Paper: Generating Molecular Conformer Fields (ICLR-2024)

Reviewer comment (Novelty):
The technical and empirical novelty is fairly limited in this paper. It is a direct application of DPFs (Zhuang et. al. 2023) to a molecular-conformer generation. The neural network also has an identical architecture, and the difference is solely to apply it to this different problem. The representation of atom/node “coordinates” (i.e. the field domain) as Laplacian eigenvectors is also a direct application of Maskey, et. al. 2022.

Author response:
In this work we generalize the problem setting in DPF to deal with functions on graphs (as opposed to functions on Euclidean spaces). Our proposed approach is conceptually simple to implement yet very powerful in its application. **Note that MCF sets the current state-of-the-art for conformer generation.** We believe that approaches that are both simple to implement and powerful in their empirical results tend to have the longest lasting impact in the field.

Reviewer comment (Evaluation):
The paper claims that explicitly enforcing domain-specific inductive biases (e.g. periodic domain of possible torsional angles in Torsion Diff) comes at a cost, but what is the cost? Torsional Diffusion seems to be performing just as well as MCF (sometimes better).

Author response:
Torsional diffusion requires breaking molecules into fragments connected by rotatable bonds. This process relies on rule-based methods that can fail, especially for complex molecules. For example, in our replica of torsional diffusion on GEOM-XL using their public codebase, we only successfully generate conformers of 77 molecules out of 102 in the database. Besides, torsional diffusion utilizes cheminformatic methods to predict the local 3D structures. Therefore, the accuracy of the model is dependent on the local models and may neglect effects of long-range interatomic interactions from other substructures. We have further clarified the cost in Introduction section. 
      * Furthermore, we trained an MCF model with updated hyper-parameters as reflected in Tab. 4. We applied updated latent dimensions as well as number of latents and trained the model for more epochs. This model now obtains state-of-the-art performance,  outperforming torsional diffusion on challenging the GEOM-DRUGS and GEOM-XL datasets (shown in updated Tab. 2 & 3).

Reviewer comment (Evaluation):
The paper claims that MCF is more scalable than other methods (e.g. Torsional Diffusion). The architecture of MCF is quite large, however, and for a system of 100 atoms, there are 200 queries to the model (100 context pairs and 100 query pairs). Is it really more scalable than Torsional Diffusion?

Author response:
First, more scalable means MCF operates directly on raw data in 3D Euclidean space without the domain-specific efforts to pre-processing the data. Such pre-processing can lead to failure cases for complex molecules and forces the conformer generation adhered to other substructure models. These priors can prevent the models from scaling up to larger and more complex molecular systems. In addition, MCF is built upon PerceiverIO, an attention-based Transformer model. PerceiverIO includes a latent array that cross attends with input context and output query. In our experiments, we set the number of latent as 512. For a molecule with 100 atoms, the model has degree of freedom up to 512 which is much greater the number of atoms. Both aspects provide the flexibility of our proposed MCF to scale up.

Reviewer comment (Evaluation):
The paper claims that the ablation study shows what factors are important for molecular-conformer generation, but this is a pretty grandiose claim. The ablation study only shows the impact of atom features and the size of the Laplacian eigenvectors, specifically on MCF performance. It does not lend insight into what factors are important for molecular-conformer generation in general.

Author response:
We thank the reviewer for pointing this out, this is a misscommunication that is clarified in the updated version of the paper. Our claim is to show what implementation factors and node features are important for MCF to obtain a good performance. We have rephrased the claim to clarify our contributions.

Reviewer comment (Experiments):
The paper claims that explicitly enforcing roto-translation equivariance is not a strong requirement for generalization, and that equivariance is not particularly useful for molecular-conformer generation. Although the authors have shown similar and comparable performance between an equivariant and non-equivariant architecture, it is not yet clear whether or not equivariance really affects performance. For example, one could ask whether MCFs could outperform Torsional Diffusion if it had used a roto-translationally equivariant architecture (not just PerceiverIO).

