Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: This work addresses the problem of learning Hamiltonian systems from data without assuming that the generalized momentum is given. This assumption limits the applicability of standard Hamiltonian Neural Networks (HNNs) because the generalized momentum is defined in terms of the Hamiltonian itself, which means it cannot typically be provided as training data for unknown Hamiltonians. This work proposes to get around this limitation by exploiting a coordinate-free formulation of the Hamiltonian and incorporating the 2-form from symplectic geometry into the structure of a HNN (either directly as a learned skew-symmetric matrix or via a learned 1-form and an exterior derivative). The resulting architecture performs favorably compared to a standard HNN and Lagrangian Neural Network (LNN) on several simple physical systems.
main_review: Positives:
- By eliminating the requirement for labeled generalized momenta at input, the core technical contribution of the paper overcomes a key limitation of HNNs and does so in a simple and elegant way. 
- The paper does an admirable job exposing the relevant literature in a way that I feel will be largely interpretable by the NeurIPS audience, despite the fact that it draws heavily from symplectic geometry. In particular, I found the discussion of 1- and 2-forms and the exterior derivative very readable.

Negatives:
- My largest concern with this paper lies in the experimental sections. Experiments were conducted with very small (single hidden-layer) networks, which are smaller than typically used in this literature. This appears to be leading to artificially bad performance by HNN and LNN baselines, as discussed in L269-272 and section D. While it's suggestive that the proposed method works well with this network, it makes the results very hard to interpret in the context of the literature. This choice isn't motivated as far as I could tell. For the method to be interpretable, it should be run in settings where HNNs and LNNs are known to perform well.
- The model state prediction should be analyzed quantitatively: how well does the model predict the state when integrated over time? It's hard to interpret the quality of the estimates without knowing how it fares over longer horizons than a single step, which is essentially what's quantified in tables 2 and 3.
- How were the energy errors in Table 3 estimated? As I understand it, these models can only recover the energy up to a constant offset.
- I believe the paper could be much more impactful if more difficult experimental settings where the proposed method can shine were explored. In particular, the proposed formulation appears to be particularly amenable to image- or other observation-based depictions of Hamiltonian systems, because the method does not depend on a representation being recovered in the Darboux coordinate system and does not require momentum to be estimated. Image-based Hamiltonian learning has been addressed in several prior works in the literature e.g. [1, 2, 3], and it would be very useful to know how the proposed method fares in those settings and how the present analysis sheds light on those experiments.
- Several of the figures could be improved:
  - Figure 1 is cluttered and hard to follow: this could probably be improved by moving many of the graphical elements to the text description, which should be expanded. I also found it hard to read the left-hand side of the figure, as the double pendulum and Lotka-Volterra systems look as if they're both being input to the system (rather than showing two example inputs).
  - Figure 2 is very hard to follow and is essentially uninterpretable without referring to the main text.
  - Figures 5, 6, and 7 are missing axis labels and do not refer to what system they evaluate.

Minor:
- L92: extra "the" at the beginning of the sentence
- L115: (4) should be (2).
- I found it a little hard to understand the structure Y_{NN} on a first read. I believe this would be easier to understand if its input and output dimensions were mentioned.
- L190: "If the time derivatives are not available, interpolated data should be used." Please rephrase this - I believe the implication is that finite-difference estimates of the temporal derivatives should be used.
- L204: Nitpick: I believe a more idiomatic way to refer to this section in a machine learning paper would be "Experiments" or "Numerical Experiments".

Overall: I believe this work makes a solid contribution to the literature on learning Hamiltonians and the weaknesses are addressable, so I weakly recommend acceptance. However, I think the paper would be much stronger if the experimental section were improved.

I am willing to raise my score if the experimental limitations pointed above are addressed, and ideally if the authors can present results on more challenging datasets.

[1] Greydanus et al - NeurIPS 2019 - Hamiltonian Neural Networks
[2] Toth et al - ICLR 2020 - Hamiltonian Generative Networks
[3] Allen-Blanchette et al - AAAI 2020 - LagNetViP: A Lagrangian Neural Network for Video Prediction
limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: Authors present a ML-based approach to learning dynamics in the space of Hamiltonian systems. The approach is based on generic formulation of Hamiltonian mechanics in the language of differential geometry, effectively avoiding the requirement of knowing the appropriate generalized coordinates beforehand. The proposed approach is based on modeling the symplectic 2-form (known to be closed) as an exterior derivative of a parametrized (learned) 1-form. This method is termed "neural symplectic form" and provides a compact parametrization of the underlying governing equations.

