Abstract Using methods of statistical physics, we analyse the error of learning couplings in large Ising models from independent data (the inverse Ising problem). We concentrate on learning based on local cost functions, such as the pseudo-likelihood method for which the couplings are inferred independently for each spin. Assuming that the data are generated from a true Ising model, we compute the reconstruction error of the couplings using a combination of the replica method with the cavity approach for densely connected systems. We show that an explicit estimator based on a quadratic cost function achieves minimal reconstruction error, but requires the length of the true coupling vector as prior knowledge. A simple mean field estimator of the couplings which does not need such knowledge is asymptotically optimal, i.e. when the number of observations is much larger than the number of spins. Comparison of the theory with numerical simulations shows excellent agreement for data generated from two models with random couplings in the high temperature region: a model with independent couplings (Sherrington–Kirkpatrick model), and a model where the matrix of couplings has a Wishart distribution.
Abstract Using methods of statistical physics, we analyse the error of learning couplings in large Ising models from independent data (the inverse Ising problem). We concentrate on learning based on local cost functions, such as the pseudo-likelihood method for which the couplings are inferred independently for each spin. Assuming that the data are generated from a true Ising model, we compute the reconstruction error of the couplings using a combination of the replica method with the cavity approach for densely connected systems. We show that an explicit estimator based on a quadratic cost function achieves minimal reconstruction error, but requires the length of the true coupling vector as prior knowledge. A simple mean field estimator of the couplings which does not need such knowledge is asymptotically optimal, i.e. when the number of observations is much larger than the number of spins. Comparison of the theory with numerical simulations shows excellent agreement for data generated from two models with random couplings in the high temperature region: a model with independent couplings (Sherrington–Kirkpatrick model), and a model where the matrix of couplings has a Wishart distribution.
Inverse Ising inference allows pairwise interactions of complex binary systems to be reconstructed from empirical correlations. Typical estimators used for this inference, such as pseudo-likelihood maximization (PLM), are biased. Using the Sherrington-Kirkpatrick model as a benchmark, we show that these biases are large in critical regimes close to phase boundaries, and they may alter the qualitative interpretation of the inferred model. In particular, we show that the small-sample bias causes models inferred through PLM to appear closer to criticality than one would expect from the data. Data-driven methods to correct this bias are explored and applied to a functional magnetic resonance imaging data set from neuroscience. Our results indicate that additional care should be taken when attributing criticality to real-world data sets.
We consider the problem of predicting the spin states in a kinetic Ising model when spin trajectories are observed for only a finite fraction of sites. In a Bayesian setting, where the probabilistic model of the spin dynamics is assumed to be known, the optimal prediction can be computed from the conditional (posterior) distribution of unobserved spins given the observed ones. Using the replica method, we compute the error of the Bayes optimal predictor for parallel discrete time dynamics in a fully connected spin system with non-symmetric random couplings. The results, exact in the thermodynamic limit, agree very well with simulations of finite spin systems.
Bayesian inference with a mismatched model can develop an overconfident phase, characterized by posterior agreement without faithful reconstruction of the true state. We study Bayesian inference from noisy bond measurements of an Ising model in its paramagnetic phase, with true inverse temperature β and measurement-noise strength Δ, while inference assumes corresponding parameters βQ and ΔQ. Using replica field theory, we analytically show that the overconfidence phase can occur when inference underestimates the measurement noise (ΔQ<Δ) or assumes a lower prior temperature than the true one (βQ>β). An exact change of variables maps the problem to a correlated random-bond Ising model, where the informative, uninformative, and overconfident phases correspond to ferromagnetic, paramagnetic, and spin-glass phases, respectively. We further perform Monte Carlo simulations of the mapped model in three dimensions and establish a finite overconfident region between the informative and uninformative phases, consistent with our analytical prediction. Finally, we show that the true model parameters can be inferred directly from the measurement records, providing a natural route to calibrating the inference 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.
When independent learners are trained on data from the same disordered system, finite and noisy samples can lead them to different descriptions of the underlying physics. We quantify the reproducibility of such descriptions through consensus, the degree of agreement among independently trained models. In a teacher-student framework on the two-dimensional random-bond Ising model, an ensemble of students each infers a single uniform effective coupling from a finite set of equilibrium configurations drawn from one quenched bond realization. Aggregating the inferred couplings across students and realizations yields the consensus distribution, whose width quantifies the agreement among learners. The variance of this distribution separates exactly into a finite-sampling contribution, bounded below by inverse of Fisher information, and a quenched-disorder contribution. Sum of these two variance is minimized near critical region, so the consensus width has a minimum near the pseudocritical temperature that sharpens with increasing disorder. The two minima arise from different mechanisms: the sampling term inherits the critical peak of the Fisher information, while the disorder term is temperature-independent at leading order and acquires its critical minimum only at fourth order in the disorder strength, through the covariance between the linear and cubic responses to the bond disorder. Because the students fit a uniform coupling to a heterogeneous lattice, the inferred coupling also carries a misspecification bias that survives in the infinite-data limit. This bias displaces the minimum of the mean-squared error above the consensus minimum, so the temperature at which independent learners agree most closely is not the temperature at which their shared answer is most faithful.
Recovering microscopic couplings directly from data provides a route to solving the inverse problem in statistical field theories, one that complements the traditional-often computationally intractable-forward approach of predicting observables from an action or Hamiltonian. Here, we propose an approach for the inverse problem that uniformly accommodates systems with discrete, continuous, and hybrid variables. We demonstrate accurate parameter recovery in several benchmark systems-including Wegner's Ising gauge theory, φ4 theory, Schwinger and Sine-Gordon models, and mixed spin-gauge systems, and show how iterating the procedure under coarse-graining reconstructs full non-perturbative renormalization-group flows. This gives direct access to phase boundaries, fixed points, and emergent interactions without relying on perturbation theory. We also address a realistic setting where full gauge configurations may be unavailable, and reformulate learning algorithms for multiple field theories so that they are recovered directly using observables such as correlations from scattering data or quantum simulators. We anticipate that our methodology will find widespread use in practical learning of field theories in strongly coupled regimes where analytical tools might fail.