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Machine Learning of Explicit Order Parameters: From the Ising Model to SU(2) Lattice Gauge Theory

Sebastian Johann Wetzel, Manuel Scherzer

arXiv:1705.05582v1cond-mat.stat-mechhep-lat

TL;DR

Neural-network decision functions can encode phase distinctions in elusive, nonlinear forms, making their underlying physical quantities difficult to read directly. The paper uses symmetry-aware correlation probing to reconstruct those functions and identifies known order parameters in the Ising model and SU(2) lattice gauge theory.

  • Problem

    Neural networks can encode phase-classification decision functions in elusive, highly nonlinear forms, raising the question of whether they can be expressed through a few spin correlations.

    Source: B. Ising Model · p. 3

  • Method

    A correlation probing neural network tests symmetry and reduces decision-function complexity, enabling reconstruction of an explicit expression for the quantity distinguishing phases.

    Source: §I · p. 1

  • Results

    The reconstructed decision functions identify magnetization or expected energy per site in the Ising model and the Polyakov loop in SU(2) lattice gauge theory.

    Source: §I · p. 1, §IV · p. 8

  • Takeaways & Limitations

    The cases show machine learning can construct formulas for physical quantities relevant to phase transitions, including in systems whose phases are not understood.

    Source: §IV · p. 8

Abstract

from arXiv · show

We present a procedure for reconstructing the decision function of an artificial neural network as a simple function of the input, provided the decision function is sufficiently symmetric. In this case one can easily deduce the quantity by which the neural network classifies the input. The procedure is embedded into a pipeline of machine learning algorithms able to detect the existence of different phases of matter, to determine the position of phase transitions and to find explicit expressions of the physical quantities by which the algorithm distinguishes between phases. We assume no prior knowledge about the Hamiltonian or the order parameters except Monte Carlo-sampled configurations. The method is applied to the Ising Model and SU(2) lattice gauge theory. In both systems we deduce the explicit expressions of the known order parameters from the decision functions of the neural networks.

I. INTRODUCTION

The paper addresses the difficulty of interpreting neural-network phase classifications by reconstructing sufficiently symmetric decision functions as explicit physical quantities without prior knowledge. It applies the approach to the Ising model and SU(2) lattice gauge theory, with the well-studied Ising model serving as a testing ground.

I. INTRODUCTION

  • The work targets the difficulty of interpreting phase classifications, despite machine-learning methods already identifying phases from Monte Carlo samples.The introduction notes that there is still no comprehensive theoretical understanding of what neural networks learn to classify phases.Source: §I · p. 1
  • The proposed correlation probing neural network reduces decision-function complexity when the function is sufficiently symmetric, enabling reconstruction of its explicit mathematical expression.Physical quantities are typically highly symmetric, making the network suitable for probing whether they underpin the learned decision function.Source: §I · p. 1

IV. CONCLUSION

  • The Ising decision functions encode magnetization or expected energy per site, while the SU(2) decision function reveals the Polyakov loop.The Polyakov loop is a non-linear, non-local order parameter.Source: §I · p. 1

I. INTRODUCTION

  • The method may apply to phase transitions that are inaccessible by other tools, including proposed studies of QCD phases and the two-dimensional Hubbard model.The introduction mentions QCD chiral or color-superconducting phases, the Hubbard-model pseudogap, and multicritical points as possible applications.Source: §I · p. 1

A. Ising Model

  • The Ising model is a well-studied, exactly solvable test system; the examined case is the square lattice with J = 1 and h = 0.In the Ising model, magnetization is zero in the paramagnetic phase and finite in the ferromagnetic phase.Source: A. Ising Model · pp. 1–2

B. SU(2) Lattice Gauge Theory

The paper studies SU(2) Yang-Mills theory using lattice gauge theory and Monte Carlo simulations. It identifies the Polyakov loop as the order parameter for the deconfinement transition and relates its behavior to center symmetry.

B. SU(2) Lattice Gauge Theory

  • Lattice gauge theory discretizes the Euclidean path integral, with lattice spacing a as a cutoff, and permits Monte Carlo simulations of gauge theories.The study uses four-dimensional lattices with temporal extent Nτ = 2 and spatial extent Ns = 8.Source: B. SU(2) Lattice Gauge Theory · p. 2
  • The simulations use the lattice version of the Yang-Mills action, and closed loops over link variables are gauge-invariant objects.Link variables live on links between neighboring sites and transform under gauge transformations.Source: B. SU(2) Lattice Gauge Theory · p. 2
  • The Polyakov loop is the order parameter for deconfinement: its expectation is zero in the confined phase and finite in the deconfined phase.Under center-symmetry transformations, the loop transforms as L → zL; the symmetry is preserved in the confined phase and broken in the deconfined phase.Source: B. SU(2) Lattice Gauge Theory · p. 2

A. Artificial Neural Networks

The paper uses a correlation probing neural network to reduce and reconstruct a sufficiently symmetric decision function, then read out the physical quantity driving phase classification. In the Ising model, the reconstructed quantities are magnetization and average nearest-neighbor spin product, the latter corresponding up to a minus sign to energy per spin.

