Unlocking Hidden Patterns: A Guide to Dimensionless Learning

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Author: Denis Avetisyan

This review explores how combining physics-inspired scaling analysis with machine learning can reveal fundamental relationships within complex datasets.

As noise increases in experimental data, learned dimensionless group coefficients demonstrably drift from their true values-specifically, away from the expected $ [1,1,0]^{T}$-highlighting a fundamental challenge in identifying these groups from noisy observations, even with uncertainty captured across twenty ensemble runs.
As noise increases in experimental data, learned dimensionless group coefficients demonstrably drift from their true values-specifically, away from the expected $ [1,1,0]^{T}$-highlighting a fundamental challenge in identifying these groups from noisy observations, even with uncertainty captured across twenty ensemble runs.

A comprehensive framework for discovering scaling laws from data, addressing the impact of noise and complexity in physical modeling and neural networks.

Despite the enduring power of dimensional analysis in physics and engineering, extracting compact physical laws directly from experimental data remains a significant challenge. This tutorial, ‘A Tutorial on Dimensionless Learning: Geometric Interpretation and the Effect of Noise’, introduces a data-driven framework that bridges classical techniques with modern machine learning to discover dimensionless numbers and scaling laws. By combining dimensional analysis with neural networks-and incorporating a novel regularization strategy-the method reveals underlying physical relationships robust to experimental uncertainties. Can this approach ultimately automate the process of physical modeling and accelerate scientific discovery across diverse fields?

The Essence of Reduction: Unveiling Hidden Structure

Many physical systems, from fluid dynamics to heat transfer, are fundamentally governed not by the absolute values of physical quantities, but by dimensionless groups – combinations of variables that yield a pure number. This often obscures the underlying relationships between these quantities, complicating analysis and making it difficult to predict system behavior. For example, understanding airflow over an aircraft wing isn’t about the absolute speed of the air or the wing’s length, but rather the ratio of these – the Reynolds number. While seemingly abstract, these dimensionless groups represent the true governing parameters, and a failure to recognize them can lead to inaccurate models and scaling issues. Consequently, researchers frequently encounter challenges in simplifying complex problems and uncovering universal scaling laws due to the hidden influence of these dimensionless quantities, requiring careful consideration of their role in any physical analysis.

Historically, determining the dimensionless groups governing a physical system has relied heavily on techniques like the Buckingham Pi Theorem. While fundamentally sound, these methods often demand substantial computational resources, particularly when dealing with systems characterized by a large number of variables. The process isn’t simply a matter of inputting parameters; it requires meticulous analysis, careful selection of fundamental dimensions, and often, considerable expertise to navigate potential ambiguities and ensure the correct identification of independent dimensionless groups. This complexity can become a significant barrier to entry for researchers lacking specialized training, and it can substantially slow down the pace of discovery, especially in fields where rapid prototyping and iterative modeling are essential. Consequently, alternative approaches that streamline this process and reduce the need for extensive manual intervention are highly sought after.

The pursuit of simplified models in complex physical systems hinges on identifying relevant dimensionless groups, which effectively reduce the number of independent variables and reveal underlying scaling laws. A novel approach leverages dimensionality reduction techniques – specifically Principal Component Analysis (PCA) and Sparse Identification of Relevant features (SIR) – to efficiently estimate the number of these dominant dimensionless groups. By aiming to capture at least 0.75 of the explained variance in the data, this method offers a robust means of discerning the most influential parameters governing a system’s behavior. This allows researchers to move beyond cumbersome traditional methods and focus analytical efforts on the essential relationships, ultimately leading to more tractable models and a deeper understanding of universal phenomena across diverse scientific disciplines.

Correlation analysis revealed that while individual dimensionless groups show scattered relationships with the output, a combined group successfully reduced dimensionality from seven inputs to a single variable, demonstrating a polynomial relationship between the output and the new dimensionless group.
Correlation analysis revealed that while individual dimensionless groups show scattered relationships with the output, a combined group successfully reduced dimensionality from seven inputs to a single variable, demonstrating a polynomial relationship between the output and the new dimensionless group.

