Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors propose a new ML force field based on GNNs that uses higher-order equivariant messages. Previous methods like Dimenet++ and Gemnet use higher-order messages (triplets or quadruplets), but they enumerated the various triplets / quadruplets, which made them computationally expensive, and limited the order of interactions. In the present work, the authors use a tensor product formulation to make the framework efficient. Moreover, the authors empirically show that only 2 MP layers are needed to get good performance with their model, which makes it quite fast. SO3 equivariance is achieved using spherical harmonics representations, similar to previous methods. The authors also perform scaling laws analysis as well as extrapolation to OOD data, and show that their model performs well.
strengths_and_weaknesses: Strengths
* The authors present a new MPNN using higher order messages. The network is equivariant which is a desirable property, for its data efficiency. The model performs well on a variety of small molecular datasets.
* The model is very fast thanks to the tensor formulation of higher order messages, and the fact that it only requires 2 layers. Efficiency for ML force fields is very useful in practice for running long MD simulations or performing structure relaxations.
* Scaling laws analysis shows desirable scaling behavior.
* The model works well for OOD data.

Weaknesses
* In section 4, the paper claims that the higher order features B_i,eta,K,L,M can be interpreted as a complete basis. Is there a reference / proof for this? The paper does not mention any.
* Experiments don't compare against newer methods like Gemnet that also use higher order message passing.


questions: * How does the cost of tensor product scale with number of nodes? In other words, can the method be applied for larger systems like materials or proteins?
* The experiments in the paper are based on small datasets. Do the authors have a sense for how MACE would perform on a much larger dataset like OC20 (opencatalyst.org)?
limitations: Seems adequate.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This work proposes a new equivariant MPNN model named MACE for force field prediction. In particular, by combining equivariant message passing with efficient many-body messages, MACE achieves both state-of-the-art performance on several benchmarks and considerable computational efficiency.
strengths_and_weaknesses: Strengths:
* The proposed method achieves state-of-the-art performance on several benchmarks.
* The proposed method is faster to train and inference than previous models, which is meaningful for modeling some macromolecular systems.
* The analysis of the impact of many-body messages on learning curves is interesting, which may inspire future works in balancing message correlation orders and network depth while designing architectures.

Weaknesses:
* The connection between tensor product and the standard many-body expansion is unclear to me. The authors should provide clearer background and technical details while presenting this core module. 
* The novelty of the proposed method is not well clarified. As mentioned in related works (lin99-109), both the theories and implementation strategies of the Multi-ACE framework have been put forward by previous works. There is no clear clarification of non-trivial modification in the proposed method, making it difficult to justify the technical contribution of the paper.
* The description of evaluation benchmarks is unclear. For example, the meaning of rMD17 is not well explained. 
* The claim about receptive fields is misleading. In line 220 of section 5.2, the authors claim that "By decoupling the increase in correlation order of the messages from the number of message passing iterations, MACE only requires two layers resulting in a much smaller receptive field." There is no doubt that a small receptive field will make the model more parallelisable. However, the big receptive field also captures more global (long-range) interactions than the small receptive field, which usually leads to better empirical performance (see Figure 1 in this paper). It seems that the authors think the correlation order of body messages is more important than the size of the receptive field, so the negative impact of small receptive fields is ignored. But there is a lack of convincing discussion about this point. Correct me if I misunderstand something.
questions: * For the evaluation benchmarks, what do you mean by the **revised** MD17 dataset?  Is there any difference between rMD17 and MD17 in other works? Why do the authors choose the 3BPA dataset to analyze the effect of higher order messages and the computational efficiency of the method, instead of the larger scale MD17 dataset?
* I encourage the authors to reorganize the methodology section to make it more readable and highlight their technical contributions.
* The authors should further clarify the potential negative impact of small receptive fields. If any, how do you deal with such impact?
* Can the authors provide a more formal mathematical description for the computation complexity for MACE and other baseline methods? This would be more direct to exhibit the advantage of MACE.
* Is there any typo in Eqn. (3)? The authors introduce the new symbol $h_i^{t+1}$ in line55 but don't interpret it in subsequent paragraphs.
limitations: Limitations are well discussed.
review_title: null

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: The paper describes a message passing NN approach that efficiently handles many-body interactions (where classical methods typically handle 2 body interactions). The method is based on a standard steerable/equivariant edge embedding and aggretation, leading to equivariant atomic environment embeddings. These per-node embeddings are then pulled through some higher order CG tensor product (like an equivariant polynomial) by which the pair-wise features start interacting with each other and effectively a higher body order is achieved. Notably, this interaction takes place on the node level (not over all possible many body combinations) which makes the method highly efficient. 

