Data-driven discovery of self-similarity using neural networks

TL;DR

This work presents a model-free, data-driven framework to uncover self-similarity in physical systems by embedding scale-transformations into neural networks and extracting the associated power-law exponents from data. By formalizing dimensionless parameters via the Buckingham Pi theorem and employing neural networks to learn invariant combinations, the method distinguishes self-similarity of the second kind and yields data collapses even in complex, multi-parameter settings. The authors demonstrate effectiveness on synthetic and experimental data for a viscoelastic impact problem, recovering exponents consistent with theory and showing robustness to noise through regularization and bootstrapping. Extensions to multi-argument scaling functions are illustrated, including a two-argument synthetic model and a Zener viscoelastic setup, highlighting the method’s broad applicability to uncovering hidden scaling laws without presupposed governing equations.

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.

Figures (20)

• Figure 1: Hierarchical structure of similarity parameters. Physical parameters $\bm z = \left( z_1, \cdots, z_{N} \right)$ are parameters that are composed of a product of a numerical number and a physical unit. They are not invariant under the scale transformation of units. $\bm \pi = \left( \pi_1, \cdots,\pi_{N_\pi} \right)$ are referred to as similarity parameters of the first class, that are composed of physical parameters and are invariant under the scale transformations of units. $\bm Z = \left(Z_1, \cdots, Z_{\bar{N}} \right)$ are the similarity parameters of the second class, that are composed of similarity parameters of the first class and are invariant under the all the scale transformations where $\bar{N} \coloneqq N_\pi - N$. The dashed lines indicate the parameters with independent dimensions of their group (group ${\rm I}$) and the solid lines indicate the remaining parameters (group ${\rm I\space I}$).
• Figure 2: Classification of self-similar solutions. The first branch asks whether data collapse is achieved by dimensional analysis. If this is the case, the problem is self-similar solution of the first kind. If not, the problem possesses self-similar solutions of the second kind. The latter case can be further divided into two types. Type A corresponds to the cases when power-law exponents of similarity parameters are constant, while in the case of Type B power-law exponents are functions of dimensionless parameters. The present neural network approach targets the determination of power-law exponents for problems under Type A.
• Figure 3: Structure of the neural network for the case where the scaling function has single argument. In the case where the scaling function has two arguments, we should modify the structure so that the second layer has two nodes.
• Figure 4: The geometrical relation of physical parameters on the collision between a spherical impactor with radius $R$, density $\rho$, the impact velocity $v_i$, and a viscoelastic board with thickness $h$, elastic modulus $E$ and viscous coefficient $\mu$. $\delta_m$ is a maximum deformation after the impact.
• Figure 5: Synthetic training data ${\psi^{(i)}}$ and neural network predictions $\psi_{\rm NN}^{(i)}$ (without the regularization terms, $\lambda = 0$) as a function of $y^{(i)} = \hat{{\bm w}}^{(0)}\cdot {\bm x}^{(i)}$, where $\hat{{\bm w}}^{(0)}$ is given as \ref{['eq:w_hat']}. Here, $r$ is the noise strength introduced in the synthetic data $\{({\bm x}^{(i)},\psi^{(i)})\}$.

Theorems & Definitions (4)

• Proposition 2.1
• proof
• Proposition 2.2
• proof