Paper:
Simulation of molecular and crystalline systems enables insight into interesting chemical properties that benefit processes ranging from drug discovery to material synthesis. However, these simulations can be computationally expensive and time consuming despite the approximations through Density Functional Theory (DFT). We propose the Valence Interaction Message Passing Neural Network (VIMPNN) to approximate DFT's potential energy calculations. VIMPNN integrates prior physics knowledge, namely the information exchange between atoms being driven and modulated by different interatomic bonds, and the relevance of specific physical properties to the problem of estimating more accurate energies. Furthermore, while many previous machine learning methods consider only stable systems, our proposed method is demonstrated on unstable systems at different interatomic distances. VIMPNN predictions can be used to determine the stable locations for atoms, a necessary step for the future simulation of crystal growth. Our method is extensively evaluated on an augmented version of the QM9 dataset that includes unstable molecules, as well as 2 new datasets of periodic and growing crystals. VIMPNN has comparable accuracy with DFT, while allowing for 5 orders of magnitude in computational speed up. It also produces more accurate and informative potential energy curves than Message Passing Neural Network.

Review:
The paper tackles the problem of estimating the electronic structure's ground state energy of a given atomic system by means of supervised machine learning, as a fast alternative to conventional explicit methods (DFT).
This is done by improving on a previous method, MPNN, Message Passing Neural Networks, and in particular by including information on bond type as input, so that the NN can learn the appropriate weight for messages going through bonds of that type.  In addition, training on several target labels (multi-regression) is attempted, with the idea that more physical outputs may help building better hidden representations (on this, there are mixed results).  Separately, training sets are enriched with non-equilibrium structures, so as to confront the NN with more diverse data.
The method is tested on 3 kind of training sets. The first is a simple but time-consuming augmentation of QM9, with inter-atomic distances varied, so as to increase the training set's size (and in particular, including non equilibrium configurations). The second consists in a periodic and thus infinite simple crystal structure (with, again, variations in inter-atomic distances, enriching the dataset). The third is a pseudo-cristalline structure with atoms randomly placed on a regular grid, forming a somewhat random structure, also named crystal (this is not a very good name).
The paper is overall rather well written, sometimes being a bit cumbersome (long sentences), but mostly it is stating clearly what is done or discovered.  The work is situated within the existing literature (that I am not familiar with at all).  The idea of using physics to guide architecture choices is gaining a lot of attention recently and seems to be well-suited to this particular problem, and well applied.  Several ways to use the bond type information have been attempted in this work, and several of them are reported and compared (a couple of them are discarded).  The results convincingly show that using bond type information indeed increases performance, both for small systems and for regular crystals.  The impact of performing multi-regression is less important, but still positive.  For large ''random crystals'', the method does not perform very well, and this represents a challenge for future work. Such a confession on the method's limitations is welcomed.
Given the idea (using physical information as bond type) is clearly and honestly presented, produces significant improvement compared with previous works, and has perspective for multiple future developments, I recommend acceptation of this paper.
There are however a number of points that could be improved.
1. There is a physical mistake that is not crucial but should be corrected, when talking about the ground state, and in particular before this sentence: ''accurate ground-state energy estimation of out-of-equilibrium molecule''. Ground state means minimal, T=0K energy level, so by definition it is at equilibrium. Thus, the sentence seems quite contradictory to a physicist.
What DFT and VIMPNN actually compute is the electronic structure's ground state's energy (at fixed positions of the atom kernels). I think this distinction should be mentioned just once, and then you could proceed with saying ground state energy. 
Because of this, I would recommend to edit the title so as to suppress ''out of equilibrium'' from it.  Otherwise readers may think the method deals with non-equilibrium electronic structures (non ground states), which clearly it does not at all, or they may think that it is especially good at estimating energies for out of equilibrium systems, which is not its primary goal.
2. I do not understand very well the training procedure.  Also there are some tests that seem to be interesting and that are not performed (as far as I understood).
Does each training set contains the 90%-150% data augmenations, for each non-augmented training configuration ?
Why is training performed separately for each kind of data set ? Wouldn't the ultimate goal be to transfer learning from a type to another, e.g. from QM9-style to crystal style, etc ?  (as far as I understood, this was not done)
Isn't it interesting to see how much training on an augmented data set (let's say QM9) improves performance on the non-augmented data (the ''true data'' in a sense) ? Although the augmented data is obtained by DFT, and comparing models trained on different data sets is unfair, I think it may be interesting to see if VIMPNN benefits more than MPNN from this strategy (so compare the performance gains of both algorithms obtained by augmenting a data set). This kind of comparison may also be done for the ''augmentation'' of a training set by the concatenation of it with another one (although in that case it may be detrimental to the test accuracy?).
If you actually did some of this, then I misunderstood and I am sorry, but then this also means you should clarify.
3. Section 3.3:
I would not call this a crystal, but more something like ''random finite structure''. This should be done everywhere in the paper.
4. Section 4.1 is a bit too short for the inexperienced reader. I suggest to be a bit more explicit on what is learned
5. Section 4.2: It is nice to say you tried other ways, keep that.
However, try to be more explicit on what is shared and what isn't, in the architecture you finally pick. Is lambda(v,w) a common value for all bonds of the same type, like C-C ? Maybe you could provide an example or some more detailed notation to make your choice fully explicit (after all this is the core of the paper).
6. Section 4.3: could you quickly comment on why you don't use more of the 13 physical observables available in QM9 ?
7. Table 1: do the three last lines correspond to ''no BT information + a single auxiliary estimate'' ?  It seems to be the case, but then you say you will continue with the auxiliaries, in addition to the BT information.  Why don't you display the result of using BT information AND the 3 auxiliary estimates ?  If you did, then I misunderstood, but it would also mean you did not explain well enough a counter-intuitive result (which would be that adding auxiliary information actually hurts the performance of the VIMPNN).
8. Please include a couple of explicit numbers of your training/test/validation sets sizes.
9. section 5.2 could be made more concise. In particular, there is no need in repeating what can be seen directly in the figures (stating numbers). It is useful to comment on the meaning of the results however (as you currently do).
10. section 5.4  is a good idea, promising, however it does not really conclude into a very strong statement, and takes essentially 1 full page, which could be used to better clarify the architecture and/or the training procedure (or to reduce towards the ideal page length).