The proposed model is trained by minimizing the MSE between estimated ground truth time derivatives of the dynamical system and predictions of the model on several datasets: mass-spring system; double pendulum; Lotka-Volterra equations (which are not captured by a traditional formulation of Hamiltonian mechanics with fixed symplectic form).

The model is compared to baselines (HNN, LNN and a naive implementation termed skew matrix learning) on test loss and errors of the conserved quantity in the predicted dynamics accumulated during simulation for a fixed time.

Authors find that the proposed model produces best results across most of the comparisons, especially on the Lotka-Volterra test case.


main_review: The article present an original approach to modeling Hamiltonian systems that uses fewer variational parameters while accurately capturing the target space of dynamical laws. When combined with symplectic integrators, such systems admit conservation laws that are highly desirable for long-term modeling. In my opinion this is a significant direction worthy further exploration.

The work is written clearly, but is definitely more accessible to readers with elementary background in exterior calculus. One of the organization caveats that in my opinion should be addressed is the silent push of the NODE model to the appendix, which often perform on-par with the proposed method.

The results section raises a few questions:
1) Some of the error-bars are comparable/larger than the mean values? Is it possible to include more experiments to make comparisons precise?
2) While summarized errors are useful it would be valueable to include the time dependence of the errors as a function of time (for both energy and MAE/MSE)
3) The networks used for modeling the system were MLP with a single hidden layer - it would be very reassuring to see that the results are not too sensitive to this choice, especially in a more expressive regime.

Overall - interesting paper worth publishing assuming that the results section can be tuned up.
limitations_and_societal_impact: The work does not have negative societal impact.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper proposes to learn conservative physical dynamics from data by directly learning a differential 1-form. The dynamics is subsequently determined by computing the exterior derivative of the learned 1-form (yielding a symplectic 2-form) and providing a coordinate-free representation of the dynamics. In addition to inherently preserving the symplectic nature of the dynamics, this procedure does not require *a priori* knowledge of the generalized momentum coordinates, i.e. only velocities need be observed so masses of the degrees of freedom in the data are not needed. The neural symplectic form methodology is compared to an approach in which the symplectic 2-form is learned directly as a skew-symmetric matrix as well the existing Hamiltonian neural network and Lagrangian neural network approaches. 
main_review: Learning the underlying Hamiltonian structure of observed trajectorial data requires methodologies that are sensitive to the conservation laws inherent in the physical structure of Hamilton's equations of motion. Without a doubt, conservation of energy is the most important property to preserve. This can be achieved by using a symplectic integration and learning with a symplectic integration scheme noticeably improves the quality of learning (cf. Chen et al. https://arxiv.org/abs/1909.13334 which should surely be cited). Indeed, a direct approach of this type requires rather complete information about the system, including knowledge of all generalized positions and momenta along a time series. This paper aims to relax this requirement by parameterizing a coordinate-free representation of the equations of motion, an elegant idea with significant potential impact.

The formulation of the algorithm relies on (1) representing the Hamiltonian $H$ as a neural network and (2) either representing the symplectic 2-form $W_u$ as a neural network or representing a 1-form as a neural network and computing its exterior derivative to compute $W_u$. These formulations constitute the core contribution of the paper. In both cases, the authors establish formally that the dynamics conserves energy (a comment in the proof of Theorem 1 could be added to indicate that the argument also applies to Theorem 2). The proof of the universal approximation theorem is rather perfunctory, but I do think it constitutes a reasonable addition to this work. The arguments for all three theorems are essentially standard results.

The motivation for preferring the neural symplectic form approach is motivated, in part, by invoking de Rham cohomology. I found this to be a bit of a distraction, likely to lead to confusion. The articulation beginning at line 153 struck me as a much clearer and conceptually better explanation of why one should prefer learning the 1-form. Of course, if this could be clearly tied to the de Rham cohomology, it could be worthwhile to discuss this. Fig. 2 uses a lot of space and did not add meaningfully to the manuscript in my opinion. I appreciated the discussion of the computational cost of the inverse of $W_u$---this is a significant limitation of the algorithm for large systems. 

