Paper: Towards Cross Domain Generalization of Hamiltonian Representation via Meta Learning (ICLR-2024)

Reviewer comment (Experiments):
The notion of $\textit{generalization}$: ideally when discussing generalization in DL we would like to obtain a single pre-trained model which can then be used for diverse applications. With respect to the presented framework, this would suggest training the model(s) on a single task and then using the same network for prediction on all held-out systems. Similar to the setting presented in Ricci et al. (2023). However, here presented results always consider a single held-out-system. Providing an ablation over the number of systems used for training will allow for strengthening the claim of generalization and applicability for real-world applications.

Author response:
> **A1**
> We appreciate the reviewer for suggesting additional experiments to strengthen our paper.
> Regarding weakness 1, it's important to clarify that our primary objective was to explore the generalization of Hamiltonian dynamics for predicting a single, unknown held-out system, underpinned by ample data. While the experiment proposed by the reviewer is indeed beneficial, it aligns more closely with further challenging scenarios involving adaptation to multiple unknown systems. Such situations extend beyond the current scope of our research, which focuses on a singular system context.

Reviewer comment (Experiments):
Baselines: it would be beneficial to extend the baselines presented in the paper in two directions: (i) optimal; Training over the tested task, using all regimes (meta-, pre-, and vanilla HNN). This will allow a better assessment of the quality of the generalized model and (ii) within system generalization; following the background presented in section 2 it will be valuable to add a comparison to frameworks that are similar in nature and allow $\textit{within}$ model generalization, e.g. CoDA (Kirchmeyer et al. 2022) or within the same functional form of the Hamiltonian, e.g. iMODE (Li et al. 2023). Here training over the same train-test splits.

Author response:
> **A1**
> For weakness 2, we originally thought that it was hard to directly compare our method with other prior baselines as the definition of the scope of generalization is different. However, we agree that adding such baselines would be beneficial. As such, we choose CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) as a representative method for domain generalization in learning the physical dynamics (as it is reported to be superior among other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised Section 4.4 and SM C.3 of our manuscript with the CoDA baseline results and included the corresponding discussion.

Reviewer comment (Experiments):
Can the authors provide additional ablation studies following the weaknesses presented above? Specifically, it will be beneficial to present the performance as a function of the number of systems used in training (see weaknesses 1.) and add additional baselines (see weaknesses 2.).

Author response:
> **A1**
> We appreciate the reviewer for suggesting additional experiments to strengthen our paper.
> Regarding weakness 1, it's important to clarify that our primary objective was to explore the generalization of Hamiltonian dynamics for predicting a single, unknown held-out system, underpinned by ample data. While the experiment proposed by the reviewer is indeed beneficial, it aligns more closely with further challenging scenarios involving adaptation to multiple unknown systems. Such situations extend beyond the current scope of our research, which focuses on a singular system context.
> For weakness 2, we originally thought that it was hard to directly compare our method with other prior baselines as the definition of the scope of generalization is different. However, we agree that adding such baselines would be beneficial. As such, we choose CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) as a representative method for domain generalization in learning the physical dynamics (as it is reported to be superior among other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised Section 4.4 and SM C.3 of our manuscript with the CoDA baseline results and included the corresponding discussion.

Reviewer comment (Experiments):
Judging from the presented results the current framework is not suitable for larger systems, could the authors suggest possible extensions that may allow? What would be the necessary refinements that could be incorporated in the meta-learning configuration to allow for that?

Author response:
> **A2**
> Although the results for the large systems are quite poor in the reported results, we want to mention that our approach can handle the problem when the degree of freedom of the prepared data is not consistent throughout the system, noting that the previous methods discussed in Section 2.2 and 2.3 cannot be used in such scenarios. Although we cannot give the exact refinements in our preliminary results, possible ways could be 1) tune the update method in meta-learning, e.g. [Lee et al. 2021](https://arxiv.org/abs/2102.11544) report that using ANIL variation gives more accuracy compared to vanilla MAML, and 2) strictly search the hyperparameter space.

Reviewer comment (Experiments):
The evaluation can be strengthened by considering comparing the proposed methods with other Few-shot Learning and Physics-informed Neural Networks methods for system domain generalization under both "consistent" and "different" physical laws.

Author response:
We first thought that it would be difficult to directly compare our method with other previous baselines, as the definition of the scope of generalization is different. However, we strongly agree with the reviewer that our evaluation can be strengthened by comparing the proposed approach with other existing methods. We added a baseline experiment with CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) which is considered a representative method for domain generalization in learning physical dynamics (as it is reported to be superior to other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised section 4.4 of our manuscript with the CoDA baseline results and discussion.

Reviewer comment (Evaluation):
In Figure 4, why does a lower CKA value of the meta-trained model suggest it learned more similar representations during adaptation? As the authors mentioned, should a low CKA value indicate different representations?

Author response:
We acknowledge that our label notation in Figure 4 could be quite misleading. The y-axis represents the 1-CKA (1 minus CKA) value, not the raw CKA value. Therefore, the lower y-axis in Figure 4 indicates a high CKA value, which leads to more similar representations during adaptation.

Reviewer comment (Novelty):
The main contribution of the paper seems to be utilizing meta-learning to efficiently adapt to new systems. However, it is not clear from the paper if it is as simple as just using off-the-shelf implementation or if there are some challenges to doing this.

