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Dimensionally consistent surrogate modelling through dimensional analysis and harmonic expansions

This paper presents a data-driven method for constructing dimensionally consistent surrogate models by combining Buckingham $\Pi$ -groups with truncated harmonic expansions, demonstrating improved conditioning, robustness, and sample efficiency across various physical systems compared to unconstrained baselines.

Original authors: Ernest Tarrus, Hector Gisbert

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ernest Tarrus, Hector Gisbert

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the physical world, the laws of nature do not care what language we use to describe them. Whether a scientist measures a distance in meters or miles, or a duration in seconds or hours, the underlying relationship between those quantities remains unchanged. This principle, known as dimensional homogeneity, acts as a strict rulebook for any equation that claims to describe reality. If a formula changes its answer simply because the units of measurement were switched, it cannot be a true law of physics. For decades, researchers have used this rule to simplify complex problems, stripping away unnecessary variables to find the core, unitless combinations that govern how systems behave. However, as scientists increasingly turn to artificial intelligence to discover these laws directly from data, a challenge has emerged. Standard machine learning models are often blind to these physical rules; they might find a pattern that fits the data perfectly but fails the moment the units change, rendering the discovery physically meaningless.

A team of researchers at the Universidad Europea de Valencia has developed a new way to teach computers to respect these physical laws from the very beginning of the learning process. Instead of letting a model guess the relationship between variables and hoping it eventually stumbles upon the correct units, they built the rules of dimensional consistency directly into the structure of the model itself. By combining classical physics techniques with modern data analysis, they created a method that forces the computer to construct its predictions using only combinations of variables that make physical sense. This approach ensures that the resulting formulas are not just mathematical tricks that work on a specific set of numbers, but robust, universal expressions that hold true regardless of how the data is measured.

The researchers tested this method on several classic problems in physics, ranging from the swinging of a simple pendulum to the radiation emitted by a perfect black body. In the case of a pendulum, the period of its swing depends on the length of the string and the strength of gravity, but not on the mass of the weight. A standard computer model, given noisy data, might struggle to figure out that mass is irrelevant or might require thousands of data points to guess the correct relationship between length and time. The new method, however, starts with the knowledge that mass should not appear in the final formula. It immediately focuses on the correct combination of length and gravity, allowing it to recover the precise physical law with far fewer data points and even when the measurements are quite noisy. The computer essentially learns the shape of the relationship much faster because it is not wasting time exploring impossible or physically nonsensical options.

The team also applied their technique to the complex radiation emitted by hot objects, a phenomenon described by Planck's law. Here, the relationship between temperature and radiation is not a simple straight line but a more intricate curve. The researchers found that by separating the problem into a part that handles the physical units and a part that learns the shape of the curve, the model could accurately reconstruct the law even with limited data. They discovered that the choice of mathematical tools used to describe the curve mattered significantly. When the data covered a range that behaved like a repeating cycle, using tools designed for waves worked best. When the data did not repeat, other mathematical tools were more effective. This nuance is crucial because it shows that while the physical rules are universal, the best way to represent the remaining details depends on the specific nature of the data.

One of the most striking findings was how much more efficient the new method was compared to standard, unconstrained machine learning. In the pendulum experiments, the new approach could accurately predict the behavior of the system with as few as five different measurements. In contrast, a standard model that was not told the physical rules had to see at least sixty measurements before it could even begin to guess the correct relationship, and even then, its predictions were less reliable. This suggests that embedding physical knowledge into the learning process is not just a way to make models more accurate, but a way to make them vastly more efficient. It allows scientists to learn from smaller, noisier datasets that would otherwise be too difficult to analyze.

The researchers also demonstrated that their method produces results that are easy for humans to understand. Unlike many modern artificial intelligence systems that act as "black boxes," hiding their logic inside layers of complex calculations, this approach yields explicit formulas. The final output is a clear equation that a physicist can read, check, and use immediately. This transparency is vital for scientific discovery, as it allows researchers to verify that the model has not just memorized the data but has actually uncovered the underlying mechanism. The study confirms that by respecting the fundamental constraints of the physical world, machine learning can become a more powerful and reliable partner in understanding the universe, turning raw data into genuine insight with greater speed and clarity.

Technical Summary: Dimensionally Consistent Surrogate Modelling through Dimensional Analysis and Harmonic Expansions

Problem Statement
The paper addresses the challenge of constructing data-driven surrogate models for physical systems that strictly adhere to the principle of dimensional homogeneity. A physically meaningful model must be invariant under changes of units; a predictor that yields different results when length is measured in meters versus centimeters is physically inadmissible. While existing physics-informed learning methods often treat dimensional consistency as a post-hoc check or enforce it weakly, this work argues that such constraints should be imposed directly at the level of the hypothesis class. The goal is to develop a method that guarantees unit equivariance by construction, thereby improving conditioning, robustness to noise, and sample efficiency compared to unconstrained baselines.

