite differences in DA coordinates; active-subspace rotation yields unique, relevance-ordered groups	(Constantine et al., 2017)
Constrained optimization and BuckiNet	Null-space-constrained exponents with nonparametric fitting, or a first neural layer that computes $T$2-groups	(Bakarji et al., 2022)
Dimensionless learning	Two-level search over null-space coefficients and predictive models for dominant, unique groups and scaling laws	(Xie et al., 2021)
Data-adaptive DA for computer experiments	FANOVA-guided basis selection, DA feature construction, and GP surrogates on dimensionless inputs	(Rodriguez-Arelis et al., 2023)
$T$3-ML	Domain-knowledge-derived $T$4 sets with XGBoost ensembles trained on non-dimensional predictors and target	(Pierzyna et al., 2023)
Tutorial neural dimensionless learning	Linear combination layer over basis $T$5-groups, hidden layers, and quantization regularization	(Gan et al., 12 Dec 2025)

In the two-level dimensionless-learning scheme, Level 1 computes the null-space basis and searches coefficient vectors within bounded, physically plausible ranges—often emphasizing small rational or integer values and fixing one coefficient for normalization—while Level 2 fits a predictive model and evaluates generalization through train/test $T$6 or criteria such as BIC. The demonstrated Level 2 models include polynomial regression up to fifth order, XGBoost, and gradient-based regression; the formulation is described as compatible with generalized linear models, Gaussian processes, random forests, neural networks, or symbolic regression (Xie et al., 2021).

BuckiNet implements a more tightly integrated architecture. Its first layer computes

$T$7

or uses a soft null-space penalty $T$8 in the loss. Downstream layers then model the input–output map, while $T$9 regularization promotes sparse, low-magnitude exponents. In parallel, the same study develops a constrained optimization method with a flexible regressor and a SINDy-based pipeline for discovering dimensionless dynamics (Bakarji et al., 2022).

In computer experiments, the data-adaptive DA approach shifts the emphasis from discovering exponents by search to choosing effective basis quantities empirically. It first fits a Gaussian process in the raw variables, computes FANOVA main and interaction effects, and then selects basis quantities that satisfy Buckingham’s representativity and independence constraints while contributing strongly to predictive variance. Dimensionless outputs and inputs are subsequently constructed from these basis variables, optionally log-transformed or augmented with pairwise sums and differences of logged dimensionless inputs, and the final surrogate is again a GP with product kernel and maximum-likelihood hyperparameter estimation (Rodriguez-Arelis et al., 2023).

The $\Theta$0-ML formulation is more application-specific. It uses dimensional analysis and domain knowledge to generate candidate $\Theta$1 sets, prunes them by constraints such as positivity of $\Theta$2 and avoidance of fractional or even powers of signed quantities, and then trains an ensemble of XGBoost models with FLAML-based hyperparameter tuning and Monte Carlo sub-sampling. Here the role of machine learning-based DA is not null-space discovery but parameterization in a similarity-variable space that is explicitly non-dimensionalized (Pierzyna et al., 2023).

4. Non-uniqueness, identifiability, and robustness

The central conceptual issue in the field is the non-uniqueness of $\Theta$3-groups. Classical DA provides a null space but not a preferred basis. Machine learning-based DA makes this ambiguity operational by introducing a data criterion. In dimensionless learning, uniqueness is defined by predictive optimality in the null space under parsimony constraints and normalization conventions; in active-subspace DA, uniqueness is defined by the eigendirections of a gradient covariance operator, up to scalar rescaling and only when eigenvalues are distinct; in the tutorial neural formulation, multiple equivalent representations may lie on a line or plane in coefficient space, especially when more than one dimensionless number is required (Xie et al., 2021, Constantine et al., 2017, Gan et al., 12 Dec 2025).

The geometric interpretation developed in the tutorial is particularly explicit. For a single $\Theta$4-group, all scalar multiples of an exponent vector define equivalent inputs, so learned solutions cluster along a line. For multiple groups, any invertible linear recombination preserves the physics, and learned coefficient vectors lie in a subspace—for two dominant groups, a plane. Quantization regularization does not remove this equivalence class; rather, it biases the solution toward simple half-integer or integer points within it. This suggests that “uniqueness” in neural dimensionless learning is often best understood as uniqueness of a physically meaningful subspace plus a regularized representative (Gan et al., 12 Dec 2025).

