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Similarity Learning with neural networks

Gabriel Sanfins, Fabio Ramos, Danilo Naiff

TL;DR

The paper tackles the data-driven discovery of scaling laws and symmetry groups in fluid mechanics by introducing Barenet, a neural-network framework that automatically uncovers incomplete and complete similarity from dimensionless data and derives associated Buckingham and renormalization-group transformations. It generalizes dimensionless constructions via Multi-Dimensionally Dependent Parameters (MDDP) and provides a linear-algebra backbone to compute similarity groups, enabling data-driven recovery of classical and novel scaling laws (e.g., Goldenfeld-type data collapse and Herschel–Bulkley incomplete similarity). Empirical demonstrations cover laminar Newtonian and non-Newtonian pipe flows and turbulent pipe flows, including the Nikuradse roughness data, with measured exponents and renormalization groups matching known or expected scaling. The work offers practical tools for experimental design and model reduction, and highlights future directions for integrating Bucknet with Barenet and for extending the theory to broader incomplete-similarity forms and symbolic regression-based discovery.

Abstract

In this work, we introduce a neural network algorithm designed to automatically identify similarity relations from data. By uncovering these similarity relations, our network approximates the underlying physical laws that relate dimensionless quantities to their dimensionless variables and coefficients. Additionally, we develop a linear algebra framework, accompanied by code, to derive the symmetry groups associated with these similarity relations. While our approach is general, we illustrate its application through examples in fluid mechanics, including laminar Newtonian and non-Newtonian flows in smooth pipes, as well as turbulent flows in both smooth and rough pipes. Such examples are chosen to highlight the framework's capability to handle both simple and intricate cases, and further validates its effectiveness in discovering underlying physical laws from data.

Similarity Learning with neural networks

TL;DR

The paper tackles the data-driven discovery of scaling laws and symmetry groups in fluid mechanics by introducing Barenet, a neural-network framework that automatically uncovers incomplete and complete similarity from dimensionless data and derives associated Buckingham and renormalization-group transformations. It generalizes dimensionless constructions via Multi-Dimensionally Dependent Parameters (MDDP) and provides a linear-algebra backbone to compute similarity groups, enabling data-driven recovery of classical and novel scaling laws (e.g., Goldenfeld-type data collapse and Herschel–Bulkley incomplete similarity). Empirical demonstrations cover laminar Newtonian and non-Newtonian pipe flows and turbulent pipe flows, including the Nikuradse roughness data, with measured exponents and renormalization groups matching known or expected scaling. The work offers practical tools for experimental design and model reduction, and highlights future directions for integrating Bucknet with Barenet and for extending the theory to broader incomplete-similarity forms and symbolic regression-based discovery.

Abstract

In this work, we introduce a neural network algorithm designed to automatically identify similarity relations from data. By uncovering these similarity relations, our network approximates the underlying physical laws that relate dimensionless quantities to their dimensionless variables and coefficients. Additionally, we develop a linear algebra framework, accompanied by code, to derive the symmetry groups associated with these similarity relations. While our approach is general, we illustrate its application through examples in fluid mechanics, including laminar Newtonian and non-Newtonian flows in smooth pipes, as well as turbulent flows in both smooth and rough pipes. Such examples are chosen to highlight the framework's capability to handle both simple and intricate cases, and further validates its effectiveness in discovering underlying physical laws from data.

Paper Structure

This paper contains 10 sections, 3 theorems, 106 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dimensional Analysis
  3. Similarity Beyond Buckingham
  4. Example: Laminar Newtonian Flows in Bulk and Pressure Coordinates
  5. Generalized Similarity Groups
  6. Learning similarity from data
  7. Laminar Herschel-Bulkley Flows: A non-trivial example of incomplete similarity.
  8. Conclusions
  9. Multiple Dimensionally Dependent Parameters (MDDP) Dimensionless Construction
  10. Buckingham's Similarity Group Invariance

Key Result

Lemma A.1

For each $\beta \in {\mathbb R}^l$ there is a unique $\alpha \in {\mathbb R}^m$ such that $d\left( \beta, \alpha \right)$ is dimensionless.

Figures (13)

  • Figure 1: Nikuradse's roughness data.
  • Figure 2: Nikuradse's roughness data collapsed into a single curve with similarity exponents proposed by Goldenfeld.
  • Figure 3: Schematics of the Barenet's architecture.
  • Figure 4: Laminar MVP in pressure drop coordinates.
  • Figure 5: Laminar MVP after renormalization with exponents found by our neural network.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Example : The log law
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Corollary A.3
  • proof