  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.
