Similarity Learning with neural networks

TL;DR

This paper presents a neural network method to automatically discover similarity relations and underlying physical laws from data, with applications demonstrated in fluid mechanics including laminar and turbulent flows.

Contribution

It introduces a neural network algorithm for identifying similarity relations and a linear algebra framework to derive associated symmetry groups, advancing data-driven discovery of physical laws.

Findings

Successfully identified similarity relations in fluid flow data.

Derived symmetry groups corresponding to physical laws.

Validated approach on complex fluid mechanics examples.

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.