Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: 
**Summary and Contributions**
The paper proposes to approximately conserve desired quantities in sequential problems, such as when predicting the physical trajectory of an object. They argue intuitively that learning the right conserved quantities would be a useful inductive bias because symmetries (which are typical inductive biases) are often related to a conserved quantity through Noether Theorem. The proposed method meta-learns the quantity to be conserved from the data.

On the theoretical side, they study a specific setting where they prove an upper bound on the generalization gap which can be decreased when conservating some quantity. This suggests that learning models that conserve the right quantities could improve generalization.

They show in simple evaluation settings that the proposed method approximately learns the correct quantity to conserve and that conserving this quantity can help generalization.
main_review: **Strengths**
- **Nice and interesting idea that will be of interest to the community**.The idea and argument for learning the quantities to conserve instead of the symmetries is very compelling and will be of interest to the symmetry/physics/ML subcommunity. I find the explanation provided on lines 36-38 particularly compelling: “whereas symmetries [… ] perturbating the actual data, conserved quantities […] only the true data”.
- **Impressive results: recovering the hamiltonian** I’m impressed by the fact that you essentially recover the mathematical formulation of the correct hamiltonian even in noisy environments. The proposed method that achieves this is interesting and a valuable contribution.
- **Novelty** to my knowledge using conservation of a quantity as an inductive bias is surprisingly novel
- all theoretical results seem correct to me.

**Weaknesses**
- **Somewhat misleading characterization of your theory in abstract and introduction** The theory sounds much stronger than it actually is. Line 10-11 (abstract): “We show theoretically […] that Noether Networks improve prediction quality”, and lines 55-56 (intro) “We theoretically characterize […] lowering the generalization gap”.  Both are a mischaracterization of the theoretical results which you provide for 2 reasons. First, the theory only holds in a constrained case where the preimages of the conserved law form parallel affine subspaces and the model is constrained to translating the input. Second, even under the aforementioned constrained, your results do not prove anything about the generalization gap, only upper bounds which could be very loose. If you wanted to use bounds to prove that models that conserve quantities have smaller generalization gaps than those which don’t, you would need to provide a LOWER bound for theorem 1 and show that this lower bound is larger than the upper bound from Theorem 2. To be clear, I do not think that section 2 is problematic, but the language in the abstract and intro should not suggest that you prove any strong theoretical results (use words such as “suggest”, “could”, “restricted setting”).
- **Could be more clear** I find that many parts of the paper were not very clear (at least not until reading all the paper). E.g,
    - Figure 1 is probably your most important figure in that most people will try to understand your method from this. I did not understand it at all before reading the paper, and even now I find it hard to understand. I would especially suggest (1) making it clear what is the difference between the top and the bottom sequences; and (2) using the caption to explain in more detail what is seen in the figure.
    - The first paragraph on page 3 is very hard to understand due to having to parse all that notation at once. I would suggest trying to explain with English words what is happening. For example, the fact that you are assuming that input space can be partitioned in subspaces in which the desired quantity is conserved (i.e., the preimages) and that these partitions are parallel affine subspaces of dimension m. Diluting the notation with some intuition / English explanations would really help the reader.
    - I find the explanation of (meta)-tailoring in the text hard to understand. Maybe having a figure would help. Your algorithm does somewhat help but it doesn't show what happens at test time. Part of the issue is that you use the language of meta-learning but say it's different because you use it in a single task setting. Maybe you could stick with standard meta-learning and say that you treat every sequence as a single "subdataset". This is for example what people do when talking about neural processes.
    - when reading the intro and abstract I originally thought that you would use the conserved quantities to learn the symmetries. But the entire paper has nothing to do with symmetries: you simply hypothesize (using the intuition behind Noether theorem) that conserving quantities is a useful inductive bias because that's related to symmetries which have been shown to be a useful inductive bias. I think it would be useful to make it explicit right after your statement of Noether's theorem that you do not do anything with symmetries.
- **Experimental results are hard to judge** besides for the first experiment, the experimental results are hard to judge as (1) the only baseline doesn't even try to add some inductive bias; (2) there are no standard benchmarks for these types of results; (3) on the provided line plots the results seem quite similar and the lack of standard errors make it hard to judge if these results are significant.

**Questions**
- I do not understand how you avoid learning a trivial/constant value to conserve (i.e., problem 2 line 121). In particular, as you are performing training and meta-training on the training set, I do not understand what is the advantage of learning a useful conservation law (I agree that it would be the case if you used a validation set). Can you please clarify?
- To obtain symbolic formulas of the Hamiltonian, I don’t really understand what you do once you have 20 candidates. Do you train all of the 20, and then select the best one on the training set?