A. Artificial Neural Networks

  • A feed-forward network approximates a probability-valued decision function, and near its decision boundary the latent prediction is linear in the learned quantity.The network output is sigmoid(ξ), and the decision boundary is where D(S) = 0.5.Source: A. Artificial Neural Networks · pp. 2–3

B. Ising Model

  • The correlation probing architecture restricts local connections to patches, averages their outputs, then predicts from the pooled representation.A 1 × 1 receptive field limits the network to functions of individual-spin correlations.Source: B. Ising Model · p. 3, Figure 1 · p. 3
  • The 1 × 1 network is less accurate than the 1 × 2 network, with transition estimates of T = 2.5 ± 0.5 and T = 2.25 ± 0.25, respectively.The 1 × 1 estimate is reported first; the authors focus on the information lost as receptive fields shrink.Source: B. Ising Model · pp. 3–4
  • The 1 × 1 network's latent prediction is perfectly correlated with the absolute average spin, identifying magnetization as its decision quantity.The authors report using only Monte Carlo configurations, without other Ising-model information, for this deduction.Source: B. Ising Model · pp. 4–5
  • The Ising decision functions identify magnetization and average nearest-neighbor spin product as the quantities distinguishing phases.The spin product is, up to a minus sign, the expected energy per spin site.Source: B. Ising Model · p. 5
  • The 1 × 2 network's latent prediction is perfectly correlated with the average product of neighboring spins, while subleading magnetization terms are negligible.Translational and rotational symmetry generalize the transverse-neighbor sum to all nearest neighbors.Source: B. Ising Model · p. 5
  • The correlation probing network reconstructs a simpler decision function from spatial correlations, allowing its phase-distinguishing quantity to be read out.The method uses the correlations identified by receptive-field-constrained networks to reduce the decision function's complexity.Source: B. Ising Model · pp. 3, 5

C. SU(2) Lattice Gauge Theory

The SU(2) lattice gauge theory analysis uses a staged machine-learning pipeline to investigate the deconfinement transition. It ends by regressing local-network results to reconstruct the correlation probing network's decision function.

C. SU(2) Lattice Gauge Theory

  • The pipeline combines PCA, neural networks, and regression to investigate the SU(2) deconfinement transition.It proceeds from unsupervised indications, to phase prediction, to spatially local samples and regression of local-network results into an explicit decision function.Source: C. SU(2) Lattice Gauge Theory · p. 5

1. Unsupervised Learning of Phase Transitions

With no prior knowledge of whether a phase transition exists, the authors use PCA to search for indicators. Reconstruction loss and principal-component norm indicate a transition region, even though the components do not correlate with the Polyakov loop.

1. Unsupervised Learning of Phase Transitions

  • The PCA search is used without prior knowledge of whether a phase transition exists, and employs two principal components.PCA is chosen because reconstruction-loss methods such as autoencoders are unsuitable for these gauge-invariant states, whose matrix representation is not unique.Source: §1 · pp. 5–6
  • PCA reconstruction loss and principal-component norm indicate a possible phase transition at β ∈ [1.8, 2.2].The reconstruction-loss curve is averaged over 100 independent runs with two principal components.Source: §1 · p. 6, Figure 3 · p. 6
  • The principal components do not correlate with the Polyakov loop, although PCA captures indicators of a phase transition.This illustrates that the components need not approximate an order parameter to provide phase indicators.Source: §1 · p. 6

2. Correlation Probing Neural Network

For SU(2), the correlation-probing networks predict a transition near β≈1.98, and smaller receptive fields reveal a scale below which phase classification loses crucial information. Simplifying the prediction network makes its decision function easier to reconstruct.

2. Correlation Probing Neural Network

  • β=1.99 ± 0.10 for the 2 × 1 × 1 × 1 network and β=1.97 ± 0.10 for the 2 × 8 × 8 × 8 network, versus β=1.880 ± 0.025 from direct calculation.The transition predictions use test configurations at β ∈ [1.3, 3.2]; training uses β ∈ [1, 1.2] and β ∈ [3.3, 3.5].Source: §2 · p. 6
  • Reducing the receptive field below 2 × 1 × 1 × 1 loses information crucial for classifying phases.Table II reports losses for networks with different receptive fields, but the supplied caption gives no individual table values.Source: §2 · p. 6, Table II · p. 6
  • With the output neuron alone, the decision function simplifies to D(S) = sigmoid(w Q(S) + b).The approach further separates the lattice into 2 × 1 × 1 × 1 patches, with functions acting on individual patches.Source: §2 · p. 6
  • The latent prediction strongly correlates with the Polyakov loop in both the 2 × 8 × 8 × 8 and 2 × 1 × 1 × 1 networks.Figure 4c shows the average prediction probability of the two networks.Source: Figure 4 · p. 7

3. Local Neural Network Regression

Local neural-network regression reconstructs the SU(2) decision function in terms of the Polyakov loop. The local expression is identified and then extended across the lattice, with the latent prediction correlating with the full-lattice loop.