I found the numerical experiments useful, concise, and sufficient to establish the efficacy of the method. 

Is learning $W_u$ directly as a 2-form distinct from learning a mass matrix in the case where the kinetic energy is assumed to be harmonic?

It would be useful for the authors to make some contact with the literature on learning potential energy functions (e.g., the work Tkatchenko and coworkers) because in that line of inquiry it has been concluded that learning with forces (learning the gradients of the Hamiltonian) is often superior to learning the potential energy. This strikes me as being analogous to learning the 1-form vs learning the 2-form and I'm not sure how to reconcile the two conclusions. 
limitations_and_societal_impact: There are a few inherent limitations when it comes to scaling this problem up to large, many-body systems. First, it is hard to imagine a realistic setting in which one could observe all the positions of a complex system with sufficient spatio-temporal resolution. Secondly, most many-body systems will decorrelate over short times due to ergodicity and the onset of chaos. In my mind, this makes the problem a rather academic one without significant applications in physics per se. That said, discovering Hamiltonian structure in ODEs (as in the Lotka-Volterra example) provides a nice counterpoint to this criticism.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This paper considers the learning of non-canonical Hamiltonian dynamics from time-series data. Canonical Hamiltonian systems in Euclidean spaces correspond to $\dot{x}=J \nabla H(x)$ where $J=[0,  I; -I, 0]$ is a constant matrix corresponding to the canonical symplectic form. Non-canonical Hamiltonian systems are in the same form, however $J$ can be $x$ dependent, and it has to be a skew-symmetric matrix satisfying Leibniz’s rule and Jacobi identity. This paper studies the Euclidean case, which has nice enough topology so that $J:\mathbb{R}^{d}\to \mathbb{R}^{d\times d}$ satisfying these conditions can be expressed using a vector field $Y: \mathbb{R}^{d}\to \mathbb{R}^d$ instead, which is no longer constrained. The proposed method represents $H$ and $Y$ by neural networks and trains them to match the data. Some numerical investigations are provided.
main_review: The method is new and interesting. Although it needs a nice cohomology like that of the Euclidean space, the setup corresponds to sufficiently many important problems. The paper is well written and the informal descriptions are very helpful to general readers.

While I like the continuous (in time) part of the paper, I hope the authors could agree that discretization matters very much when the method is actually implemented. This is something that the current version can improve upon. For example, the formulation of the method is summarized inbetween Line 189 and 190, but how is $du/dt$ estimated? After the continuous dynamics is learned, how is prediction done? In fact, these questions are central to many of the recent advances in this area, some of them already cited, such as [20, 21], but two important references are missed. Namely, [Jin et al. SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Neural Networks 2020] and [Chen & Tao. Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps. ICML 2021]. These aspects should be discussed for the sake of the readers.

In addition, [12] was cited but not sufficiently discussed. However, it is closely related work and their similarity and difference should be clarified.

Continuing along the line of discretization, it was written that “A numerical integration of the Hamiltonian system is known to destroy the symplectic structure and does not conserve the Hamiltonian H.” This is incorrect. Symplectic integrators exist; see for instance [Hairer Lubich Wanner. Geometric Numerical Integration. 2006] for separable Hamiltonians (and much more) and [Tao. Explicit symplectic approximation of nonseparable Hamiltonians. 2016] for the rest of the picture. Regarding conserving the Hamiltonian, though, one cannot have simultaneous exact conservation of symplecticity and energy in constant step integrations, as the Ge-Marsden theorem gives.

In addition, I would like to understand two things:

Firstly, the proposed method is better than skew matrix learning and much better than HNN, according to the mass spring experiment and the double pendulum experiment, but why? For these two problems, the coordinates are already canonical, so I was expecting the new method to degenerate to HNN and give comparable performance, but it is not the case. What are the learned symplectic 2-forms in these two cases? If it is because of different network architecture and/or training details, the comparison needs to be made fair.

Secondly, how is the invertibility of $\tilde{W}$ ensured?

limitations_and_societal_impact: N/A
review_title: null

Meta-review: All ratings were "accept". The reviewers raised various suggestions which the authors responded to thoroughly, with additional experiments that reinforced their results. This should be a solid contribution.
Decision: Accept (Spotlight)