Robustness has been studied under several forms of uncertainty. Dimensionless learning reports performance on real, noisy literature datasets rather than synthetic data, using train/test splits such as $\Theta$5 and out-of-sample $\Theta$6 as the main metric. Its robustness is attributed to the strong reduction of the feature search space to dimensionally admissible combinations, bounded exponent search, and explicit sensitivity analyses showing that omission of a key variable can severely cap accuracy (Xie et al., 2021). The tutorial extends this by injecting controlled $\Theta$7–$\Theta$8 noise into synthetic benchmarks and showing that the learned $\Theta$9 drifts as noise increases, while the quantization regularizer stabilizes solutions and creates clusters that can be filtered by $\pi$0; it also reports that SIR maintains robust first-direction explained variance above $\pi$1 up to high noise levels, whereas PCA degrades more rapidly (Gan et al., 12 Dec 2025).

The active-subspace framework highlights a different dependency: the learned unique and relevance-ranked groups depend on the probability density $\pi$2 over inputs. This is not presented as a defect of the algorithm but as the price of resolving non-uniqueness in a regime-specific manner. A change in the support or weighting of the input distribution changes the sensitivity metric and therefore the preferred rotation within the DA subspace (Constantine et al., 2017). In computer experiments, the GP-based data-adaptive approach argues that DA features compare variables rather than use absolute magnitudes, making them less dependent on the training ranges and thereby improving extrapolation, but it also stresses that poor basis choices or unstable ratios can degrade performance (Rodriguez-Arelis et al., 2023).

5. Representative applications and empirical performance

The dimensionless-learning study provides three widely cited engineering case studies. In turbulent Rayleigh–Bénard convection, with variables including $\pi$3, $\pi$4, $\pi$5, $\pi$6, $\pi$7, $\pi$8, and $\pi$9, the method rediscovered the classical Rayleigh number

$n$0

identified a unique optimum in exponent space, and achieved out-of-sample $n$1 for $n$2 versus the discovered $n$3 using a fifth-order polynomial model; when the dimensional heat flux $n$4 was used as output, additional $n$5’s and a nearly linear log-log scaling were found. In laser-metal interaction, it recovered the keyhole number

$n$6

and learned

$n$7

with out-of-sample $n$8; omitting $n$9 reduced the maximum achievable $r$0 below $r$1. In laser powder bed fusion porosity, classical energy density yielded only $r$2, whereas two discovered groups—NED and NEP-based quantities—reached $r$3 up to $r$4 across materials and operating regimes (Xie et al., 2021).

The data-adaptive DA study addresses deterministic computer experiments and emphasizes interpolation and extrapolation. For gravity displacement, FANOVA-guided basis selection produced a near-perfect predictor and markedly improved extrapolation, including to gravitational values associated with Jupiter and Saturn. For the borehole function, FANOVA DA with expanded log-inputs achieved normalized test RMSE of approximately $r$5 at $r$6 and approximately $r$7 at $r$8 in interpolation, and approximately $r$9 at $k=n-r$0 and approximately $k=n-r$1 at $k=n-r$2 in extrapolation. For heat transfer in a solid sphere, FANOVA DA sustained extrapolation accuracy with N-RMSE of approximately $k=n-r$3 at $k=n-r$4 and approximately $k=n-r$5 at $k=n-r$6, compared with approximately $k=n-r$7 for non-DA and approximately $k=n-r$8 for T-DA at $k=n-r$9 (Rodriguez-Arelis et al., 2023).

The active-subspace literature and the Buckingham-$q_0=f(\pi_1,\ldots,\pi_k)$00 constrained learning literature supply additional exemplars. In viscous pipe flow, active-subspace DA recovered one dominant group aligned with Reynolds number in the laminar regime, discovered two unique rotated combinations of $q_0=f(\pi_1,\ldots,\pi_k)$01 and $q_0=f(\pi_1,\ldots,\pi_k)$02 in turbulent flow, and recovered roughness dominance at very high Reynolds number. In the rotating-hoop and Blasius boundary-layer examples, BuckiNet, constrained optimization, and dimensionless SINDy recovered canonical groups such as $q_0=f(\pi_1,\ldots,\pi_k)$03, $q_0=f(\pi_1,\ldots,\pi_k)$04, and the Blasius similarity variable $q_0=f(\pi_1,\ldots,\pi_k)$05, while the Rayleigh–Bénard convection normal-form study identified $q_0=f(\pi_1,\ldots,\pi_k)$06 and a pitchfork normal form with $q_0=f(\pi_1,\ldots,\pi_k)$07 (Constantine et al., 2017, Bakarji et al., 2022).

