Out-of-distribution Neural Inference in Dynamical Ising Models
Authors: Yuan-Bin Zhu, Shuang Qiao, Shi-Ju Ran
Organizations: Center for Quantum Physics and Intelligent Sciences, Department of Physics, Capital Normal University, Beijing 10048, China
Abstract
Neural networks are increasingly used to infer hidden physical structure from dynamical observations, yet it remains unclear whether their out-of-distribution performance reflects transferable physical rule learning. We address this question in a controlled inverse problem: reconstructing interaction graphs of a kinetic Ising model from Glauber magnetization trajectories. Across convolutional, graph, Transformer, and hybrid architectures, we find that data-driven training produces distinct and reproducible statistical strategies under topology and temperature shifts. Edge-population diagnostics reveal that Transformer-based models tend to preserve the link density of the training ensemble, whereas convolutional models can collapse toward sparse- or no-link predictions that appear out-of-distribution stable by exploiting the majority no-link class. Thus, high in-distribution accuracy and apparent out-of-distribution robustness do not necessarily imply a learned dynamics-to-structure rule. Instead, neural reconstruction can be governed by architecture-dependent statistical priors. Our results identify a concrete failure mode of standard data-driven learning in physical inverse problems and motivate rule-guided principles for machine-learning-assisted scientific discovery.
Thermodynamic computing devices based on the Ising model show great promise for low-power AI inference and edge computing, but scalable methods for training large models for such hardware remain limited. Prior theory shows that the time-averaged behavior of high-temperature Gibbs-sampled Ising systems can implement feed-forward neural inference. We turn this theoretical correspondence into a scalable and purely backpropagation-based algorithm for training deep convolutional networks for thermodynamic inference on Ising machine hardware. Our image classification models achieve accuracies of 94.9% on CIFAR-10 and 76.0% on CIFAR-100 under binary Gibbs sampling. We then develop and experimentally validate a mathematical theory relating inference cost to accuracy and controlling autocorrelation times. Subsequently, we calculate asymptotic results showing that inference cost is bounded by a well-controlled tradeoff with performance and exhibit algorithms for computing optimal inference schedules. Finally, we discuss implications for hardware development and the future of high-temperature thermodynamic AI models.
We study how unsupervised autoencoders trained on microscopic spin configurations from the Ising model learn macroscopic, theory-relevant variables underlying the data-generating process. Without embedding domain knowledge, we mimic a typical discovery setting: We quantify learning across multiple spatial (coarse-graining) scales and reveal two distinct dynamical regimes controlled by main hyperparameters (model depth, width, and learning rate) -- a magnetization-dominated regime and an energy-dominated regime characterized by trade-offs in their representation quality. The first regime is a transitory state exhibiting dynamical scaling and fluctuations that follow an ordering-to-scale; the second gradually shifts resolution towards smaller scales relevant for the energy representation. Deep models trained at moderate and fast rates become arrested before reaching these regimes. With a novel analysis of recursive-dynamic trajectories, we demonstrate that prediction errors induce flow fields that produce a common trajectory topology across all representation spaces. A dynamical viewpoint of learning is established in which intrinsic properties expose the effects of forced changes in representation during training. We utilize the intuition that learning operates as a process driven far from equilibrium by fluctuations from the training data and optimizer to provide an interpretive basis grounded in both the physical world and the machine models that represent it.
Marginal inference in discrete graphical models forces a choice between exactness and scalability: exact algorithms are intractable for high-treewidth graphs, while iterative approximations (Belief Propagation, variational methods) sacrifice convergence guarantees on frustrated topologies. We argue that this dichotomy stems from a mismatched inductive bias: iterative methods abandon the sequential elimination structure that makes exact inference correct. We introduce In-Context Graphical Inference (ICG-I), an autoregressive Graph Transformer that restores this structure by mimicking Variable Elimination with learned, Tensor- Train-compressed intermediate factors, paired with a Dirichlet output layer and Weighted Conformal Prediction for calibrated, distribution-free coverage guarantees under topological shift. We prove that TT compression errors propagate at most lincarly through the autoregressive chain, that the Dirichlet-Multinomial loss is a proper scoring rule, and that WCP maintains coverage with a quantifiable degradation under estimated density ratios. We conducted intensive experiments to evaluate ICG-I and achieved state-of-the-art performance across all benchmarks. ICG-I reduces MAE from 0.041 (best baseline) to 0.020 on standard instances and achieves 0.048 on N=500 frustrated spin glasses where BP diverges entirely.