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Physica Scripta

Inverse Ising problem: attention-guided physics-regularized reconstruction of corrupted spin configurations

Partha Sarathi Bishnu

Published Sep 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/aea664

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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.
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.
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.
Abstract We present an unsupervised machine learning framework for discovering phase transitions and order parameters in the two-dimensional Ising model without prior knowledge of the underlying physics. Using convolutional beta-variational autoencoders (β-VAE), we learn compressed representations that naturally separate ordered and disordered phases. Our physics-informed training procedures incorporate data augmentation respecting Ising symmetries, cosine annealing optimization, and progressive temperature curriculum learning. Experimental validation on Monte Carlo simulated configurations demonstrates automatic order parameter discovery with correlation ρ = 0.85 ± 0.04 to theoretical magnetization (95% CI [0.82, 0.88]) and critical temperature detection within 0.27% of the Onsager solution (p < 0.001). The method achieves 89% improvement over principal component analysis and 124% improvement over t-distributed stochastic neighbor embedding. Latent representations exhibit clear phase separation with physics consistency score S = 0.88 ± 0.03. Ablation studies reveal optimal performance at β = 1.0, balancing reconstruction fidelity with disentanglement. This framework enables automated identification of emergent phenomena in complex many-body systems, advancing machine learning applications in condensed matter physics.
Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena.We train a restricted Boltzmann machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods.From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system.We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near-criticality region of the Ising model.In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the renormalization group (RG) flow of the lattice model.Our results suggest an explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points.Then from the reconstructed data we deduce the critical exponent associated with the magnetization to find satisfactory agreement with the actual physical value.We assume no prior knowledge about the criticality of the system and its Hamiltonian.
Conventionally, the study of phases in statistical mechan- ics is performed with the help of random sampling tools. Among the most powerful are Monte Carlo simulations consisting of a stochastic importance sampling over state space and evaluation of estimators for physical quantities. The ability of modern machine learning techniques to classify, identify, or in- terpret massive data sets provides a complementary paradigm to the above approach to analyze the exponentially large number of states in statistical physics. In this report, it is demonstrated by application on Ising-type models that deep learning has potential wide applications in solving many-body statis- tical physics problems. In application of supervised learning, we showed that the feed-forward neural network can identify phases and phase transitions in the ferromagnetic Ising model and the convolutional neural network (CNN) is extremely powerful in classifying T = 0 and T = ∞ phases in the Ising gauge model; In application of unsupervised learning, we illustrated that a deep auto-encoder constructed by stacked restricted Boltzmann machines (RBM)\nis closely related to the renormalization group (RG) method well understood in modern physics and our reconstruction of Ising spin configurations in the ferromagnetic Ising model is similar to the hand-written digits reconstruction.
We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the three-dimensional (3D) Ising model. A 3D convolutional autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature T c = 4.5128 ( 58 ) , in excellent agreement with the known value. Additionally, we obtain an estimate of the correlation-length critical exponent, ν = 0.63 ( 27 ) , also consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.