🤖 AI Summary
This work addresses the bias introduced by prior model dependence in discovering self-similarity in complex physical systems. We propose a model-free, data-driven symmetry learning framework. Its core innovation is the explicit incorporation of power-law structure—i.e., parameterized scale-invariance—as an architectural inductive bias into neural networks, achieved via structured embedding and power-law parameterization to directly learn the governing scaling exponents. The method integrates supervised and unsupervised learning with scale-invariance constraints for optimization. It accurately infers scaling exponents on both synthetic and real experimental datasets, demonstrating robustness. To our knowledge, this is the first approach to automatically discover scaling laws and underlying conserved structures solely from observational data, enabling interpretable modeling of symmetries and conservation laws in complex systems.
📝 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.