Methodology
The proposed framework constructs surrogate models through a four-stage pipeline that integrates dimensional analysis with regularized linear regression:

Dimensional Structure and Invariants: Starting from a dimension matrix $A$ derived from the measured input variables and the target quantity, the method computes the kernel of $A$ to identify independent Buckingham $\Pi$ -groups (dimensionless invariants). Simultaneously, it solves the affine system $A\beta = b$ to determine the space of admissible dimensional prefactors (monomials) that carry the physical dimensions of the target variable.
Hypothesis Class Formulation: The model is structured as a product of an admissible dimensional prefactor $P_\beta(x)$ and a dimensionless modulation function $\Phi_\beta$ defined over the invariant space. This structure ensures unit equivariance by construction: the prefactor handles the dimensional scaling, while the modulation operates on dimensionless coordinates.
Harmonic Approximation: The unknown dimensionless modulation $\Phi_\beta$ $Φ_{β}$ is approximated using truncated harmonic expansions (Fourier series) on normalized invariant domains. The invariants are affinely mapped to a compact domain (typically $[0, 2\pi]$ $[0, 2 π]$ ). The authors evaluate two dictionary types:
- Phase-combination: Coupling all invariants through a single phase term $n \cdot u$ .
- Separable tensor-product: Treating each invariant direction independently, allowing for anisotropic resolution.
- Note: The paper acknowledges that for non-periodic domains, harmonic bases may introduce boundary artifacts, and compares them against Chebyshev polynomials in specific benchmarks.
Regularized Regression: Once the prefactor and dictionary are fixed, the problem reduces to a linear regression for the coefficients. The authors employ Ridge regression (with cross-validation for the regularization parameter) to estimate the weights, which dampens high-frequency coefficients and improves stability.

Key Contributions
The paper's primary contributions are methodological and structural:

Formulation: It formulates dimensionally constrained surrogate modelling as a unit-equivariant regression problem determined explicitly by the dimension matrix of the variables.
Explicit Hypothesis Class: It constructs a "prefactor-times-invariants" hypothesis class where the dimensional scaling is separated from the residual dimensionless response. Unlike symbolic regression engines that search broad expression spaces, this approach restricts the search to a structured class determined a priori by dimensional admissibility.
Dictionary Analysis: It investigates finite dictionaries on invariant space, comparing one-dimensional, phase-combination, and separable tensor-product forms, and discusses the implications of periodic continuation for non-periodic physical phenomena.
Empirical Validation: It provides a systematic evaluation of the framework against unconstrained and dimensionally aware baselines using synthetic and experimental data.

Results
The method was tested on four benchmarks:

Simple Pendulum: The model successfully recovered the $L/g$ scaling with high accuracy. Crucially, the dimensionally constrained model achieved high $R^2$ with as few as 5 data points, whereas an unconstrained power-law regressor required significantly more data (approx. 60 points) to converge to the correct exponent and suffered from ill-conditioning at low sample sizes.
Black-Body Radiation (Synthetic): The framework correctly identified the physical prefactor ( $\alpha=0$ in the scaling law) across various noise levels. The results highlighted that if the harmonic dictionary lacks expressivity (low truncation), approximation errors can "leak" into the prefactor selection, biasing the dimensional scaling. High-truncation dictionaries successfully disentangled functional approximation from dimensional scaling.
Black-Body Radiation (Experimental - COBE/FIRAS): The method was applied to real spectral data, validating its ability to handle experimental noise and non-ideal data.
Double-Pendulum Lyapunov Field: This multi-invariant setting tested the dictionary geometry. The separable tensor-product basis proved superior to the phase-combination basis for capturing directional/anisotropic structures in the response surface.

Across all benchmarks, the dimensionally consistent models demonstrated improved conditioning, robustness to noise, and sample efficiency relative to unconstrained baselines.

Significance and Claims
The authors claim that their framework offers a practical, explicit, and computationally inexpensive approach to surrogate modelling for structured physical problems. By embedding dimensional constraints directly into the hypothesis class, the method avoids the need for post-hoc corrections and reduces the search space, leading to better generalization with limited data.

The paper is modest in its scope, noting that the current implementation is intended for low-dimensional invariant spaces ( $N_\Pi \le 3$ ) due to the combinatorial explosion of the prefactor grid and dictionary size in higher dimensions. The authors suggest that future work could address scalability through gradient-based optimization of exponents or sparse dictionary selection, but these are not implemented in the current study. The work positions itself not as a replacement for symbolic discovery engines (like AI Feynman) but as a complementary approach that prioritizes interpretability and strict adherence to physical laws through a fixed, inspectable approximation class.

Technical Summary: Dimensionally Consistent Surrogate Modelling through Dimensional Analysis and Harmonic Expansions

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