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arXiv:2510.06508 · Machine Learning Science and Technology

Application of deep neural networks for computing the renormalization group flow of the two-dimensional ϕ 4 field theory

Yueqi Zhao, M. M. Fogler, Yi‐Zhuang You

Published Feb 23, 2026• 0 citations
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Abstract

Abstract We introduce RGFlow, a deep neural network–based real-space renormalization group (RG) framework tailored for continuum scalar field theories. Leveraging generative capabilities of flow-based neural networks, RGFlow autonomously learns real-space RG transformations from data without prior knowledge of the underlying model. In contrast to conventional approaches, RGFlow is bijective (information-preserving) and is optimized based on the principle of minimal mutual information. We demonstrate the method on two examples. The first one is a one-dimensional Gaussian model, where RGFlow is shown to learn the classical decimation rule. The second is the two-dimensional φ 4 theory, where the network successfully identifies a Wilson–Fisher-like critical point and provides an estimate of the correlation-length critical exponent.

DOI
10.1088/2632-2153/ae493b

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Abstract We introduce RGFlow, a deep neural network–based real-space renormalization group (RG) framework tailored for continuum scalar field theories. Leveraging generative capabilities of flow-based neural networks, RGFlow autonomously learns real-space RG transformations from data without prior knowledge of the underlying model. In contrast to conventional approaches, RGFlow is bijective (information-preserving) and is optimized based on the principle of minimal mutual information. We demonstrate the method on two examples. The first one is a one-dimensional Gaussian model, where RGFlow is shown to learn the classical decimation rule. The second is the two-dimensional φ 4 theory, where the network successfully identifies a Wilson–Fisher-like critical point and provides an estimate of the correlation-length critical exponent.
The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters.However, finding a systematic way to construct RG transformation for particular systems remains difficult due to the many possible choices of the weight factors in the RG procedure.Here we show, by identifying the conditional distribution in the restricted Boltzmann machine and the weight factor distribution in the RG procedure, that a valid real-space RG transformation can be learned without prior knowledge of the physical system.This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents.Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way for further interdisciplinary research.
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.
At low energies, the microscopic characteristics and changes of physical systems as viewed at different distance scales are described by universal scale invariant properties investigated by the Renormalization Group (RG) apparatus, an efficient tool used to deal with scaling problems in effective field theories. We employ an information-theoretic approach in a deep learning setup by introducing an artificial neural network algorithm to map and identify new physical degrees of freedom. Using deep learning methods mapped to a genuine field theory, we develop a mechanism capable to identify relevant degrees of freedom and induce scale invariance without prior knowledge about a system. We show that deep learning algorithms that use an RG-like scheme to learn relevant features from data could help to understand the nature of the holographic entanglement entropy and the holographic principle in context of the AdS/CFT correspondence.
On the lattice, a renormalization group (RG) flow for two-dimensional partition functions expressed as a tensor network can be obtained using the tensor network renormalization (TNR) algorithm [G. Evenbly, G. Vidal, Phys. Rev. Lett. 115 (18), 180405 (2015)]. In this work we explain how to extend TNR to field theories in the continuum. First, a short-distance length scale 1/Λ1/Λ1/Λ is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Λ1/Λ1/Λ. An infinitesimal coarse-graining step is then generated by the combined action of a rescalingrescalingrescaling operator LLL and a disentanglingdisentanglingdisentangling operator KKK that implements a quasi-local field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory.
We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these 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.