A domain-specific but influential parameterization example is $q_0=f(\pi_1,\ldots,\pi_k)$08-ML for optical turbulence in the atmospheric surface layer. Using $q_0=f(\pi_1,\ldots,\pi_k)$09 variables, $q_0=f(\pi_1,\ldots,\pi_k)$10 base dimensions, and a selected $q_0=f(\pi_1,\ldots,\pi_k)$11 set containing $q_0=f(\pi_1,\ldots,\pi_k)$12, $q_0=f(\pi_1,\ldots,\pi_k)$13, $q_0=f(\pi_1,\ldots,\pi_k)$14, $q_0=f(\pi_1,\ldots,\pi_k)$15, $q_0=f(\pi_1,\ldots,\pi_k)$16, $q_0=f(\pi_1,\ldots,\pi_k)$17, and $q_0=f(\pi_1,\ldots,\pi_k)$18, an ensemble of $q_0=f(\pi_1,\ldots,\pi_k)$19 XGBoost models achieved out-of-sample $q_0=f(\pi_1,\ldots,\pi_k)$20 and log-space RMSE $q_0=f(\pi_1,\ldots,\pi_k)$21 on a middle test interval. A reduced-order model using only $q_0=f(\pi_1,\ldots,\pi_k)$22 still achieved $q_0=f(\pi_1,\ldots,\pi_k)$23 and $q_0=f(\pi_1,\ldots,\pi_k)$24, and permutation feature importance identified $q_0=f(\pi_1,\ldots,\pi_k)$25 as the dominant feature (Pierzyna et al., 2023).

6. Limitations, related methods, and open directions

Despite the strong empirical results, the literature is explicit about the method’s limitations. Variable completeness is critical: if important physics is omitted from the candidate variable list, the best achievable performance may degrade substantially, as demonstrated by the omission of thermal diffusivity $q_0=f(\pi_1,\ldots,\pi_k)$26 in the laser keyhole problem (Xie et al., 2021). Narrow operating ranges, collinearity, or irrelevant variables can produce multiple near-optimal $q_0=f(\pi_1,\ldots,\pi_k)$27 sets or make distinct groups difficult to distinguish. Computational cost also grows with null-space dimension and search bounds; pattern-search and gradient-based optimizers are presented as mitigations, but the tutorial notes that the search space expands sharply for multiple $q_0=f(\pi_1,\ldots,\pi_k)$28-groups, making $q_0=f(\pi_1,\ldots,\pi_k)$29–$q_0=f(\pi_1,\ldots,\pi_k)$30 challenging (Xie et al., 2021, Gan et al., 12 Dec 2025).

Noise, discrete sampling, and architecture choice remain active concerns. The tutorial reports that data characteristics—noise level, sample count, dynamic range, sparsity, and correlation structure—strongly influence recovery and calls for more systematic study. It also identifies automated input selection as an open challenge and notes that broader user-friendly tooling is still needed, despite a Streamlit interface called PyDimension (Gan et al., 12 Dec 2025). In computer experiments, DA-space FANOVA is described as difficult because DA features induce irregular domains, so basis selection is performed in raw space and validation is used to choose among DA feature sets (Rodriguez-Arelis et al., 2023).

The relation to adjacent methods is increasingly well defined. SINDy discovers sparse governing equations, while DA-constrained SINDy restricts the candidate library to dimensionally homogeneous terms and can recover minimal dimensionless normal forms (Xie et al., 2021, Bakarji et al., 2022). Symbolic regression can be used as a Level 2 model class in dimensionless learning, but DA differs from symbolic-regression-style equation discovery because it first imposes unit consistency on the representation (Xie et al., 2021). The tutorial also positions dimensionless learning relative to SLAW, active subspaces, DimensionNet (HiDeNN, 2021), BuckiNet, units-equivariant models, and information-theoretic dimensionless learning, treating these as complementary or antecedent approaches rather than interchangeable formulations (Gan et al., 12 Dec 2025).

A plausible implication of the combined evidence is that machine learning-based DA is best viewed not as a replacement for classical DA, but as a set of inference procedures defined on the DA manifold. The shared contribution across the literature is to transform dimensional invariance from a static algebraic constraint into an optimization principle: predictive performance, sensitivity structure, sparsity, extrapolation behavior, and coefficient simplicity all become criteria for selecting among equivalent dimensionless representations. Within that perspective, the field’s main open questions are not whether dimensionless groups exist, but how to identify the most useful ones under noise, limited data, multiple regimes, and incomplete prior knowledge (Constantine et al., 2017, Xie et al., 2021, Rodriguez-Arelis et al., 2023)