The paper then systematically shows what components of equivariant architectures are important (spherical harmonic degree or max body order, or both?). To a large degree it is the max body order, and spherical harmonic degree helps but to a lesser extend. The paper further shows that with high-body order, the number of layers can be greatly reduced (to 2) whilst reaching state of the art.
strengths_and_weaknesses: **Strengths**
* The paper is very well written.
* Literature review is coherent and the storyline is clear
* Despite though material, the paper does an excellent job in presenting the method in a concise, precise and comprehendible way
* Experiments are systematically setup as to allow to specifically test individual components and in order to draw appropriate conclusions.

**Weakness**
* I found it hard to assess the impact of some of the ablation studies, in particular those pertaining to MD17.  I wonder if anything can be set about how well these results generalize to large datasets and molecules of larger size? I am wondering how representative is the aspirin molecule for the large amount of challenging problems in computational chemistry?
* The work could still benefit from a more in depth discussion on handling higher-order interactions (e.g. via simplicial neural networks or via Dimenet-like methods). It is clear that these approaches are different, is there anything to say about what difference is decisive? (e.g. eq 10 is based on products

**Small notes**
I do not see the point of the MLP in equation 13. All are linear layers, except for the last one, why? The last one is apparently a one layer MLP, is the only difference then an activation function? (otherwise a one-layer MLP == a linear layer)

questions: 1. What can there be said about which observations would hold in general, in which would be limited to single (and small?) molecule analysis, versus systems of molecules (maybe like OC20?) or large molecules like proteins?
2. Dimenet and other higher-order methods like simplicial neural networks maintain a representation of the “many bodies” (e.g. dimenet evolves edge features). Another difference is the encoding of higher order interactions via geometric invariants s.a. angles, whereas MACE relies on products of invariants and covariants. I have the feeling that this product structure  (instead of e.g. summing over bodies) is important, but cannot quite put my finger on it. It is evident that the presented approach works well, but what precisely makes it successful? E.g. if Dimenet uses 3 body interactions, and this work does as well, MACE (mu=2) seems to be better, but why? Related to my puzzlement w.r.t. Dimenet comparison, if L=0 and mu=2, we get 3-body interactions through multiplication (?) of invariants like distances. Is there any interpretation to this?
3. Equation 10 reminds me of polynomical activation functions, akin to consideration in Kondor [31] that such (higher order) tensor products are non-linearities in themselves (viewing them as activation functions). I am used to the idea of such equivariant polynomial activation functions (through works like [F21]), simply as necessary non-linearities for neural networks to work. But should I change my viewpoint and consider it more than just a non-linearity, but rather see it as a way to achieve higher-body interactions?

[F21] Franzen, D., & Wand, M. (2021). General Nonlinearities in SO (2)-Equivariant CNNs. Advances in Neural Information Processing Systems, 34, 9086-9098.

limitations: Experiments are very well organized and conclusions are appropriate. It is however slightly unclear how well this method generalizes to other types of problems in computational chemistry (involving more or large molecules).
review_title: null

Meta-review: metareview: This paper proposes a novel equivariant message passing network for modeling atomistic systems based on the popular Atomic Cluster Expansion formalism. The method relies on a clever factorization of higher order terms into products of two-body terms. This allows MACE to be fast while also taking into account many-body effects. Experimental results seem strong with intriguing scaling properties. Neural network potentials for atomistic systems is a rapidly growing subfield and this seems like a great contribution. 

All reviewers supported acceptance noting the strong experimental performance, the fast training and inference speed, and demonstrated scaling with dataset size. 
confidence: Certain
Decision: Accept