Dreaming up scale invariance via inverse renormalization group

TL;DR

This paper demonstrates that minimal neural networks can probabilistically invert the RG coarse-graining process in the 2D Ising model, capturing critical scale-invariant distributions with very few parameters.

Contribution

It shows that even extremely simple neural networks can generate critical configurations that reproduce key RG properties and scale invariance in the Ising model.

Findings

Neural networks with as few as three parameters can generate critical configurations.

Generated configurations reproduce scaling behavior of observables like susceptibility and heat capacity.

Increasing network complexity does not significantly improve the inversion performance.

Abstract

We explore how minimal neural networks can invert the renormalization group (RG) coarse-graining procedure in the two-dimensional Ising model, effectively ``dreaming up'' microscopic configurations from coarse-grained states. This task - formally impossible at the level of configurations - can be approached probabilistically, allowing machine learning models to reconstruct scale-invariant distributions without relying on microscopic input. We demonstrate that even neural networks with as few as three trainable parameters can learn to generate critical configurations, reproducing the scaling behavior of observables such as magnetic susceptibility, heat capacity, and Binder ratios. A real-space renormalization group analysis of the generated configurations confirms that the models capture not only scale invariance but also reproduce nontrivial eigenvalues of the RG transformation. While the inversion is necessarily imperfect, these minimal models robustly reproduce the RG-relevant structure of the critical distribution. Surprisingly, we find that increasing network complexity by introducing multiple layers offers no significant benefit. These findings suggest that simple local rules, akin to those generating fractal structures, are sufficient to encode the universality of critical phenomena, creating an opportunity for efficient generative models of statistical ensembles in physics.