Paper: PDED: Revitalize physics laws submerged in data information for Traffic State Estimation (ICLR-2024)

Reviewer comment (Experiments):
As shown in **Experiments settings** on Page 6, all of the hyper-parameters are set to 1. It seems they don’t have large variances in magnitudes. However, the test datasets focus on $x, t$ dimensions, which can be regarded as 1D PDEs in physics-informed learning. 1D PDEs are relatively easy to learn, and the optimization issues are not severe. I am wondering if the authors could test on more challenging 2D datasets in traffic modeling.

Author response:
**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.

Reviewer comment (Novelty):
On Page 2, the authors claim that they discovered the optimization issue of physics-informed learning, where many existing research works have identified this problem [1-2]. I don’t think that is one of the contributions of this paper.

Author response:
**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.

Reviewer comment (Writing):
On Page 5, “As mentioned by Theorem 2.1, the effectiveness of ensemble teacher model is also determined by the the accuracy of individual student model.” There is a double “the” in this sentence.

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

Reviewer comment (Evaluation):
This paper considers relative L2 errors, as commonly seen in physics-informed learning research. However, in the domain of traffic prediction, researchers also consider RMSE and MAPE to measure the errors from different perspectives. Is it possible to add more evaluation metrics since the traffic data has some rush hour phenomena, where extreme statistics are needed?

Author response:
**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.

| Loop     | 2      | 6      | 10     | 15     | 18     |
| :------- | :----- | :----- | :----- | ------ | ------ |
| PIDL-LWR | 0.0351 | 0.0307 | 0.0291 | 0.0281 | 0.0278 |
| PDED-LWR | 0.0336 | 0.0287 | 0.0259 | 0.0236 | 0.0230 |

Reviewer comment (Experiments):
Regarding the noise setting in Section 3.4, is it common to set the noise ratio by using a small covariance of 0.01/0.04 in the Gaussian distribution? Can the authors add some references here?

Author response:
**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].

Reviewer comment (Novelty):
**[Limited Novelty].** The proposed model is a combination of a physics informed neural network (PINN) employed with a relatively simple PDE, in addition to another standard Bayesian Neural Network (BNN) paradigm combined with knowledge distillation (KD). These three paradigms (PINN, BNN, KD) are all extremely well studied, well-understood and the current paper doesn’t propose any novel extensions of the actual paradigms or increase the characteristic understanding of any of the aforementioned paradigms. It is simply an exercise in the application (specifically, the combination) of these paradigms in a (somewhat) creative way to address the problem of traffic state estimation.

Author response:
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.

Reviewer comment (Novelty):
**[Important Related Work Missing].** The paper completely misses mentioning operator learning paradigms which are more recent updates to traditional PINNs which learn families of PDEs as opposed to single instances of PDEs (albeit in a slightly different manner). The reviewer believes that a brief description of operators like [1, 2] would contextualize the current work in the physics informed deep learning space.

Author response:
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.

Reviewer comment (Experiments):
**[Incomplete Performance comparison].** A related paper Trafficflowgan [6] that employs physics information, uncertainty aware GAN for traffic state estimation has not been compared against.

Author response:
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.

|                |               | Loop   |        |        |
| :------------- | :------------ | :----- | :----- | ------ |
| Model          | Traffic State | 2      | 6      | 10     |
| TrafficFlowGAN| v             | 0.1925 | 0.1851 | 0.1757 |
| PDED           | v             | 0.1237 | 0.0531 | 0.0417 |
| TrafficFlowGAN| $ \rho $        | 0.3095 | 0.2858 | 0.2547 |
| PDED           | $ \rho $       | 0.2045 | 0.1724 | 0.1572 |

Reviewer comment (Experiments):
Why has TrafficflowGAN [6] not been compared despite being a related / physics-informed + uncertainty aware model for traffic state estimation?

Author response:
Thank you for your suggestion. As you suggest, we fine-tune the parameters of TrafficFlowGAN again. Since it may result in gradient explosion as mentioned in their paper, we give the best performance of TrafficFlowGAN for 8000 epochs. Their source code and data are available at https://github.com/ZhaobinMo/TrafficFlowGAN, and **you can also rerun the code if you think it is not a representative comparison between the two models.** 

|                |               | Loop   |        |        |
| :------------- | :------------ | :----- | :----- | ------ |
| Model          | Traffic State | 2      | 6      | 10     |
| TrafficFlowGAN | v             | 0.1719 | 0.1621 | 0.1573 |
| PDED           | v             | 0.1237 | 0.0531 | 0.0417 |
| TrafficFlowGAN | $\rho$        | 0.2884 | 0.2631 | 0.2462 |
| PDED           | $\rho$        | 0.2045 | 0.1724 | 0.1572 |

Reviewer comment (Novelty):
Using Traffic State Relation Distillation, Ensemble Distillation, and anomaly detection via BNN lacks motivation. The author's approach seems to amalgamate all available techniques into the model to ascertain if performance can be improved, resembling a combination of existing strategies. At least within this paper, the author fails to provide a rationale for choosing this particular strategy, thereby giving readers an impression of a brute-force stacking of various established methods.

Author response:
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.