PDED: Revitalize physics laws submerged in data information for Traffic State Estimation

Q: I don’t think that is one of the contributions of this paper.

A: Sorry for the confusion. The first contribution is not finding the optimization issue of physics-informed learning, and it is the motivation to lead to our novel solution. We focus on the way to solve this problem that we propose to first disentangle and then assemble theory to manage physical knowledge and data information.

Q: There is a double “the” in this sentence.

A: Sorry for the mistake. We will delete one 'the' in the revised paper.

Q: Is it possible to add more evaluation metrics since the traffic data has some rush hour phenomena, where extreme statistics are needed.

A: Thank you for your suggestion. In fact, the reason we do not use other metrics is that almost all the existing methods for traffic state estimation only use the L2 error [1-5]. As you suggest to add more evaluation metric, we make efforts to add the RMSE for traffic state $\rho$ of our method and one best baseline for US-101. The results can be seen in the following table. It can be seen that the proposed method PDED has better performance. We will test RMSE for all baselines in our future work.

Q: Can the authors add some references here?

A: Thank you for your suggestion. It is common to set the noise ration by using a small covariance, please see the references [5-7].

Refs:

[1]Archie J Huang and Shaurya Agarwal. Physics informed deep learning for traffic state estimation. In 2020 IEEE 23rd International Conference on Intelligent Transportation Systems (ITSC), pp. 1–6. IEEE, 2020.

[2]Rongye Shi, Zhaobin Mo, and Xuan Di. Physics-informed deep learning for traffic state estimation: A hybrid paradigm informed by second-order traffic models. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 540–547, 2021a.

[3]Rongye Shi, Zhaobin Mo, Kuang Huang, Xuan Di, and Qiang Du. Physics-informed deep learning for traffic state estimation. arXiv preprint arXiv:2101.06580, 2021b.

[4]Archie J Huang and Shaurya Agarwal. Physics-informed deep learning for traffic state estimation: illustrations with lwr and ctm models. IEEE Open Journal of Intelligent Transportation Systems, 3:503–518, 2022.

[5]Zhaobin Mo, Yongjie Fu, Daran Xu, and Xuan Di. Trafficflowgan: Physics-informed flow based generative adversarial network for uncertainty quantification. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 323–339. Springer, 2022.

[6]Di X, Shi R, Mo Z, et al. Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook. Algorithms, 2023, 16(6): 305.

[7]Yang M, Foster J T. Multi-output physics-informed neural networks for forward and inverse PDE problems with uncertainties. Computer Methods in Applied Mechanics and Engineering, 2022, 402: 115041.

Thank you for your constructive comments and suggestions, and they are exceedingly helpful for us to improve our paper.

Please see our responses below. We have started working on the corresponding changes, and will incorporate all of them in the final revision. If you have any further concerns, we would be keen to address them as well.

Q: I am not sure of the magnitude differences between those loss terms in physics loss and data loss in Eq. (14) and (15).

A: Sorry for the confusion. The magnitude differences exist between physics loss and data loss, as we proved in Theorem A.1 in Appendix A.1. However, the magnitude differences may not exist if the losses are all physics losses, or all data losses. The losses $L_{pde}$ and $L_{pde-st}$ in Eq.(14) are both physics losses, while the losses $L_{mse-d}$, $L_{mse-st}$ and $L_r$ in Eq.(15) are all data losses. Thus, losses in Eq.(14) have relative same magnitude, and losses in Eq.(15) have relative same magnitude. Because we optimize Eq.(14) and Eq.(15) separately, the large magnitude differences between physics loss and data loss would not affect each other so that we can get the better performance. This is one of novelty for our model thanks to the idea of disentangling the data-driven model and physics-based model as students to fit their own objective functions.

Q: The question about Experiments settings on Page 6.

A: Sorry for the confusion. As mentioned in above answers, the losses in Eq.(14) have relative same magnitude, while losses in Eq.(15) have relative same magnitude. Thus, we can set all of the hyper-parameters in Eq.(14) and Eq.(15) to 1 instead of spending lots of time to find proper hyper-parameters in existing methods. Actually, if the hyper-parameters in Eq.(14) and Eq.(15) are determined by a grid search during the training process, it may get better performance than the results shown in our paper. However, it seems not necessary since it will take a lot of time, and our model has already achieved better performance when they are set to 1 compared to the previous work.

Moreover, the fact that the hyper-parameters in Eq.(14) and Eq.(15) are set to 1 has no direct relationship with whether it is 1D PDEs or 2D PDEs. As we disentangle the data-driven model and physics-based model as students to fit their own objective functions, they are optimized separately. So the hyper-parameters set to 1 is also applied to 2D PDEs. However, the existing methods just simply add physics loss and data loss, even for 1D PDEs which are relatively easy to learn, the magnitude differences between physics loss and data loss are huge (see left figure in Figure 1), and thus the hyper-parameters can not be set to 1. In conclusion, it is our framework of first disentangling and then assembling that makes the hyper-parameters in Eq.(14) and Eq.(15) set to 1 whether it is 1D PDE or 2D PDE, which strongly indicates the novelty of our framework.