Author response:
We have updated Tab. 2 after training an MCF model with updated hyper-parameters (see Tab. 4). MCF now surpasses all baseline models by a large margin on GEOM-DRUGS. The latest results further demonstrate that roto-translational equivairance is not a strong requirement for achieving superior performance on molecular conformer generation. Our model, which is built on a general-purposed Transformer-based architecture, can achieve better performance than other models which are roto-translational equivariant. 
      * Also, we would like to underline that to the best of our knowledge, our work is the first to highlight the question whether equivariance is required for solving molecular conformer generation problem. Previous works have focused on building equiavriant neural networks and baking inductive biases (e.g. modeling torsional angles) for conformation generation. Indeed current work doesn’t completely answer the question whether equivariance is needed for conformer generation. But we hope this work can shed a light on future investigations of the role equivariance plays in such tasks.

Reviewer comment (Experiments):
The datasets seem to be split into training/validation/test uniformly at random, but this can lead to train/test leakage. Many molecules in the datasets share very similar scaffolds, and it is well known that prediction between similar scaffolds is much easier. To really assess the quality of the model, the datasets should be split into scaffold-aware subsets to avoid leakage. Although this issue is almost certainly inherited from previous works (e.g. Torsional Diffusion) and is not counted against the evaluation of this paper, it is highly recommended to explore better dataset splits to improve evaluation.

Author response:
We agree with the review that random splitting the molecular data can lead to “leakage” in a way, namely molecules with similar scaffolds can be found in both training and test sets. In this work, we use the exact same random splitting  as in previous works (e.g., Torsional Diffusion, GeoMol, etc.) so that we can have an apples-to-apples comparison with the baseline models. 
   - Besides experiments on standard random splitting of GEOM-QM9 and GEOM-DRUGS datasets. We also benchmark on GEOM-XL (see Tab. 3), a dataset that contains molecules with number of atoms greater than 100 collected by [1]. MCF trained on GEOM-DRUGS achieves superior performance on GEOM-XL than the baselines. Notice the median number of atoms in GEOM-DRUGS is 40. The results demonstrate that our proposed MCF can generalize to molecules with different scaffolds.

Reviewer comment (Evaluation):
Figure 6 and the exploration of the continuity of MCFs does not seem very strong or insightful. The analysis only shows that the model learns to “round” coordinates toward atom centers. There is no indication that the model actually learns any information about molecular bonds (and not to put atom density on them). One could claim that the model will always output some set of atomic coordinates which are not too close to each other regardless of the input, even if the network has no understanding of bonds.

Author response:
We thank the reviewer for raising this point. Our goal is to show that one can evaluate MCF with an interpolated eigen-function coordinate. Our observation is that if one has supervision for what are corresponding 3D points for interpolated eigen-functions (eg. simulating the PDF of electron clouds between atoms) those could be used during during training. In addition, we see that MCF does not completely snap generated 3D coordinates to atom positions and there are a few 3D points generated in between atoms (colored in light blue in Fig 6.). This is surprising because MCF was never trained to produce any logical output for interpolated eigen-functions.

Reviewer comment (Evaluation):
It is not clear what the benefit is to being able to predict on interpolated atom coordinates. If the goal is to input the interpolated eigenvector halfway between two nodes and generate the 3D coordinate that is halfway between the two atoms, then this analysis shows the opposite. If the goal is to input the interpolated eigenvector halfway between two nodes and probabilistically generate the 3D atomic coordinate of either endpoint atom, then this analysis somewhat shows that, but it is unclear why this latter question is interesting or useful.

Author response:
We thank the reviewer for raising this point. Our goal is to show that one can evaluate MCF with an interpolated eigen-function coordinate. Our observation is that if one has supervision for what are corresponding 3D points for interpolated eigen-functions (eg. simulating the PDF of electron clouds between atoms) those could be used during during training. In addition, we see that MCF does not completely snap generated 3D coordinates to atom positions and there are a few 3D points generated in between atoms (colored in light blue in Fig 6.). This is surprising because MCF was never trained to produce any logical output for interpolated eigen-functions.

Reviewer comment (Writing):
More background on DPF would be nice; this work is so dependent on DPFs, that it would be good to have more background on how they work.

Author response:
We thank the reviewer for the suggestion and have added the DPF background to section 3.1 of Preliminaries in the revised main text.

Reviewer comment (Writing):
It would be good to include the definitions of performance metrics like RMSD recall (in the supplement) instead of needing to refer to the Torsional Diffusion paper.

Author response:
We thank the reviewer for bring this up. We have added the definitions of the metrics to Appendix A.2.4.

Reviewer comment (Writing):
Notation should be cleaner: if a molecule’s conformer is a function \( f: \mathcal{G}\rightarrow\mathbb{R}^{3} \), this implies the function maps an entire molecular graph to a single 3D coordinate.

