# Data-driven discovery of self-similarity using neural networks

arXiv: https://arxiv.org/abs/2406.03896 · HTML: https://mlsanity.com/paper/2406.03896/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 6 Jun 2024.

## Builds on (most cited first, 5 of 5)

- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [Symmetry meets AI](https://mlsanity.com/paper/2103.06115/) (2021)

## Built on by (most cited first, 1 of 1)

- [Hierarchical Dimensionless Learning (Hi-π): A physics-data hybrid-driven approach for discovering dimensionless parameter combinations](https://mlsanity.com/paper/2507.18332/) (2025)

## Abstract (arXiv)

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.