Data-Driven Insight: Identifying Key Dimensionless Groups

Dimensionality reduction techniques, specifically Sliced Inverse Regression (SIR) and Principal Component Analysis (PCA), are utilized to determine the essential dimensionless groups governing a system. SIR operates by regressing the input variables onto the output variables, identifying directions in the input space that maximize the variance of the predicted output; these directions correspond to the dominant dimensionless groups. PCA, conversely, focuses on identifying orthogonal components in the input data that capture the maximum variance, effectively reducing the dimensionality while preserving key information. Both methods rely on identifying combinations of physical parameters that explain the largest portion of the system’s behavior, thus revealing the governing dimensionless parameters without requiring a priori knowledge of their form. The resulting dimensionless groups are expressed as ratios of forces, allowing for effective scaling and generalization of experimental or simulation results.

Dimensionality reduction techniques, such as Sliced Inverse Regression (SIR) and Principal Component Analysis (PCA), function by analyzing data to determine which linear combinations of input variables account for the greatest proportion of the observed variance in the output. These methods do not require a priori knowledge of the governing physics; instead, they statistically identify relationships within the data itself. The resulting combinations of variables effectively represent the key dimensionless parameters influencing the system, as these parameters are responsible for capturing the most significant trends and behaviors. By focusing on the directions of greatest variance, these techniques isolate the essential parameters from noise and redundancy, providing a data-driven means of model reduction and simplification.

The Dimension Matrix, constructed from the input data, establishes the parameter space for identifying dominant dimensionless groups. This matrix, with dimensions representing the number of data samples and the number of input parameters, is central to the process of Null Space analysis. The Null Space, derived from the Dimension Matrix, contains vectors representing linear combinations of the input parameters that explain minimal variance in the data. These vectors are then used as Basis Vectors to construct the dimensionless groups. To ensure the robustness and reliability of the identified groups, a minimum explained variance ratio of 0.75 is required, indicating that at least 75% of the total data variance is captured by the selected Basis Vectors and therefore adequately represented by the identified dimensionless parameters.

Correlation analysis reveals that while individual dimensionless groups fail to capture output structure, a two-dimensional manifold defined by the target dimensionless numbers Π₁ and Π₂ effectively reduces data dimensionality and reveals underlying relationships.
Correlation analysis reveals that while individual dimensionless groups fail to capture output structure, a two-dimensional manifold defined by the target dimensionless numbers Π₁ and Π₂ effectively reduces data dimensionality and reveals underlying relationships.

Neural Networks: Mapping Complexity to Prediction

Neural networks are employed to model the functional mapping between dimensionless groups – derived through dimensional analysis to reduce problem complexity – and a target output variable. This approach allows the network to approximate potentially non-linear relationships without requiring explicit analytical solutions. The dimensionless groups serve as inputs to the network, while the output variable represents the predicted value. Training involves iteratively adjusting the network’s internal weights based on a dataset of known input-output pairs, enabling it to learn the underlying relationship and generalize to new, unseen data. The effectiveness of this method relies on the quality and representativeness of the training data and the network’s architecture, which must be sufficient to capture the complexity of the system being modeled.

Model training utilizes the Adam optimizer, a stochastic gradient descent method that computes adaptive learning rates for each parameter. This optimization process aims to minimize the Mean Squared Error ($MSE$) between the network’s predicted outputs and the actual observed values. The $MSE$ is calculated as the average of the squared differences between predicted and actual values, providing a quantifiable measure of model accuracy. Minimizing this error function iteratively adjusts the network’s weights and biases, improving its predictive capability on the training dataset.

Quantization Regularization was implemented to improve the interpretability of the neural network and mitigate overfitting during training. This technique encourages the network to produce sparse solutions by penalizing large weights, effectively simplifying the model and reducing its reliance on individual features. Empirical results demonstrated that a regularization strength of 0.05 yielded the optimal balance between model accuracy and sparsity, resulting in both improved predictive performance and increased diversity in the learned parameters. This value was determined through hyperparameter tuning, evaluating performance on a validation dataset to minimize the $Mean Squared Error$ between predicted and actual values.

Quantization regularization effectively constrains learned coefficient vectors to a discrete set of solutions-clustering at [-1,-1,0], [-0.5,-0.5,0], and [0.5,0.5,0]-enhancing interpretability of dimensionless groups without compromising predictive performance.
Quantization regularization effectively constrains learned coefficient vectors to a discrete set of solutions-clustering at [-1,-1,0], [-0.5,-0.5,0], and [0.5,0.5,0]-enhancing interpretability of dimensionless groups without compromising predictive performance.

