As I mentioned, up to my knowledge, the idea of using Hamiltonian equations as loss functions of NNs is new, interesting and easy to follow. However I am not convinced that it can be applied to a large set of physical problems. The major draw back is that the Hamiltonian equations should be known in advance by the designers of the model rather than learned from data. Another short-coming is their trivial experimental results. As a matter of fact, I do not find much point in the presented toy tasks 1, 2 and even 3, as the maximum information that the network is potentially able to learn is to estimate the noise parameter as otherwise the provided prior knowledge is sufficient to solve these tasks (and therefore no neural net is needed). The last task is much more interesting because the NN learns to link the raw data (pixels) into the quantities for which the Hamiltonian is defined. However even this task is in a sense too simple and does not convince me that such an approach can be applied to any real world problem.

This paper is very well written, nicely motivated and introduces a general principle to design neural network for data with conservation laws using Hamiltonian mechanics. Contrary to what the authors state, including energy conservation into neural networks and optimizing its gradients is now common procedure in this domain, for example: - Pukrittayakamee et al. Journal of Chemical Physics, 130(13). 2009 - Behler. Physical Chemistry Chemical Physics, 13(40). 2011 - Gastegger. Journal of Chemical Theory and Computation, 11(5), 2187-2198. 2015 - Schuett et al, NeurIPS 30 / Journal of Chemical Physics 148(24). 2017 - Yao et al., Chemical science 9(8). 2018 The proposed approach constitutes a generalization of neural networks for high-dimensional potential energy surface by including the momentum in the input to arrive at the Hamiltonian formalism. For classical systems, as presented in this paper, it seems that this addition is rather counter-productive: while the change of momentum is described by the potential (see references above), the change of positions directly follows from the equations of motion and does not require an additional derivative of the network. This is both more computationally efficient and generalizes by design to all initial momenta (provided the corresponding positions stay close to the training manifold). On the other hand, I am not convinced that the proposed architecture would still work when applying a trained model to a different energy level. The mentioned property of reversibility is not convincing as it is not described how this property can help to make back-propagation more efficient. Reversing the trajectory only allows to discard the intermediate time step, which are already present in the training set. For the network itself, one still needs to use the full back-propagation of each time step to get the gradient w.r.t. the parameters. Beyond that, the mentioned reversibility is given for all predictions that are by-design conservative force fields, such as the earlier mentioned neural network potentials. A strong point of the paper is the application to the pixel pendulum. This could inspire new and creative applications of neural networks that encode conservation laws. I would still guess, that similar results can be obtained from a pure potential approach without momentum inputs. It would be interesting to see future, more general applications, where this is no longer possible. In the author feedback, my main issue with the paper has not been addressed sufficiently. The Hamiltonian formalism is more general than predicting just the potential energy + derivatives (which amounts to energy conservation from classical mechanic), but this is not taken advantage of. The author's response, that potential energy approaches need potential energy labels is not correct: they can also be trained solely on derivatives (e.g., see JCP ref above). A concrete example: A consequence of the more general Hamiltonian formalism is that the network would need to be retrained for a pendulum with a different total energy since the impulse at the same pendulum position would be different. In contrast, a network using potential energy derivatives + equations of motions can handle different energy levels, since it does not depend on the impulse and takes only positions. Apart from this, the approach is a valuable contribution with great potential in other areas than classical mechanics.

1. Originality: To the best of my knowledge, modeling hamiltonian of a dynamic system using NN is novel. Though there are concurrent works with the similar theme, this paper is, from my point of view, the most clear and thorough one among them. The related works are well cited. 2. Quality This paper is technically sound. This work is self-contained and did a good job to prove the concept it introduces. The evaluation is thorough, yet a bit simple, but is powerful enough to prove the concept of this paper. 3. Clarity: This paper is well written. It give a gentle introduction to hamiltonian mechanics for readers who may not have the proper background. 4. Significance: This work provides a novel, concrete and practical methodology for learning dynamic systems in a physically grounded fashion. Moreover, this modeling strategy has great potential since it may not rely on the exact coordinate frame which the system is defined, as hamiltonians can be defined on any valid generalized frame which suit the constraint. This direction definitely requires more thoughts and efforts and should be of significance to the community.