Author response:
Here $f: \mathcal{G} \rightarrow \mathbb{R}^3$ represents an implicit function/field that maps  points on the graph $\mathcal{G}$ to  Euclidean space $\mathbb{R}^3$. In this case, $f$ doesn’t denote the mapping from a whole molecular graph to its 3D conformer but rather a node in the molecular graph. We have added extra explanations in the updated manuscript to clarify the concept.

Reviewer comment (Writing):
“equivairant” in first paragraph of Section 2 is a spelling error.

Author response:
We thank the reviewer for pointing the typos and we have fixed them in the updated manuscript.

Reviewer comment (Writing):
“In contraposition” should be “In contrast”.

Author response:
We thank the reviewer for pointing the typos and we have fixed them in the updated manuscript.

Reviewer comment (Experiments):
Is the dataset split the same as in previous works? Is the RMSD cutoff \( \delta \) the same as previous works for precision/recall? It is crucial that the dataset splits (and RMSD precision/recall \( \delta \) value) are the same here compared to prior works, because the performance numbers in the tables are directly copied from those papers.

Author response:
Yes, we use the exact same data splitting as reported in GeoMol and Torsional Diffusion. And following Torsional Diffusion, we use 0.5 and 0.75 as coverage cutoff for GEOM-QM9 and GEOM-DRUGS respectively. We have added the clarification to Appendix A.2.4.

Reviewer comment (Reproducibility):
What atom features are concatenated? How are bond features incorporated (e.g. double vs triple bonds)?

Author response:
We include atomic number, aromatic, degree, hybridization, number of rings, formal charge, and implicit valence in the atom feature following [1]. We have added an extra section A.2.2 in Appendix to clarify the atomic features used in this work. In MCF, we don’t explicitly include bond features since the bond information is encoded by the atom features (e.g., degree, aromatic, formal charge, etc) and the topological information from graph Laplacians. As we shown in Appendix A.2.2 and Tab. 5, adding extra atom or bond features doesn’t lead to significant performance improvement.

Reviewer comment (Novelty):
The proposed method looked novel to me at first, but after I read the DPF paper (Zhuang et al., 2023), I found the novelty is actually limited. The authors compute normalized graph Laplacian as the "index" of atom in the graph, an analogy of the pixel (x, y) index of the image. The main formulation and model architecture are same to DPF.

Author response:
1. **The main formulation and model architecture are same to DPF**
   - In this work we generalize the problem setting in DPF to deal with functions on graphs (as opposed to functions on Euclidean spaces). Our proposed approach is conceptually simple to implement while being very powerful in its application. **Note that MCF sets the current state-of-the-art for conformer generation.** We believe that approaches that are both simple to implement and powerful in their empirical results tend to have the longest lasting impact in the field.

Reviewer comment (Writing):
In terms of writing, I think more high-level introduction to DPF is needed in the background / preliminary section. The construction of context pairs and query pairs appear at a sudden, without any motivation behind them, which makes the paper hard to read.

Author response:
2. **More explanation about background of DPF**
   - We have included a sub-section 3.1 in the Preliminaries introducing diffusion models and diffusion probabilistic fields.

Reviewer comment (Theory):
Why should the context pair also be corrupted? Is there any motivation behind it?

Author response:
3. **Why should the context pair also be corrupted? Is there any motivation behind it?**
   - During training, the context pair provides the information about the field/function at a particular timestep $t$. Drawing an analogy to the image case, the input to the model at timestep $t$ is a noisy image. Similarly, the input to MCF at timestep $t$ is a set of noisy context pairs that explicitly encode the field. At inference time, sampling from MCF starts from gaussian samples for both context and query pairs. Therefore, noisy pairs are required to effectively model the distribution of fields/functions.

Reviewer comment (Theory):
In Sec. 3.2 the notation looks problematic. How can a function be used as the variable of Gaussian distribution?

Author response:
4. **How can a function be used as the variable of Gaussian distribution?**
   - In our case, we sample a conformer field $f_0 \sim q(f_0)$. Here, $q(f_0)$ denotes the ground truth distribution of conformer fields, namely the implicit functions that map points from molecular graph space to 3D Euclidean space, which is not a Gaussian distribution. We sample noise from Gaussian distribution $\epsilon \sim N(0, I)$ to corrupt the signal (i.e., 3D atomic coordinates) in context and query pairs rather than the field itself.