Beyond Prediction: Towards a More Principled Understanding

This automated approach to model simplification offers substantial benefits across a wide spectrum of scientific inquiry. By systematically reducing the number of parameters needed to accurately represent a system, researchers can drastically lower computational demands without sacrificing predictive power. This is particularly valuable when dealing with complex simulations-such as those found in climate modeling, fluid dynamics, or materials science-where even minor reductions in computational cost can enable exploration of larger parameter spaces or longer timescales. The methodology’s adaptability extends beyond mere efficiency; it facilitates a clearer understanding of the core mechanisms governing a phenomenon, ultimately leading to more robust and generalizable models capable of accurately forecasting behavior in diverse scenarios. Consequently, this framework empowers scientists to tackle previously intractable problems and accelerate discovery in fields reliant on complex simulations and data analysis.

The automated identification of relevant dimensionless groups represents a significant advancement in understanding complex physical phenomena. By distilling governing equations into these non-dimensional forms, researchers can move beyond simply predicting outcomes and begin to elucidate the fundamental mechanisms at play. This process effectively reduces the dimensionality of a problem, highlighting the key parameters that control system behavior and revealing hidden relationships between variables. Consequently, the discovery of new scaling laws – mathematical relationships that describe how a system changes with scale – is greatly accelerated, offering a more generalized and insightful understanding that transcends specific experimental conditions. This capability has broad implications, enabling scientists to develop more robust models, design more efficient experiments, and ultimately, gain a deeper appreciation for the underlying principles governing the natural world.

Continued development centers on broadening the applicability of this methodology to encompass more complex, higher-dimensional datasets – a significant challenge given the exponential increase in computational demands. To address this, researchers are actively exploring strategies for incorporating domain-specific knowledge, effectively acting as prior constraints to guide the identification of the most relevant dimensionless parameters and accelerate the discovery process. The robustness of the current framework is already demonstrated through ensemble training, utilizing between 5 and 20 independent runs to capture solution diversity and ensure reliable performance across varying initial conditions and data configurations. This approach not only enhances the accuracy of identified scaling laws but also provides a valuable measure of confidence in the resulting simplified models, paving the way for more efficient and insightful scientific exploration.

This learning pipeline transforms high-dimensional data through a series of modules to establish concise, scalable relationships.
This learning pipeline transforms high-dimensional data through a series of modules to establish concise, scalable relationships.

The pursuit of understanding, as demonstrated in this exploration of dimensionless learning, often benefits from a stripping away of unnecessary complication. The article champions a method for discerning fundamental scaling laws from data-a process inherently reliant on identifying core relationships unburdened by specific units or scales. This echoes the sentiment of Henri Poincaré, who observed, “It is through science that we learn to control the forces of nature.” The work presented systematically reduces complexity, revealing underlying principles applicable across varied physical modeling contexts, and in doing so, offers a pathway to more robust and generalizable machine learning models. The focus on noise reduction and accessibility further exemplifies this commitment to clarity and reveals the essential structure within the data.

What Remains?

The pursuit of dimensionless learning, as presented, clarifies a path – but every clearing reveals a deeper wood. The framework offers a method for extracting scaling from data, a laudable reduction of complexity. Yet, the true limitation isn’t in the technique itself, but in the data. Noise, predictably, remains a persistent adversary. Mitigation strategies are presented, but the inherent ambiguity of real-world observation introduces a fundamental, irreducible uncertainty. The signal, one suspects, is always weaker than the static.

Future work will inevitably focus on more robust noise handling, and perhaps, adaptive dimensional analysis – a system that refines its initial assumptions based on observed data. More intriguing, however, is the question of interpretability. Discovering a scaling law is one thing; understanding why it exists is another. The current approach excels at the former, but offers little guidance on the latter. True progress demands a bridge between data-driven discovery and physical insight.

Ultimately, the value of this work lies not in its completeness, but in its precision. It does not offer a universal solution, but a rigorously defined set of tools. And in a field often obscured by algorithmic complexity, a little clarity – a subtraction of unnecessary layers – is a rare and welcome thing.

Original article: https://arxiv.org/pdf/2512.15760.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-20 18:59