**Suggestions / specific issues**
- Please remove the main text from the supplementary material.
- In the future, please do not remove numbered lines for reviewing (the first paragraph of page 3 has no numbers)
- For theorem 1 and theorem 2, the loss will often be the cross-entropy and so $C= \infty$, right? If so, it should at least be pointed out that the bounds are vacuous for log loss.
- Please be consistent with your notation. For example, $\phi$ in section 2 is a function, but a parameter for the (same?) function $g_\phi$ in section 3. As another example you are using $f_{theta}(x)$ in section 2, $f_{theta}(x,t)$ in section 3.3 and then back to $f_{theta}(x)$ in Algorithm 1.
- In section 4.2. please mention the incurred computational overhead. In particular, it seems to be very large for finding symbolic formulas, right?
- Please add in the main text that you had to remove some datasets from Greydanus because the DSL couldn’t find the form of the symbolic formula.  
- Lines 515-517 of the appendices show an important advantage of your method, which I would briefly mention in the main text.
- In the proof of theorem 2, I would suggest stopping after line 501 and simply saying “the rest of the proof is the same as for theorem 1 but uses $\mathcal{}(r,\mathcal{Z}) \leq (2R \sqrt{m}/r)^m$". The redundancy of the current proof forces the reader to reread that part of the proof to understand how the result is different.

**Minor Suggestions**
- Line 30 / 33 / 114: i.e -> i.e.,
- Line 59: improve -> improving
- Top Paragraph of page 3: In the first occurrence of $i$ and $n$ please explain that it refers to the dataset/samples.
- Line 100: ensure satisfies -> ensure that it satisfies
- Line 100: bias -> unsupervised loss
- Line 136: described -> describe
- Line 242: cos q -> cos(q)
- Appcs line 457: remove “at”
- Appcs line 460: t he -> the
- Appcs equation 19: this is exactly the same as equation 15, I’m not sure what you wanted to add.
limitations_and_societal_impact: Limitations and societal impact are adequately discussed.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The authors present a meta-learning approach for identifying conserved quantities in time and leveraging them in time series predictions. They demonstrate the utility of their approach on a series of physical systems and real-world settings.
main_review: The idea of formulating a loss based on time conservation is very interesting in the context of time series, as it allows to account for additional symmetries in a data driven fashion. Identifying those symmetries directly can be a significantly harder task using the original data.

The experiments conducted show that their approach can successfully identify existing conserved quantities and improve predictive performance in physical setups and video prediction tasks. Although the improvements in the latter tasks are minor, they still serve as a demonstration of the algorithms utility, since in these cases the functional form of the conservation law was learned in a completely data driven fashion. As such, an analysis of the learned function (e.g. the filters of the CNN) would be of high interest to the reader, which is currently missing from the text. Another point which would warrant a more detailed discussion is how learned symmetries can be applied to the initial model. Currently this is done via a single gradient update of the weights, however this might not be the best strategy in all scenarios (e.g. due to the strong non-linear dependence of a function on the weights). Evidence for this can be seen in the pendulum control experiments, where it proved to be more beneficial to adjust the activations.

The algorithm, experiments and results are described in a clear fashion and theoretical background is provided on when it can be beneficial to account for conservation laws.

On the whole, the approach presented in the paper can be considered an interesting step towards learning conserved quantities and has the potential to serve as a conceptual basis for future developments, although a more in-depth analysis would be desirable.
limitations_and_societal_impact: Potential limitations and societal impact have been sufficiently addressed.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: **Summary**: 
This paper is concerned with the problem of automatic symmetry discovery, by meta-learning conserved quantities that are useful for predictions. They propose Noether Networks, where the learned symmetries take the form of a meta-learned conservative loss optimised inside the prediction function during test time. They demonstrate experimentally that the proposed approach can improve prediction quality and discover useful inductive biases in sequential problems.

**Contributions**:
- This paper proposes a new algorithm that automatically learns its inductive biases in the form of a meta-learned conservative loss optimised inside the prediction function.
- (In a simplified setting) They theoretically show when and why enforcing conservative laws brings benefits.
- They demonstrate through experiments that the proposed algorithm can recover known physical laws, and improve prediction performance for video data.

main_review: **Originality**: 
Firstly, meta-learning symmetries is a relatively new and under-explored problem, and this paper offers a simple yet effective framework. Secondly, the idea of meta-learning conservation loss optimised inside the prediction function during test time has not been explored before, and the authors show empirically that this is actually a viable direction. Thirdly, this paper demonstrates the success of meta-learning continuous symmetries and learning symmetries that are useful even in noisy or partial observation settings, which I think is a big obstacle for wider applications of symmetries.

**Quality**:
Although under a simplified setting, the theoretical analysis in section 2 still offers some insight on when and why enforcing conservation laws is beneficial. The conclusion is sensible and expected, but I did not check the mathematical statements rigorously.

The methodology is simple and clean. I’d be curious to hear the following because I think it is less clear in the manuscript:
- When a conservation loss is learned, does it always correspond to certain continuous symmetries? (the opposite of Noether’s theorem) How does it compare to learning a regulariser in general?
- To enforce the conservation laws, why did the authors choose tailoring and optimise conversion loss during test time. What is the motivation behind this choice?

