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Physical review. E

Temperature-conditioned deep generative framework for scalable Ising spin configuration synthesis with physics-informed constraints

Abhishek Kumar, Partha Sarathi Bishnu, Debabrata Deb

Published Aug 29, 2025• 2 citations
Publisher

Abstract

This study presents a temperature-conditioned deep generative approach for efficiently generating spin configurations of the two-dimensional (2D) Ising model. The method employs a convolutional generator network, conditioned on spin states, temperature embeddings, and random noise, to generate thermodynamically accurate spin configurations on a 2D square lattice. Physics-based constraints, derived from magnetization, energy, and nearest-neighbor correlations obtained via Monte Carlo simulations, are integrated as auxiliary loss terms during training to ensure physical consistency. The model accurately captures phase transition behaviors near the critical temperature and supports the scalable generation of synthetic configurations for further analysis. Investigations across lattice sizes L=16, 32, and 64 demonstrate close alignment with Monte Carlo baselines, with generated configurations achieving mean absolute errors as low as 0.0016 for magnetization, 0.0063 for energy, and 0.0016 for correlation in the L=64 case. Phase classification using standard machine learning models further validates the quality of the generated data, with ensemble classifiers yielding F1 scores above 0.99. These findings highlight the proposed framework as a scalable, computationally efficient solution for generating spin configurations in an Ising model.