3. Local Neural Network Regression

  • A second-order polynomial performs best among the tested regression orders, approximating the latent prediction with 561 terms.Regression uses 1% of the dataset and another 1% for validation.Source: §3 · p. 7, Table III · p. 7
  • The reconstruction keeps leading contributions, treating differences between them as approximation errors and allowing errors on the scale of next-to-leading coefficients.Overall factors and intercepts can be absorbed into the neural-network weights and biases, allowing the parameter to be rescaled.Source: §3 · p. 7
  • The reconstructed local quantity is the Polyakov loop on a single spatial lattice site.Source: §3 · p. 7
  • The reconstructed decision function uses the Polyakov loop on the full lattice, whose latent prediction correlates perfectly with that loop.The fitted single-site expression is promoted to act across the full lattice.Source: §3 · p. 7

IV. CONCLUSION

The paper presents a pipeline to identify phases and predict transition positions, then reconstructs the neural networks’ decision functions explicitly. The reconstructed quantities include Ising magnetization or expected energy per site and the SU(2) Polyakov loop.

IV. CONCLUSION

  • The pipeline combines machine-learning analyses to identify phases and predict transition positions, while the correlation-probing network helps reconstruct decision functions and reveal their physical quantities.Source: §IV · pp. 7–8
  • The reconstructed decision functions contain the Ising magnetization or expected energy per site and the SU(2) Polyakov loop, a nonlinear, non-local order parameter.The authors identify this explicit calculation as the work’s most important result.Source: §IV · p. 8
  • The approach could be applied to strongly correlated systems with unknown phases, including high-density QCD and the two-dimensional Hubbard model at finite chemical potential.Source: §IV · p. 8
  • Future work includes extracting order parameters without reduction to a semi-local one, selecting features, extending the method to unsupervised learning, and probing neural networks in other fields.Source: §IV · p. 8

Appendix A: Monte Carlo Simulations

Appendix A describes Monte Carlo sampling, the Ising and SU(2) data settings, and the limits imposed by finite sampling and ergodicity. Appendix B specifies the network architecture and training choices.

Appendix A: Monte Carlo Simulations

  • Monte Carlo configurations are sampled from the equilibrium Boltzmann distribution, and observables are estimated by averaging over those configurations.The sampling sequence is stochastic and begins from a random initial configuration.Source: Appendix A · p. 8
  • Finite samples introduce statistical error, while ergodicity may require long simulations when phase space has distinct regions.The expectation value is approximated using large N; enough configurations are needed to achieve ergodicity.Source: Appendix A · p. 8
  • The Ising dataset contains 55,000 configurations of size 28 × 28, equally distributed over 11 temperatures from 0 to 5.Configurations were generated with Metropolis-Hastings and simulated annealing.Source: Appendix A · p. 8
  • The SU(2) dataset contains 15,600 decorrelated configurations across 26 β_latt values from 1 to 3.5, but unit-matrix initialization leaves large-β_latt sampling non-ergodic.If neural networks were used to extract the transition position from these samples, the non-ergodicity would shift the critical β_latt; more simulations or simulated annealing or overrelaxation can retain ergodicity.Source: Appendix A · p. 8

Appendix B: Neural Network Architecture

  • The networks use selected filter counts and kernel sizes to set receptive fields, with nC = 1 sufficient for these order-parameter structures.Training uses ReLU hidden activations, a sigmoid output, binary cross-entropy, RMSprop, up to 300 epochs, and a validation set of 20%; no regularization is used.Source: Appendix B · pp. 8–9, Table IV · p. 9, Table V · p. 9

Appendix C: Regression of the Polyakov Loop

The appendix reconstructs the SU(2) decision function by fitting the prediction and localization stages, then identifies the local function with the Polyakov loop and extends it across space.

Appendix C: Regression of the Polyakov Loop

  • A degree-1 polynomial fit maps the prediction network's latent output to the averaging-layer output, approximating ξ.The slope and intercept can be absorbed into the output neuron's weight and bias.Source: Appendix C · p. 9
  • The localization network is evaluated patch by patch using transferred weights, and a degree-2 regression with 561 parameters fits its local input-output relation.The local network uses patches of size 2 × 1 × 1 × 1 and retains the full network's number of weights and biases.Source: Appendix C · p. 9
  • The local function is identified as the Polyakov loop on a single lattice site and promoted to depend on space in the decision function.The reconstruction absorbs overall factors and the intercept and neglects subleading terms.Source: Appendix C · pp. 9–10

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