The experiments are all sensible and well-conducted. But I would be more convinced if the authors can show on video data what would happen when the unsupervised loss does not take the form of conservation loss. Is it obviously inferior compared to a conservation loss?

**Clarity**:
The paper is well written and it is enjoyable to read. The methodology and theoretical analysis are clean and well-paced.

**Significance**:
This paper presents a simple yet effective framework to discover continuous symmetries in the form of a meta-learned conservation loss. It has been demonstrated to be effective for discovering continuous symmetries, and can be applied to challenging settings where the conservation law is not exact due to noise, or only partial information is available. I believe it is an interesting direction for future exploration.

limitations_and_societal_impact: **Limitations**:
I think the authors already address some of their limitations. One suggestion would be: A theoretical or empirical analysis of why the learned conservation loss must correspond to symmetries rather than a generic form of regularisation.

**Societal Impact**:
I think the authors already have a good discussion of potential societal impact.
review_title: null

final_score: rating: 5: Marginally below the acceptance threshold
review_content: summary: The authors tackle the challenge of automatically discovering symmetries in dynamical systems. As continuous symmetries are equivalently described by conserved quantities--as stated by Noether's theorem--they introduce a method for learning (approximately) conserved quantities. They do so by building on the `meta-tailoring' framework which allows to fine-tune the model according to the learned conservation laws at test time. For problems with few and interpretable coordinates (e.g. position, momentum, etc) they propose a symbolic approach that generate many valid (i.e. with compatible physical units) expressions that are parametrized with learnable weights. For raw-pixel data, they parametrize and learn the function to be conserved with a neural network. In both cases, the learned conservation function is used to guide the model towards physically 'realistic' solutions. They empirically assess their method on pendulum and spring systems, but also on real data from the Physics 101 dataset.
main_review: 
## Strengths
The proposed algorithm is of relevance to the community as incorporating physical inductive bias by hand is not always possible. Empirical assessment shows that the model is able to recover the true conserved quantity and that generalization performance is indeed improved. Authors also show that enforcing a `soft conservation' is useful as dissipative systems would only approximately conserve energy.
The theoretical evidence from Section 2 strengthen the motivation of the proposed approach.

## Weaknesses
The two main weaknesses that I can identify are;
1/ clarity, as developed later.
2/ the symbolic approach to learn the conserved expression. It is an interesting approach and it seems to work well in practice, but it has a bit of a heuristic flavour and has a non marginal computational overhead. Also, I am not entirely convinced that there exist problems of interest where there is a closed-form conserved quantity that we wouldn't know about. I may be wrong though.
Also, I am not really convinced by the Physics 101 experiment as the improvement seems marginal (hard to tell without confidence intervals) and there is no visualisation of the dataset, predictions or learned embeddings.

## Clarity
Generally, I feel that even though the paper is fairly well written some of the core ideas could and should be better conveyed. For instance, readers that aren't familiar with `meta-tailoring may struggle at Section 3.1. Since this is a core aspect of the proposed method, it would be worth adding an equation or a diagram guiding the reader.
Also, regarding Section 2, the first paragraph is quite packed with notations. I am not certain that it is necessary to introduce general notations as a special case is eventually considered. Also, a simple diagram could be useful to illustrate this setting and give intuition to the reader on the improved generalization gap. If I understand correctly, the main idea of this section is that, if the solution space is an affine subspace, and this is enforced, the the generalization gap bound grows w.r.t. the dimension of the subspace instead of the embedding space. I think that if this is indeed the core message, it should be stressed more.
Algorithm 1 is extremely welcomed as it is really helpful to understand the proposed approach.

## Reproducibility
No code has been provided in the supplementary materials, yet there seems to be quite some experimental details in Section 4. I am not certain that this is enough to reproduce the results.

## Additional feedback
- Links are broken.
- Lack of standard errors / confidence intervals.
- 53: "are"?
- Section 2: Is this setting motivated by a physical problem? Or because it is simple enough so that such analysis can be done.
- 65-66: What is the difference between $\phi$ and $G$?
- Equations 1 and 2: It should be stressed that the key difference is that $d$ is replaced by $m$, which is great if $m \le d$ (as stated on line 67).
- 76: It may not be immediate for the reader to understand what are the dimensions $d$ and $m$. A drawing may help?
- Figure 2 & Figure 3: Looks a bit blurry and the font is quite small.
- 246-161: It would be quite helpful to see what the 'Physics 101 ramp data' look like. Would there be a way also to visualize the learnt embeddings? Am whether it is correlated to the `true' conserved quantities (i.e. energy) in this setting?

limitations_and_societal_impact: Limitations are discussed.
review_title: null

Meta-review: All reviewers were happy in the end to accept this paper: I vote to accept. The biggest changes the reviewers wanted to see in the camera ready include: (i) improving the clarity, (ii) additional visualizations/experiments. Making these changes as well as others suggested by the reviewers will make a good paper a great one and a welcome addition to the conference.
Decision: Accept (Poster)