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: An improved method to approximately solve the Schroedinger equation is described, which combines ideas from the PauliNet and FermiNet papers. A variety of ablation studies are performed. good performance is achieved.
strengths_and_weaknesses: Strengths:
- good results
- thorough study and recombination of approaches of prior works, without reinventing the wheel
- ablation studies
- no grandiose claims
- good discussion of limitations

Weaknesses:
- I found the the description of the method quite unclear, and some notation was not very clearly defined where it appeared in the text 
- one could argue the paper is "only" a recombination of prior works, but if we take this argument, AlexNet would have needed to be rejected as well.
questions: Would a study of the transition states of the butadiene system (as in the paulinet paper) be instructive?



limitations: yes
review_title: null

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 present and analyze a list of improvements that significantly increase accuracy and reduce the computational cost of variational Monte Carlo methods based on deep learning.
strengths_and_weaknesses: # Strengths

- Overall, the paper is very well-written and clear, even for someone with little domain expertise.
- The introduction is helpful and allows non-experts to onboard (to the extent one can reasonably expect).
- The key contributions are laid out.
- Differences and similarities to previous methods (especially PauliNet) are exhibited.
- Thoughtful experimental design.
- Impressive experimental results.

# Weaknesses

- A (minor) weakness is the lack of immediate relevance of the model improvements to the broader ML community.
questions: - The units used in the equation before Eq. (1) should be clarified.
- The (probably well-known) Rayleigh-Ritz principle mentioned in L57 warrants a citation for the interested reader.
- How is $H \psi(r)$ evaluated?
- In Eq. (2), $k$ seems to be, next to $i$, an index for an electron. In the previous equations, you use $j$. Why?
- The concrete form of Eq. (1) is not obvious. I assume that $\lambda$ is actually a matrix of shape $n_{el} \times n_{el}$?
- Below Eq. (3), you have another $k$. Is that related to the $k$ from before?
- I didn't fully understand the challenges you (and the community, I assume) are facing in Section 2.2. How are these requirements different from, let's say, an ML force field.
- Do you mention anywhere the total computational cost of each method?
limitations: The authors discuss the limitations of their work.
review_title: null

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: This paper focuses on empirical improvements of variational quantum Monte Carlo estimation of molecular ground state energy. In particular, the authors focus on evaluating the various design choices in the FermiNet and the PauliNet, including dense determinant, hyper parameter tuning, the choice of envelope function and the pretraining strategies. On top of these, the authors propose to use a SchNet-like embedding blocks and an input feature transformation to make the input feature to make the feature to be invariant, local and expressive. 

The authors empirically evaluate all these design choices on molecules with up to ~40 electrons and obtain a model which improves upon the existing methods both in terms of accuracy and speed. The authors also conduct ablation studies by removing each design choice one by one. 

From the experiments, although the factors are not totally independent, it seems the most improvements come from the use of dense determinant, hyper parameters tuning, and the SchNet-like embedding. The proposed feature transformation only has minor effects.
strengths_and_weaknesses: **Strength:** 

The paper is well written and easy to follow. The experimental results are good and the ablation studies are extensive and well presented. The effects of different design choices can be easily identified. Since PauliNet and FermiNet differ in many aspects while both give high quality quantum Monte Carlo results, it is interesting to compare and clarify the effects of these differences. Making improvements by fusing these two methods is well motivated.

**Weakness:** 
* Since the paper essentially integrates FermiNet and PauliNet, the novelty of proposed methods are very limited. In particular, the proposed coordinate transform only has sight effects. 
* Although the overall improvement and speedup are clear, for some claims it is difficult to find reference in the main paper. E.g., "40-70% lower energy error at 8x lower computational cost" as stated in the abstract. It would be clearer to collect these comparisons in one place.
questions: * Some results in Figure 4 are lower than the reference energy, especially for larger molecules where the differences are significant (>100mHa) compared to the chemical accuracy (~2mHa). To which extent are we certain that the QMC results are correct? Although the variational principe could grantee the variational energy to be upper bound but doesn't it assume the sampling to be accurate in the first place? For example, in the Figure 3 of the FermiNet paper [Pfau et al. 2020], during VMC training, the energy could overshot when the MCMC step size is not large enough.

* Under which setting the claimed "40-70% lower energy error at 8x lower computational cost" is obtained? As shown in Table 5, the runtime per epoch improves from ~6s to 3.6s. How is the convergence time defined?

[Pfau et al. 2020] Ab initio solution of the many-electron Schrödinger equation with deep neural networks. Physical Review Research, 2020. 

------
**After rebuttal:**

The authors' response clarifies the concern about the uncertainty and the speedup. Overall I think the proposed method is well motivated and results are convincing. Hence I will increase my score from 5 to 7 and will recommend acceptance of this paper.
limitations: The authors state the limitations adequately.
review_title: null

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: This paper proposes a deep-learning architecture to solve ground state of many-electron systems. The proposed model combines PauliNet-like neural network and envelope function in FermiNet, with additional improvements on embedding, input feature and parameter initialization of previous methods. Experimental results show that the proposed method can reduce errors by 40-70% with 4-8x lower computational cost. This paper also establish a new benchmark of several deep learning based and classical methods over a number of different atoms and molecules. Authors also research into the reason for improvements and find out that including too much physical prior knowledge can deteriorate the accuracy.
strengths_and_weaknesses: **Strengths:**
- *originality:* Related works are adequately cited and it is clear to see the difference from these works. This work is a novel combination of PauliNet and FermiNet along with additional improvements.  
- *quality:* This work provides solid experiments to prove the accuracy and efficiency improvements. Also, this work systematically breaks down which changes cause the improvements, and gives insights that including physical prior knowledge can hinder the optimization. 
- *clarity*: This paper is overall well written and easy to follow. The method and results are clearly presented.
- *significance:* This work achieves the best results for the numerical solution of the electronic Schrodinger equation and establishes a new benchmark for current most accurate methods on a number of molecules and atoms. The proposed method is 4-8x faster than FermiNet in terms of optimization.  

**Weaknesses:**
This paper demonstrates great experimental results, but lacking the necessary theoretical explanation to address the accuracy improvement under reduced computational cost. Section 4 discussed in detail the improvements obtained by each individual change and their combined effects, but the discussion is still restricted within the experimental point of view. Adding theoretical explanation on why the proposed method can obtain accuracy better than the classical as well as existing deep learning method can be beneficial. 

questions: Please see weaknesses.
limitations: Limitation of this work is well addressed.
review_title: null

Meta-review: metareview: There is a clear consensus among the reviewers that this is a quality paper and worthy of acceptance (in fact, this may be the first time I've ever seen 4 reviewers give the exact same score), so I recommend accept.

I do however have one additional comment. I find the current title somewhat unwieldy and wonder if it would be possible for the authors to condense it at all. This is not a critical issue, of course, but one that the authors may want to consider (if the program chairs allow it).
confidence: Certain
Decision: Accept