Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 8: Accept
review_content: Summary:

The paper improves upon Hamiltonian Neural Networks to model physical systems from observed trajectories. Specifically, the authors propose to use (i) better integrator, (ii) multi-step learning, and (iii) initial state optimization. Authors experimentally show that all these improvements are beneficial in modelling complex and noisy Hamiltonian systems.


My Comments:

I come from RNNs background and I am not an expert on physical systems. But the paper is extremely well written that I can easily follow. Experiments are convincing.

I do not have any issues with the claims.

Are the authors willing to release their code to reproduce the results?

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Post rebuttal: I stand by my decision.

review_title: Official Blind Review #3

final_score: rating: 8: Accept
review_content: This paper proposes to represent a Hamiltonian model of a physical system by a neural network. The parameters are then adjusted, so that the observations are considered maximally likely under a probabilistic model. The novelty is to consider a symplectic Leapfrog integration scheme for the Hamiltonian system, which is known to conserve important quantities such as volume in the state space. The proposed approach is shown to outperform the recent work "Hamiltonian Neural Networks" by a large margin on mass-spring chain dynamics and three body systems. The approach can even handle stiff dynamical systems such as bouncing billiards. 

Overall, the work is solid contribution and a reasonable improvement over the recent work on HNNs is demonstrated. Therefore I recommend acceptance of the paper. However, I have some fundamental doubts on the motivation on this line of works. This might be, because I'm not too familiar with the subject, and I'm willing to increase my score if the doubts are cleared.

In the shown examples, to best of my knowledge, the "exact" Hamiltonians describing the physics of the system are well-known. Therefore, I'm unsure what is the advantage of trying to learn physical laws, that are already well understood. The paper argues that the learned Hamiltonian will correct for errors in the discretization, but one could instead use a better integration scheme or a finer time-discretization, based on classical theory which has been developed over the last 50 years which comes with strong convergence guarantees and error bounds. I would have liked to see a stronger motivation, why it is interesting to learn an Hamiltonian of a system, where the exact Hamiltonian is already known. It would also be enlightening to see some plots, which illustrate how "far" the learned Hamiltonian is from the analytical one.

Of course, one might argue that the ultimate goal is to have a learning based approach discover physical laws so far unknown to humans,  just from observations. But it is unclear why the inductive bias that the observations are generated by a Hamiltonian might be reasonable. It could very well be, that the law cannot be described by a Hamiltonian system. 

From a high-level point of view, one might even argue that it is not too surprising that one can fit a parametrized Hamiltonian to observations generated by a Hamiltonian system better than a general purpose function approximator without such an inductive bias or better than a system based on a naive/unsuitable non-conservative integrator.

As a remark, often the exact Hamiltonian is known to be (strictly) convex. I'm wondering whether convex function approximators such as convex neural networks could provide an even stronger inductive bias. But it might be that a general purpose RNN can account better for the discretization errors. 
review_title: Official Blind Review #2

final_score: rating: 6: Weak Accept
review_content: This paper introduces SRNN to model the hamiltonian of a dynamical system. Authors break down the design choices made in the algorithm and validate each one through experiments. 

I like the motivation of the paper of solving general physics problems using neural networks. The paper is well-written and the ideas are communicated properly.

However, I have some concerns regarding the experimental results:

In Figure 1, even though the fifth mass seems to follow the exact pattern of observations for single-step L-L H-NET, the L2 error is increasing after each time-step suggesting that the fifth mass might not be the best one to consider for comparison between different algorithms here. Would be nice if a comparison between all trajectories was presented and discussed (perhaps in the Appendix)

The idea tested in section 4.2 seems not novel since previous works have already used recurrence to mitigate problems faced using one-step training. (https://arxiv.org/abs/1902.09689) 

I like the optimization method proposed in Section 4.3 and the results in Table 1 and 2 seems to justify its effect. Does this approach also handle degenerate cases in which different initial states can lead to the same trajectories after some amount of time? To me, it seems like a boundary on the noise variance should be assumed. Otherwise, no amount of optimization would be actually able to retrieve the correct initial state. Is this true or am I missing something?



-minor comments:

1. The conclusion section is missing from the paper. It would be nice to recap your findings there.
2.  In section 2, the "universality property" of Niu's recurrent model should be explained or referenced. 
3. In Figure 1, the second "Left:" should be changed to "Right:".
4. In Table 2, the Error std. for O-NET (E-L) should be bold not H-NET (L-L) unless it was intended to highlight the values for the model with the lowest mean error. (please clarify this in the paper.)
5. Section 6: "We focus on this section" should be changed to "We focus in this section".

 
Overall I think the work is interesting but it lacks some justifications regarding the claims made (as mentioned above), and although it is generalizable to other tasks and systems, it does not have sufficient novelty in its algorithm and approach.

As of now, I am recommending a rejection, but I am willing to reconsider my score should the authors address the above concerns.


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Updates:

Thank you for your response and addressing my concerns in the revised version of the paper. I also see new updates to the text which has improved the readability considerably. Thank you for your work.
I have updated my score accordingly. 



review_title: Official Blind Review #1

Meta-review: This paper proposes a novel architecture for learning Hamiltonian dynamics from data. The model outperforms the existing state of the art Hamiltonian Neural Networks on challenging physical datasets. It also goes further by proposing a way to deal with observation noise and a way to model stiff dynamical systems, like bouncing balls. The paper is well written, the model works well and the experimental evaluation is solid. All reviewers agree that this is an excellent contribution to the field, hence I am happy to recommend acceptance as an oral.
Decision: Accept (Spotlight)