Parameter diagnostics of phases and phase transition learning by neural networks

Abstract: We present an analysis of neural network-based machine learning schemes for phases and phase transitions in theoretical condensed matter research, focusing on neural networks with a single hidden layer. Such shallow neural networks were previously found to be efficient in classifying phases and locating phase transitions of various basic model systems. In order to rationalize the emergence of the classification process and for identifying any underlying physical quantities, it is feasible to examine the weight matrices and the convolutional filter kernels that result from the learning process of such shallow networks. Furthermore, we demonstrate how the learning-by-confusing scheme can be used, in combination with a simple threshold-value classification method, to diagnose the learning parameters of neural networks. In particular, we study the classification process of both fully-connected and convolutional neural networks for the two-dimensional Ising model with extended domain wall configurations included in the low-temperature regime. Moreover, we consider the two-dimensional XY model and contrast the performance of the learning-by-confusing scheme and convolutional neural networks trained on bare spin configurations to the case of preprocessed samples with respect to vortex configurations. We discuss these findings in relation to similar recent investigations and possible further applications.

• The paper investigates how shallow neural networks learn and utilize physical parameters to classify phases and detect transitions in condensed matter models.
• The study employed CNNs on Ising and XY models, demonstrating their ability to extract relevant local physical parameters for accurate phase classification.
• Key findings include the effectiveness of 'learning-by-confusion' and parameter diagnostics to infer learned physical quantities and estimate critical temperatures.

Parameter Diagnostics of Phases and Phase Transition Learning by Neural Networks

The study presented in "Parameter diagnostics of phases and phase transition learning by neural networks" by Philippe Suchsland and Stefan Wessel provides an in-depth analysis of neural networks applied to phases and phase transitions within the framework of theoretical condensed matter research. The focus is on shallow neural networks with a single hidden layer, which have demonstrated efficiency in classifying phases and identifying phase transitions across basic model systems.

Key Findings and Analysis

The paper examines the internal mechanisms of these networks, particularly by investigating the weight matrices and the convolutional filter kernels resulting from the learning processes. Such examination aims to uncover any underlying physical parameters that the neural networks learn during classification tasks.

The authors employ both fully-connected and convolutional neural networks (CNNs) to study the two-dimensional Ising and XY models. In the case of the Ising model, the inclusion of extended domain wall (EDW) configurations in low-temperature regimes presents additional complexity. The shallow CNNs, with filter kernels capable of identifying local magnetization and configurational energy, adapt effectively to these challenges, showcasing robust classification accuracy.

In contrast, the XY model, characterized by a Kosterlitz-Thouless (KT) transition, presents unique challenges due to its topological nature driven by vortices. Directly inputting spin configurations or pre-processed data involving vorticity demonstrates different learning outcomes. The CNNs extract relevant parameters such as local energy estimates, which play a significant role in their classification success.

Methodological Insights

The study introduces the learning-by-confusion scheme, an approach to estimate unknown critical temperatures by evaluating neural networks' classification performance as a function of a guessed transition temperature $T^*$. This method, alongside a proposed threshold-value classification based on physical observables, offers diagnostic tools to infer the relevant parameters guiding the network's learning process.

For the Ising model, the test accuracy closely follows classifications based on magnetization and energy, highlighting these as key learned features. However, in the XY model, the energy gradient derived from local angle differences becomes a significant, albeit non-unique, classifier. This difference indicates the complexity introduced by the vortex-driven nature of the phase transition in the XY system.

Implications and Future Directions

This research contributes significantly to understanding machine learning applications in condensed matter physics. By dissecting the inputs and transformations within neural networks, it clarifies how physical parameters reflect in machine learning classifications. Such insights are crucial for advancing neural network methodologies for complex systems where traditional theoretical frameworks may not straightforwardly apply.

For future developments, extending this analysis to deep learning architectures could uncover hierarchical representations of physical phenomena, potentially resolving existing limitations in identifying transition temperatures accurately—especially in systems similar to the XY model where traditional identifiers may not align with transition phenomena.

Additionally, challenges highlighted by this study, such as the misalignment between specific heat peaks and true transition temperatures in the XY model, call for further research. This work implies that deeper networks or alternative preprocessing methods might reconcile these issues. Moreover, cross-examination of neural network output activity could provide empirical indicators of phase transitions in previously inaccessible or poorly understood systems.

Overall, Suchsland and Wessel's work not only contributes to theoretical advancements but also sets a foundation for explorations into unexplored territories of condensed matter physics with machine learning as a pivotal tool.