Smallest Neural Network to Learn the Ising Criticality

Abstract

Learning with an artificial neural network encodes the system behavior in a feed-forward function with a number of parameters optimized by data-driven training. An open question is whether one can minimize the network complexity without loss of performance to reveal how and why it works. Here we investigate the learning of the phase transition in the Ising model and find that having two hidden neurons can be enough for an accurate prediction of critical temperature. We show that the networks learn the scaling dimension of the order parameter while being trained as a phase classifier, demonstrating how the machine learning exploits the Ising universality to work for different lattices of the same criticality within a single set of trainings in one lattice geometry.

References in corpus (49)

• An efficient, multiple range random walk algorithm to calculate the density of states
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Machine learning phases of matter
• Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram
• Many-body quantum state tomography with neural networks
• Learning phase transitions by confusion
• Discovering Phase Transitions with Unsupervised Learning
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders
• Quantum Entanglement in Neural Network States
• Machine learning quantum phases of matter beyond the fermion sign problem
• Discovering Phases, Phase Transitions and Crossovers through Unsupervised Machine Learning: A critical examination
• Efficient Representation of Quantum Many-body States with Deep Neural Networks
• Restricted-Boltzmann-Machine Learning for Solving Strongly Correlated Quantum Systems
• A Finite Size Scaling Study of Lattice Models in the three-dimensional Ising Universality Class
• Quantum Loop Topography for Machine Learning
• Machine Learning Topological Invariants with Neural Networks
• Learning Thermodynamics with Boltzmann Machines
• Machine Learning Phases of Strongly Correlated Fermions
• Self-Learning Monte Carlo Method
• Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
• Equivalence of restricted Boltzmann machines and tensor network states
• Approximating quantum many-body wave-functions using artificial neural networks
• Neural-Network Quantum States, String-Bond States, and Chiral Topological States
• Mutual Information, Neural Networks and the Renormalization Group
• Machine learning vortices at the Kosterlitz-Thouless transition
• Detection of phase transition via convolutional neural network
• Unsupervised machine learning account of magnetic transitions in the Hubbard model
• Machine Learning of Explicit Order Parameters: From the Ising Model to SU(2) Lattice Gauge Theory
• Probing many-body localization with neural networks
• Kernel methods for interpretable machine learning of order parameters
• Deep Learning the Quantum Phase Transitions in Random Two-Dimensional Electron Systems
• Identifying Quantum Phase Transitions with Adversarial Neural Networks
• Machine Learning ${\mathbb{Z}}_2$ Quantum Spin Liquids with Quasi-particle Statistics
• Discriminative Cooperative Networks for Detecting Phase Transitions
• Self-Learning Monte Carlo Method in Fermion Systems
• Parameter diagnostics of phases and phase transition learning by neural networks
• Deep Learning Topological Invariants of Band Insulators
• Scale-invariant Feature Extraction of Neural Network and Renormalization Group Flow
• Deep Learning the Ising Model Near Criticality
• Self-Learning Determinantal Quantum Monte Carlo Method
• Deep Learning the Quantum Phase Transitions in Random Electron Systems: Applications to Three Dimensions
• First-order phase transition and tricritical scaling behavior of the Blume-Capel model: a Wang-Landau sampling approach
• Self-Learning Monte Carlo Method: Continuous-Time Algorithm
• Principal Component Analysis for Fermionic Critical Points
• Can Boltzmann Machines Discover Cluster Updates ?
• Deep neural networks for direct, featureless learning through observation: the case of 2d spin models
• Identifying Product Order with Restricted Boltzmann Machines
• Simulating the Ising Model with a Deep Convolutional Generative Adversarial Network

Cited by in corpus (36)

• Machine Learning for Quantum Matter
• Machine learning of quantum phase transitions
• Machine Learning for Condensed Matter Physics
• Discovering Symmetry Invariants and Conserved Quantities by Interpreting Siamese Neural Networks
• Extensive deep neural networks for transferring small scale learning to large scale systems
• Drawing Phase Diagrams of Random Quantum Systems by Deep Learning the Wave Functions
• The critical temperature of the 2D-Ising model through Deep Learning Autoencoders
• Learning the Ising Model with Generative Neural Networks
• Unsupervised learning of topological phase transitions using Calinski-Harabaz index
• Deep Learning on the 2-Dimensional Ising Model to Extract the Crossover Region with a Variational Autoencoder
• Interpretable machine learning for inferring the phase boundaries in a nonequilibrium system
• Machine Learning Application to Two-Dimensional Dzyaloshinskii-Moriya Ferromagnets
• Absorbing phase transition with a continuously varying exponent in a quantum contact process: a neural network approach
• A perspective on machine learning and data science for strongly correlated electron problems
• Phase transition encoded in neural network
• Learning quantum phase transitions through Topological Data Analysis
• Markov Chain Monte Carlo Methods in Quantum Field Theories: A Modern Primer
• Machine-learning detection of the Berezinskii-Kosterlitz-Thouless transition and the second-order phase transition in the XXZ models
• Unsupervised machine learning approaches to the $q$-state Potts model
• Machine learning generated configurations in presence of a conserved quantity: a cautionary tale
• Can a CNN trained on the Ising model detect the phase transition of the $q$-state Potts model?
• On the generalizability of artificial neural networks in spin models
• Machine learning phase transitions of the three-dimensional Ising universality class
• Unsupervised Machine Learning of Quenched Gauge Symmetries: A Proof-of-Concept Demonstration
• Short sighted deep learning
• Unsupervised Learning of Rydberg Atom Array Phase Diagram with Siamese Neural Networks
• Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition
• Supervised Training of Neural-Network Quantum States for the Next Nearest Neighbor Ising model
• Application of the Variational Autoencoder to Detect the Critical Points of the Anisotropic Ising Model
• Comments on the minimal training set for CNN: a case study of the frustrated $J_1$-$J_2$ Ising model on the square lattice
• Global exploration of phase behavior in frustrated Ising models using unsupervised learning techniques
• Discovering quasiorder parameters in the Potts model: A bridge between machine learning and critical phenomena
• Randomized-gauge test for machine learning of Ising model order parameter
• Machine-Learning Detection of the Berezinskii-Kosterlitz-Thouless Transitions
• Identifying phase transitions in physical systems with neural networks: a neural architecture search perspective
• Learning disentangled representation for classical models