Conditional generative models for sampling and phase transition indication in spin systems

Submission summary

 As Contributors: Mathias Scheurer Preprint link: scipost_202103_00010v2 Date accepted: 2021-08-20 Date submitted: 2021-07-28 18:04 Submitted by: Scheurer, Mathias Submitted to: SciPost Physics Academic field: Physics Specialties: Artificial Intelligence Condensed Matter Physics - Computational Statistical and Soft Matter Physics Approaches: Theoretical, Computational

Abstract

In this work, we study generative adversarial networks (GANs) as a tool to learn the distribution of spin configurations and to generate samples, conditioned on external tuning parameters or other quantities associated with individual configurations. For concreteness, we focus on two examples of conditional variables---the temperature of the system and the energy of the samples. We show that temperature-conditioned models can not only be used to generate samples across thermal phase transitions, but also be employed as unsupervised indicators of transitions. To this end, we introduce a GAN-fidelity measure that captures the model’s susceptibility to external changes of parameters. The proposed energy-conditioned models are integrated with Monte Carlo simulations to perform over-relaxation steps, which break the Markov chain and reduce auto-correlations. We propose ways of efficiently representing the physical states in our network architectures, e.g., by exploiting symmetries, and to minimize the correlations between generated samples. A detailed evaluation, using the two-dimensional XY model as an example, shows that these incorporations bring in considerable improvements over standard machine-learning approaches. We further study the performance of our architectures when no training data is provided near the critical region.

Published as SciPost Phys. 11, 043 (2021)

List of changes

Change #1. Section 5. Conclusion. Paragraph 3.
One could also explore interpretable ML models [Murdoch et al.] to extract the crucial physical aspects, such as order parameters or defect proliferation, underlying the phase transition.

Change #2. Section 5. Conclusion. Paragraph 4.
In the future, we are also planning to further test and refine the ImplicitGAN approach, by applying it to other classical models and systematically studying the behavior of observables and $\mathcal{F}_{\text{GAN}}$ with increasing system size. Additional directions include developing models conditioned on system size and exploring quantum mechanical systems.

Change #3. Appendix A. Ablation analysis. Paragraph 2.
Thus, discontinuous jumps of $\theta_i$ at $2\pi$ and not taking into account periodic boundary conditions and spin-rotation symmetry seem to be important factors causing the bad performance of C-GAN$_1$ and C-GAN$_2$. Consistent with [34], we observed that this was not a serious problem when (unconditioned) GANs were trained only for a single temperature.

Change #4. Section 4.5.2 Detecting phase transitions. Paragraph 2
The line "A more detailed finite-size scaling analysis would be required to address this issue" is removed and the line "We leave a detailed, system-size-dependent study of these aspects for future work" is added later in this section.

The first line of third paragraph of this section now reads as "Due to these shortcomings of $\mathcal{D}$ for detecting the BKT transition in our GAN architecture, we here focus on the second measure---the GAN-fidelity---introduced in Eq. (14) with corresponding plot in Fig. 3b, using $\Delta T = 0.0625$.

(Minor) Change #5. Section 5. Conclusion. Paragraph 2.
The line We demonstrate that this can be used to generate configurations near criticality..."" is modified as We demonstrate that this can be employed for generating configurations near criticality...''

(Minor) Change #6. Section 5. Conclusion. Paragraph 3.
The line Most importantly, we propose a GAN fidelity measure that can be readily ...'' is modified as
Most importantly, we propose a GAN fidelity measure $\mathcal{F}_{\text{GAN}}$ that can be readily ...''

Submission & Refereeing History

Resubmission scipost_202103_00010v2 on 28 July 2021
Submission scipost_202103_00010v1 on 6 March 2021

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Dear Editor,
My comments and suggestions have been carefully considered in the latest manuscript.
Meanwhile, authors’ responses are accurate and clear to the questions.
Thus, I recommend to accept the paper as an formal article on SciPost.

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Dear Editor
I have reviewed the changes made to the manuscript. I consider that in its current form the paper should be accepted for publication.

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