Author response:
> **A1, A2**
> We presume that weaknesses 1 and 2 arise from the lack of clarity that we have written in the paper. Existing previous methods have clear limitations; 1) they do not guarantee generalization across diverse systems, and 2) to do so, they need to be flexible with respect to the varying degrees of freedom of the systems. Our approach aims at resolving these challenges by jointly using GNNs and meta-learning. Accordingly, we have revised section 2.3 to clearly state the points mentioned by the reviewer in Q1 and Q2.

Reviewer comment (Novelty):
I would like to see some discussion around why meta-learning is preferred over other representation learning methods e.g. Domain Generalization, and why optimization-based methods surpass other approaches in meta-learning.

Author response:
> **A1, A2**
> We presume that weaknesses 1 and 2 arise from the lack of clarity that we have written in the paper. Existing previous methods have clear limitations; 1) they do not guarantee generalization across diverse systems, and 2) to do so, they need to be flexible with respect to the varying degrees of freedom of the systems. Our approach aims at resolving these challenges by jointly using GNNs and meta-learning. Accordingly, we have revised section 2.3 to clearly state the points mentioned by the reviewer in Q1 and Q2.

Reviewer comment (Experiments):
The experiments are not sufficient. First, there is no comparison with existing baseline models in domain generalization or meta-learning.

Author response:
> **A3**
> - First, we acknowledge the insufficient comparison with existing baselines. Originally, we considered it to be hard to directly compare our method with other prior baselines due to the different definitions of the scope of generalization, we added a baseline experiment with CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) which is considered a representative method for domain generalization in learning physical dynamics (as it is reported to be superior among other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised Section 4.4 and SM C.3 of our manuscript with the CoDA baseline results and discussion.
> - Second, while we appreciate the reviewer's perspective, our work is at the cornerstone in our scope of the problem definition, and thus to show both the possibility and the effectiveness of the unified representation of Hamiltonian dynamics, we chose as baselines the 1) pre-trained model that used the same total number of gradient steps during meta-training, and the 2) randomly initialized model, following the approach of the prior work of [Finn et al. 2017](https://arxiv.org/abs/1703.03400) and [Lee et al. 2021](https://arxiv.org/abs/2102.11544).
> - Third, we respectfully ask the reviewer to check Section 4.2, where the dynamics of the predicted systems are discussed using Figure 3. However, we acknowledge that the presentation of the predicted dynamics written in the paper should be more explicit and accordingly we have updated the title of Sections 4.1, 4.2 and the caption of Figure 3.

Reviewer comment (Experiments):
The comparison of the meta-model and pre-trained model is not fair, and I would suggest the author fine-tune the pre-trained model on the K-shot support set.

Author response:
> **A3**
> - First, we acknowledge the insufficient comparison with existing baselines. Originally, we considered it to be hard to directly compare our method with other prior baselines due to the different definitions of the scope of generalization, we added a baseline experiment with CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) which is considered a representative method for domain generalization in learning physical dynamics (as it is reported to be superior among other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised Section 4.4 and SM C.3 of our manuscript with the CoDA baseline results and discussion.
> - Second, while we appreciate the reviewer's perspective, our work is at the cornerstone in our scope of the problem definition, and thus to show both the possibility and the effectiveness of the unified representation of Hamiltonian dynamics, we chose as baselines the 1) pre-trained model that used the same total number of gradient steps during meta-training, and the 2) randomly initialized model, following the approach of the prior work of [Finn et al. 2017](https://arxiv.org/abs/1703.03400) and [Lee et al. 2021](https://arxiv.org/abs/2102.11544).
> - Third, we respectfully ask the reviewer to check Section 4.2, where the dynamics of the predicted systems are discussed using Figure 3. However, we acknowledge that the presentation of the predicted dynamics written in the paper should be more explicit and accordingly we have updated the title of Sections 4.1, 4.2 and the caption of Figure 3.

Reviewer comment (Presentation):
There is no visual comparison of the predicted dynamics and the ground truth, making the conclusion less convincing.

Author response:
> **A3**
> - First, we acknowledge the insufficient comparison with existing baselines. Originally, we considered it to be hard to directly compare our method with other prior baselines due to the different definitions of the scope of generalization, we added a baseline experiment with CoDA ([Kerchmeyer et al. 2022](https://arxiv.org/abs/2202.01889)) which is considered a representative method for domain generalization in learning physical dynamics (as it is reported to be superior among other related methods according to the CoDA authors). As expected, CoDA failed to adapt to an unseen system with the same number of steps, even for sufficient steps up to 10000. We have revised Section 4.4 and SM C.3 of our manuscript with the CoDA baseline results and discussion.
> - Second, while we appreciate the reviewer's perspective, our work is at the cornerstone in our scope of the problem definition, and thus to show both the possibility and the effectiveness of the unified representation of Hamiltonian dynamics, we chose as baselines the 1) pre-trained model that used the same total number of gradient steps during meta-training, and the 2) randomly initialized model, following the approach of the prior work of [Finn et al. 2017](https://arxiv.org/abs/1703.03400) and [Lee et al. 2021](https://arxiv.org/abs/2102.11544).
> - Third, we respectfully ask the reviewer to check Section 4.2, where the dynamics of the predicted systems are discussed using Figure 3. However, we acknowledge that the presentation of the predicted dynamics written in the paper should be more explicit and accordingly we have updated the title of Sections 4.1, 4.2 and the caption of Figure 3.