We propose a general framework to extract microscopic interactions from raw configurations with deep neural networks. The approach replaces the modeling Hamiltonian by the neural networks, in which the interaction is encoded. It can be trained with data collected from Ab initio computations or experiments. The well-trained neural networks give an accurate estimation of the possibility distribution of the configurations at fixed external parameters. It can be spontaneously extrapolated to detect the phase structures since classical statistical mechanics as prior knowledge here. We apply the approach to a 2D spin system, training at a fixed temperature, and reproducing the phase structure. Scaling the configuration on lattice exhibits the interaction changes with the degree of freedom, which can be naturally applied to the experimental measurements. Our approach bridges the gap between the real configurations and the microscopic dynamics with an autoregressive neural network.
We propose a general framework to extract microscopic interactions from raw configurations with deep neural networks. The approach replaces the modeling Hamiltonian by the neural networks, in which the interaction is encoded. It can be trained with data collected from Ab initio computations or experiments. The well-trained neural networks give an accurate estimation of the possibility distribution of the configurations at fixed external parameters. It can be spontaneously extrapolated to detect the phase structures since classical statistical mechanics as prior knowledge here. We apply the approach to a 2D spin system, training at a fixed temperature, and reproducing the phase structure. Scaling the configuration on lattice exhibits the interaction changes with the degree of freedom, which can be naturally applied to the experimental measurements. Our approach bridges the gap between the real configurations and the microscopic dynamics with an autoregressive neural network.
We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once. Training of the network employs the policy gradient approach in reinforcement learning, which unbiasedly estimates the gradient of variational parameters. We apply our approach to several classic systems, including 2D Ising models, the Hopfield model, the Sherrington-Kirkpatrick model, and the inverse Ising model, for demonstrating its advantages over existing variational mean-field methods. Our approach sheds light on solving statistical physics problems using modern deep generative neural networks.
We propose a general method for solving statistical mechanics problems defined on sparse graphs, such as random graphs, real-world networks, and low-dimensional lattices. Our approach extract a small feedback vertex set of the sparse graph, converting the sparse system to a strongly correlated system with many-body and dense interactions on the feedback set, then solve it using variational method based on neural networks to estimate free energy, observables, and generate unbiased samples via direct sampling. Extensive experiments show that our approach is more accurate than existing approaches for sparse spin glass systems. On random graphs and real-world networks, our approach significantly outperforms the standard methods for sparse systems such as belief-propagation; on structured sparse systems such as two-dimensional lattices our approach is significantly faster and more accurate than recently proposed variational autoregressive networks using convolution neural networks.
Identifying phase transitions in complex many-body systems traditionally necessitates the definition of specific order parameters, a task often requiring prior knowledge of the statistical model and the symmetry-breaking mechanism. In this work, we propose a framework for detecting phase transitions directly from raw (experimental) data without requiring knowledge of the underlying model Hamiltonian, parameters, or pre-defined labels. Inspired by generative modeling in machine learning, our method utilizes autoregressive networks to estimate the normalized probability distribution of the system from raw configuration data. We then quantify the intrinsic sensitivity of this learned distribution to control parameters (such as temperature) to construct a robust indicator of phase transitions. This indicator is based on the expectation of the change in absolute logarithmic probability, derived entirely from the raw data. Our approach is purely data-driven: it takes raw data across varying control parameters as input and outputs the most likely estimate of the phase transition point. To validate our approach, we conduct extensive numerical experiments on the 2D Ising model on both triangular and square lattices, and on the Sherrington-Kirkpatrick (SK) model utilizing raw data generated via Markov Chain Monte Carlo and Tensor Network methods. The results demonstrate that our generative approach accurately identifies phase transitions using only raw data. Our framework provides a general tool for exploring critical phenomena in model systems, with the potential to be extended to realistic experimental data where theoretical descriptions remain incomplete.
Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.
Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.
Determining the microscopic Hamiltonian of a quantum magnet from inelastic neutron scattering measurements remains a central challenge in condensed matter physics. While the dynamical spin structure factor contains, in principle, sufficient information about the underlying interactions, extracting Hamiltonian parameters from experimental spectra constitutes a highly nontrivial inverse problem due to the large parameter space and the presence of experimental noise. Here we demonstrate that neural networks can efficiently solve this inverse problem for a broad class of quantum magnets. As a benchmark, we consider an eight-parameter family of honeycomb lattice Heisenberg spin models in the fully polarized phase, where the dynamical spin structure factor can be computed exactly within linear spin-wave theory. Using a large synthetic dataset of neutron scattering spectra, we train neural networks to infer the underlying Hamiltonian parameters directly from the dynamical response. We compare the performance of three different architectures---fully connected neural networks (FCNNs), one-dimensional convolutional neural networks (CNN1Ds), and two-dimensional convolutional neural networks (CNN2Ds)---and find that all achieve high predictive accuracy. Remarkably, the trained models remain robust in the presence of substantial experimental uncertainty, reliably recovering the Hamiltonian parameters even when the input spectra are contaminated by random noise with amplitudes reaching 10% of the signal intensity. Our results establish machine learning as a powerful framework for quantitative Hamiltonian reconstruction from neutron scattering data and provide a practical route toward automated characterization of quantum magnetic materials.