Rapid Measurements and Phase Transition Detections Made Simple by AC-GANs

Response to the second referee:
We thank the referee for the careful reading of our manuscript and his/her support on publication of our work. Below are our responses (R) to the referee's comments (C).

C1: The authors only use simple observables to evaluate the performance of the models. If possible, it could be worthwhile to include other measures such as earth mover distances of the distributions.
R1: We thank the referee for the suggestion. In the revised manuscript, we have added the discussion of the Earth mover's distance and the Jensons-Shannon divergence in Appendix C. Figure 9 shows the Earth Mover’s distance and Jenson-Shannon (JS) divergence measuring the difference in the distribution of the MCMC samples and the AC-GAN generated samples. Both the distance measures are small throughout the simulated temperature range, providing further evidence of the validity of our method.

C2: Did the authors test whether their approach suffers from mode collapse – a well-known issue of GANs?
R2: Our model also suffers from certain degree of mode collapse depending on the system’s driving parameters. Nevertheless, as shown in Fig.9 of the revised manuscript, the small earth mover distance and JS divergence suggest the sampled distribution from GANs is sufficiently close to that from MCMC. Therefore, the mode collapse did not affect the use of GAN to sample physical quantities. We have added a comment on this point in the last paragraph of Sec. 3 of the revised manuscript.

C3: I do not fully understand in which sense using Eq. (7) is beneficial compared to other ways people have studied for the detection of phase transitions. It seems, one needs both MCMC data and train a GAN. Relatedly, in the last paragraph before the conclusion starts, the authors write “It is also different from other machine learning methods in detecting phase transitions where the machine works as a black-box. Our method has a physical interpretation that the phase transition is accompanied by the change in the configuration distribution.” I likely do not understand their approach well enough, but I am not so sure in which sense their approach is less of a black-box than other machine-learning techniques studied before and I think previous approaches are also based on changes in the configuration distributions, see, for instance, Phys. Rev. E 99, 062107 (2019) or also that in Ref. 32.
R3: We thank the referee for the questions. The order parameters that detect the phase transitions in condensed matter models may not be always known and can be difficult to obtain. Therefore, in most cases, even with extensive Monte Carlo data, it is challenging to locate the transition points accurately. By using Monte Carlo data, AC-GANs data, and Equation 7, our method can effectively locate the transition point without the need of the order parameters, thus demonstrating the practical value of our approach. We have added a comment on this point in the last paragraph of Sec.5.1 in the revised manuscript.

The term ‘black box’ generally refers to the issue that deep learning methods have, i.e., what makes the machine effectively learn the phase transition is unknown. In Physical Review E 99, 062107 (2019), the authors use confusion of the machine for phase transition detection and provide a rigorous and mathematical definition of the confusion. However, the reason for why the machine experiences confusion near the transition point was not addressed. In comparison, our method does not rely on confusion but on the changes in the distributions during the learning process to detect the transition. It provides a more physically intuitive way to capture the phase transition from the perspective of configuration distributions. Nevertheless, to avoid confusion, we have removed the statement about the ‘black box’, and further explain the benefits of our approach in the last paragraph of Sec. 5.2.

C4: I also noticed a few typos: (i) page 2, “To let the Generator performs …”, (ii) page 3, “… besides the random matrix z, constrains are added …”, and (iii) in the caption of Fig. 2, it seems that column and rows are swapped.
R4: We thank the referee for carefully reading our manuscript and pointing out these typos. We have corrected them in the revised manuscript.

Response to the first referee:
Below are our responses (R) to the referee's comments (C).
C1: although results seem consistent and the research done has value, I believe this article is not suitable to publish in SciPost physics.
Many of the concepts and or techniques used in the present article are equal or small modifications of https://www.scipost.org/SciPostPhys.11.2.043/pdf
Moreover, the clarity of the written English is far from acceptable,
R1: We strongly disagree with the referee’s comment that our work is just a minor modification of SciPost Phys. 11, 043 (2021) and his criticism on the presentation of our manuscript. The second referee also commented that our manuscript is “very well written and presented. It also addresses an important question – how to improve sampling algorithms using generative models.”. Our work is distinguished from the mentioned work in the following ways. First, we propose the advanced integration of self-attention layers, which significantly improves the performance of the GAN model. Especially some of the physical quantities such as the vorticity, can be estimated with much higher accuracy, as shown in Fig. 8c in Appendix A of our manuscript. Second, we also discuss how the symmetries of the lattice can be utilized to augment the training dataset, allowing us to train GANs more efficiently with much less data. Third, we proposed a novel quantity of distribution overlap to capture the phase transition. The quantity provides a physically intuitive way to detect the phase transition point, and it is especially useful when the order parameter in the model is unknown. We sincerely suggest the referee read our manuscript in more detail.

C2: more details of the network architectures should be added.
R2: In the revised manuscript, we have included a more detailed architect of the deep learning model as Fig.7 in Appendix A to further clarify our methodology.