Data-driven discovery of self-similarity using neural networks

TL;DR

This paper introduces a neural network method to discover self-similarity in physical data without relying on predefined models, enabling the identification of underlying scale-invariance properties.

Contribution

It presents a novel neural network approach that directly uncovers self-similarity and power-law exponents from observed data, bypassing traditional model-based methods.

Findings

Successfully identifies self-similarity in synthetic data

Validates approach with experimental physical data

Extracts power-law exponents indicating scale invariance

Abstract

Finding self-similarity is a key step for understanding the governing law behind complex physical phenomena. Traditional methods for identifying self-similarity often rely on specific models, which can introduce significant bias. In this paper, we present a novel neural network-based approach that discovers self-similarity directly from observed data, without presupposing any models. The presence of self-similar solutions in a physical problem signals that the governing law contains a function whose arguments are given by power-law monomials of physical parameters, which are characterized by power-law exponents. The basic idea is to enforce such particular forms structurally in a neural network in a parametrized way. We train the neural network model using the observed data, and when the training is successful, we can extract the power exponents that characterize scale-transformation symmetries of the physical problem. We demonstrate the effectiveness of our method with both synthetic and experimental data, validating its potential as a robust, model-independent tool for exploring self-similarity in complex systems.