Thank you for your constructive comments and suggestions. They are exceedingly helpful for us to improve our paper.

Please see our responses below. We have started working on the corresponding changes and will incorporate all of them in the final revision. If you have any further concerns, we would be keen to address them as well.

Q: Limited Novelty.

A: Thank you for your comments. Here is the novelty of the proposed model:

1. Though the problem that physics losses and data-driven loss don’t always play well together isn’t new and has been well researched in some papers, all their frameworks are the same that they still use the combination by adding the physics losses and data-driven loss together. We do some important experiments in Introduction Section to show that these simple integration way has many problems: large magnitude differences, varying convergence rates, and conflicting directions of the gradients so that the physics law can not work as expected. By finding these issues, we are motivated to propose our model to solve these problems.
2. Moreover, we theoretically analyze that the above issues will general exist in the previous methods in Appendix A.1 and A.2. The existing methods can not deal with the above problem simultaneously, since the framework of adding the physics losses and data-driven loss is the root problem. Thus, breaking the original integration way is urgent and innovative.
3. Therefore, the superiority of the proposed method is that we naturally propose an overall framework to disentangle the data-driven model and physics-based model as students to fit their own objective functions for the first time, whose rationale is analyzed in Theorem 2.1. Then we combine them in a more elegant way by assembling their embeddings to construct a competitive model. The key point is our whole framework where each parts have their own physical meanings. Our principle is utilizing concise and effective tools to construct the overall framework for better performance instead of designing the complex structure in each part, and the results show the superiority of our model.

Q: Important Related Work Missing.

A: Thank you for your suggestion. In fact, we focus on designing a model for a certain situation with a fixed boundary condition, thus we do not describe the operators method in related work. But we will follow your suggestion to give a brief review of operators in our revised paper.

Q: Incomplete Performance comparison.

A: Thank you for your suggestion. We have noticed the paper TrafficFlowGAN, and we originally intended to compare with it. There are the following reasons that we do not compare with TrafficFlowGAN:

1. The results summarized in their paper are shown in figures instead of tables. Thus, we do not know the exact metric values for different situations.

2. Because our experiment setting has changed, we need to rerun the codes of TrafficFlowGAN to make fair comparison. However, we can not get the results with the given parameters by authors for 20000 epochs, since it results in gradient explosion as mentioned in their paper, though the author has made some improvement.

As you mention the TrafficFlowGAN, we make efforts to achieve it. Since it may occur gradient explosion and it takes a long time to run, we can only give the best performance of TrafficFlowGAN for 8000 epochs with the parameters the author provided for US-101 dataset in limited time. It can be seen that the results have lower performance than our model.

Thank you for your constructive comments and suggestions. They are exceedingly helpful for us to improve our paper.

Please see our responses below. We have started working on the corresponding changes and will incorporate all of them in the final revision. If you have any further concerns, we would be keen to address them as well.

Q: Could you elaborate on the difference/superiority between the proposed method and existing works on combining physical losses and standard ML losses (e.g., MSE)?

A: Sorry for the confusion. Here are the motivation and superiority of the proposed method:

1. We do some important experiments in Introduction Section to show that the existing works on adding physical losses and standard MSE losses have many problems: large magnitude differences, varying convergence rates, and conflicting directions of the gradients so that the physics law can not work as expected. By finding these issues, we are motivated to propose our model to solve these problems.
2. Moreover, we theoretically analyze that the above issues will general exist in the previous methods in Appendix A.1 and A.2. The existing methods can not deal with the above problem simultaneously, since the framework of adding the physics losses and data-driven loss is the root problem. Thus, breaking the original integration way is urgent and innovative.
3. Therefore, the superiority of the proposed method is that we naturally propose an overall framework to disentangle the data-driven model and physics-based model as students to fit their own objective functions for the first time, whose rationale is analyzed in Theorem 2.1. Then we combine them in a more elegant way by assembling their embeddings to construct a competitive model. The key point is our whole framework where each parts have their own physical meanings. Our principle is utilizing concise and effective tools to construct the overall framework for better performance, and the results show the superiority of our model, which is not a brute-force stacking of various established methods.

Q: It's inconclusive whether the NN and PIDL models, if augmented with uncertainty quantification, would demonstrate weaker noise robustness than PDED.

A: Sorry for the confusion. We add more experiments to show that our model has better noise robustness not only because of the BNN module but also other parts such as assembling features of student models and distilling knowledge from teacher model. PDED-LWR w/o BNN denotes that PDED use NN instead of BNN in data-driven model. The results of L2 relative error for traffic state v on a 30% penetration rate in the synthetic dataset with different data noise ratios (Gaussian noise with variance 0.01) are shown as follows:

As shown in above table, we can see that even without BNN module, the prediction error of our framework also increases slowly than NN and PIDL models, which indicates that our model has better noise robustness not only because of using the BNN module. It also proves the novelty of our framework.