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arXiv:2506.04016 · Physical review. E

Dreaming up scale invariance via inverse renormalization group

Adam Rançon, Ulysse Rançon, Tomislav Ivek, Ivan Balog

Published Apr 2, 2026• 1 citation
arXivHTMLPDF

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.

DOI
10.1103/3njc-5tlx

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Machine learning offers a possible route to data-driven real-space renormalization when the relevant observables are nonlocal and difficult to prescribe explicitly. We explore this idea for two-dimensional site percolation developping a supervised, scale-shared neural architecture. The model recursively applies the same learned coarse-graining rule across scales, producing a latent field from which the crossing probability is predicted, while a corresponding fine-graining decoder reconstructs the largest-cluster mask. Trained only on small lattices, the model extrapolates to substantially larger systems, recovers the spanning cluster with high fidelity, and produces observables obeying the expected finite-size scaling near the critical point. We observe that to get such performance it is key that the learned latent representation exhibits critical fluctuations and scale-dependent flows consistent with the renormalization-group structure of percolation.
We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers.
The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent ν. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.
Abstract We develop a machine-learning renormalization group (MLRG) algorithm to explore and analyze many-body lattice models in statistical physics. Using the representation learning capability of generative modeling, MLRG automatically learns the optimal renormalization group (RG) transformations from self-generated spin configurations and formulates RG equations without human supervision. The algorithm does not focus on simulating any particular lattice model but broadly explores all possible models compatible with the internal and lattice symmetries given the on-site symmetry representation. It can uncover the RG monotone that governs the RG flow, assuming a strong form of the c-theorem. This enables several downstream tasks, including unsupervised classification of phases, automatic location of phase transitions or critical points, controlled estimation of critical exponents, and operator scaling dimensions. We demonstrate the MLRG method in two-dimensional lattice models with Ising symmetry and show that the algorithm correctly identifies and characterizes the Ising criticality.
Numerical simulations of critical lattice systems are fundamentally limited by critical slowing down, as long-range correlations are typically established through slow temporal equilibration. A physically constrained generative framework that replaces temporal relaxation with a spatial projection mechanism for critical systems is proposed. Using the two-dimensional Ising model at criticality as a benchmark, we introduce an energy-constrained kernel that synthesizes large-scale configurations from compact equilibrated seeds by enforcing Hamiltonian-level observables. The generated configurations are projected onto the nearest-neighbor energy manifold, ensuring thermodynamic consistency while retaining universal critical properties. We show that the resulting configurations reproduce scale-invariant spin correlations, Binder cumulants, and isotropic structure factors for lattice sizes exceeding 10,000, without iterative Monte Carlo equilibration. While not a strict renormalization group transformation, and motivated by renormalization ideas, the method provides a practical inverse mapping that retains universal features of criticality and enables efficient GPU-parallel generation of ultra-large critical ensembles.