DOI
10.1103/7qbv-4y2s

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This study presents a temperature-conditioned deep generative approach for efficiently generating spin configurations of the two-dimensional (2D) Ising model. The method employs a convolutional generator network, conditioned on spin states, temperature embeddings, and random noise, to generate thermodynamically accurate spin configurations on a 2D square lattice. Physics-based constraints, derived from magnetization, energy, and nearest-neighbor correlations obtained via Monte Carlo simulations, are integrated as auxiliary loss terms during training to ensure physical consistency. The model accurately captures phase transition behaviors near the critical temperature and supports the scalable generation of synthetic configurations for further analysis. Investigations across lattice sizes L=16, 32, and 64 demonstrate close alignment with Monte Carlo baselines, with generated configurations achieving mean absolute errors as low as 0.0016 for magnetization, 0.0063 for energy, and 0.0016 for correlation in the L=64 case. Phase classification using standard machine learning models further validates the quality of the generated data, with ensemble classifiers yielding F1 scores above 0.99. These findings highlight the proposed framework as a scalable, computationally efficient solution for generating spin configurations in an Ising model.
Abstract Objective. Reconstructing clean spin configurations from corrupted or partially observed lattice data is an important inverse problem in statistical physics and scientific machine learning. This paper proposes an attention-guided physics -regularized deep learning framework to reconstruct corrupted two-dimensional Ising spin configurations. Approach. The proposed framework receives a degraded spin lattice together with its corresponding observation mask and jointly reconstructs the unknown spin configuration while estimating its global thermodynamic state. A convolutional encoder–decoder with residual connections and spatial attention is employed to recover the underlying clean spin field. Physical consistency is enforced during training through a composite objective combining reconstruction fidelity with constraints on energy ( E ), absolute magnetization ( | M | ), second-neighbor correlation ( C 2 ) at lattice separation r l = 2 , and binary spin consistency, together with temperature regression and phase classification. Main results. Experiments on Monte Carlo generated datasets for lattice sizes L show that the proposed model achieves a high reconstruction accuracy of approximately 0.88 , with an observed-spin accuracy around 0.90 and a missing-spin accuracy around 0.83 . The model also attains a low temperature prediction error, with the mean absolute error decreasing from 0.197 ( L = 32 ) to 0.088 ( L = 128 ), together with strong phase classification performance ranging from 0.985 to 0.999 . Significance. The reconstructed configurations preserve key thermodynamic observables across temperature, including magnetization, energy, correlation, and heat capacity, demonstrating that the framework successfully maintains physical consistency while performing inverse reconstruction from corrupted Ising data.
Abstract In this work, we explore the statistical system's configuration generation with generative models from Deep Learning to go beyond conventional Monte Carlo methods. Specifically, we devise a conditional generative adversarial network (cGAN) for the Ising system spin configuration generation, and we demonstrate it's working outside the training range of temperature for the ensemble of configurations. Being different from the original GAN design, we add a further recognizer network for the constraint on the conditional parameters (in our case temperature) and also provide a diversity benefit for the generative model. We showed that the newly proposed cGAN here can learn the distribution of the Ising model for different temperatures, which can generate spin configuration efficiently with correct (within a probability distribution) temperature estimation for the microscopic configurations. Moreover, without information about criticality provided inside the training data set, it is also shown that the developed cGAN can generate Ising spin configurations around the phase transition point with the order parameter (mean magnetization) reasonably well-match to the conventional MCMC simulation but with parallel sampling advantage. We also compared typical spin configurations from cGAN with specified conditional temperature to be the critical temperature with samples simulated by MCMC, which visibly is not distinguishable. This thus can possibly help to avoid critical slowing-down as shown in the traditional Monte Carlo method.
Quantum spin networks represent systems in which quantum spins are arranged on a topological lattice, where the nature of spin interactions and the influence of external magnetic fields can lead to complex collective behaviors. The Hamiltonian governing such systems determines the energy landscape and phase transitions under various thermodynamic conditions. In this paper, we focus on the two-dimensional Ising spin model with periodic boundary conditions subjected to a uniform external magnetic field. Initially, we employ the Metropolis Monte Carlo (MP-MN) algorithm to simulate the system and identify its phase transitions at different magnetic field strengths. The emergence of ordered and disordered phases in response to thermal fluctuations and magnetic interactions is systematically analyzed. Subsequently, we explore the application of convolutional neural networks (CNNs), a powerful class of deep learning models, to detect and classify the phases of the Ising model based on spin configurations generated at a fixed temperature. The CNN is trained using labeled data representing different magnetic field values, and its performance in phase prediction is quantitatively evaluated. The results demonstrate that CNNs can successfully learn complex spin patterns and provide accurate classification of spin phases, highlighting their potential for analyzing phase transitions in statistical physics models.
This paper presents the investigation of convolutional neural network (CNN) prediction successfully recognizing the temperature of the nonequilibrium phase transitions in two-dimensional (2D) Ising spins on a square lattice. The model uses image snapshots of ferromagnetic 2D spin configurations as an input shape to provide the average output predictions. By considering supervised machine learning techniques, we perform Metropolis Monte Carlo (MC) simulations to generate the configurations. In the equilibrium Ising model, the Metropolis algorithm respects detailed balance condition (DBC), while its nonequilibrium version violates DBC. Violating the DBC of the algorithm is characterized by a parameter −8<ε<8. We find the exact result of the transition temperature Tc(ε) in terms of ε. If we set ε=0, the usual single spin-flip algorithm can be restored, and the equilibrium configurations generated with such a set up are used to train our model. For ε≠0, the system attains the nonequilibrium steady states (NESS), and the modified algorithm generates NESS configurations (test dataset). The trained model is successfully tested on the test dataset. Our result shows that CNN can determine Tc(ε≠0) for various ε values, consistent with the exact result.
In this work, we study generative adversarial networks (GANs) as a tool to learn the distribution of spin configurations and to generate samples, conditioned on external tuning parameters or other quantities associated with individual configurations. For concreteness, we focus on two examples of conditional variables---the temperature of the system and the energy of the samples. We show that temperature-conditioned models can not only be used to generate samples across thermal phase transitions, but also be employed as unsupervised indicators of transitions. To this end, we introduce a GAN-fidelity measure that captures the model’s susceptibility to external changes of parameters. The proposed energy-conditioned models are integrated with Monte Carlo simulations to perform over-relaxation steps, which break the Markov chain and reduce auto-correlations. We propose ways of efficiently representing the physical states in our network architectures, e.g., by exploiting symmetries, and to minimize the correlations between generated samples. A detailed evaluation, using the two-dimensional XY model as an example, shows that these incorporations bring in considerable improvements over standard machine-learning approaches. We further study the performance of our architectures when no training data is provided near the critical region.
Machine learning is becoming widely used in condensed matter physics. Inspired by the concept of image super-resolution, we propose a method to increase the size of lattice spin configurations using deep convolutional neural networks. Through supervised learning on Monte Carlo (MC) generated spin configurations, we train networks that invert real-space renormalization decimations. We demonstrate that super-resolution can reproduce thermodynamic observables that agree with MC calculations for the one- and two-dimensional Ising model at various temperatures. We find that it is possible to predict thermodynamic quantities for lattice sizes larger than those used in training by extrapolating the parameters of the network. We use this method to compute the critical exponents of the 2D Ising model, finding good agreement with theory.