Dreaming up scale invariance via inverse renormalization group
This work addresses the challenge of efficiently generating statistical ensembles in physics, offering a method to encode universality in critical phenomena with minimal models, though it is incremental as it builds on existing renormalization group concepts.
The paper tackled the problem of inverting the renormalization group coarse-graining procedure to generate microscopic configurations from coarse-grained states in the two-dimensional Ising model, demonstrating that neural networks with as few as three parameters can learn to produce critical configurations that reproduce scaling behaviors like magnetic susceptibility and heat capacity.
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. 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, opening the door to efficient generative models of statistical ensembles in physics.