Reviewer comment (Experiments):
I don't understand why the proposed method can outperform GeoDiff, both of which are based on DDPM and aim to solve the same task. Does the improvement come from the DPF formulation, the more powerful PerceiverIP architecture, or something else?

Author response:
5. **Why can the proposed method outperform GeoDiff?**
   - As the reviewer mentioned, the superior performance of MCF comparing to GeoDiff roots from both the field formulation and the relaxation of using equivariant constraints in the architecture. MCF models the distribution of fields instead of distributions over discrete graphs. In addition, our backbone architecture (ie. PerceiverIO), is an efficient Transformer version, which can model the global interactions between atoms. On the other hand, GeoDiff is built upon an equivariant graph neural network which is known to suffer from over-smoothing and hard to model long-range interactions. Besides, our transformer backbone provides more flexibility of scaling up the number of parameters. In our case, we implement the number of latents as 512 which widens the bandwidth and degree of freedom in modeling the conformer field beyond number of atoms (which is usually in the order of 10).

Reviewer comment (Novelty):
One related work [1] on conformation generation is missing.  

[1] Guan, J., Qian, W. W., Ma, W. Y., Ma, J., & Peng, J. Energy-inspired molecular conformation optimization. ICLR 2022.

Author response:
6. **Missing related works**
   - We thank the reviewer for point us to the related work, we have included that in the revised manuscript.

Reviewer comment (Experiments):
In Section A.3.3, which pertains to the chemical properties experiment, this paper adopts experimental results from "Torsional Diffusion." However, there's a discrepancy in the subsets used. Specifically, the paper does not use the identical subset as "Torsional Diffusion," implying that they have selected different molecules. This raises concerns about the validity of their experimental comparison. For it to be persuasive, it should be more convincingly aligned.

Author response:
We have added experimental results of MCF and reproduced Torsional Diffusion on same subset in Appendix A.3.3. It indicates that on the aligned subset, MCF generate conformers that are closer to the ground truth than Torsional Diffusion in terms of the ensemble properties. This further validates that MCF learns to model the distribution of physically/chemically valid molecular conformers.

Reviewer comment (Theory):
In section 1, page 2, for example, the quality of conformers from GeoMol (Ganea et al., 2021) and Torsional Diffusion (Jing et al., 2022) depends on the local substructure prediction model which is not differentiable.

Author response:
We thank the reviewer for pointing out this. We have revised the description regarding GeoMol in section 1 and 2.

Reviewer comment (Experiments):
The GEOM-DRUGS dataset is considered a more challenging metric in this field. The method proposed in this paper demonstrates enhanced performance only regarding recall. Recall measures the capability to locate ground-truth conformers within the generated ones, and this metric is susceptible to influences from the training data. Notably, Torsional Diffusion depends on the local generated by RDKit, utilizing only 30 normalized conformers for each molecule during training. Although both the proposed method and Torsional Diffusion use the same dataset and split, the mean count of conformers in the GEOM-DRUGS dataset is approximately 100. This suggests that the boost in recall could result from the proposed method being trained on over 300% more conformers and having double the training epochs rather than the inherent efficacy of the method.

Author response:
We have updated the results on GEOM-DRUGS in Table 2, after training MCF with updated hyper-parameters (see Tab. 4). MCF now surpasses Torsional Diffusion on all metrics regarding recall and precision. On more challenging datasets like GEOM-DRUGS, MCF shows its advantage over other baselines. 
   - Also, we did not use the all the conformers in GEOM-DRUGS dataset during training. We follow the setting of GeoMol [1] and randomly pick at most 10 and 20 conformers for each molecule in training set of GEOM-QM9 and GEOM-DRUGS, respectively. Compared with 30 conformers per molecule in training Torsional Diffusion, our MCF is actually trained on less number of conformers. Trained on less data while achieving superior performance, MCF demonstrates the effectiveness in molecular conformer generation.

Reviewer comment (Evaluation):
In terms of precision, the proposed method is worse than Torsional Diffusion, especially evident when Torsional Diffusion undergoes 50 denoising steps, as shown in [Table.9] of the supplemental, and MCF requires 1000 denoising steps.

Author response:
Following the reviewer’s suggestion, we have included experiments with DDIM sampling [2] with small number of timesteps in Appendix A.4 and Tab. 9. Our MCF demonstrates competitive performance using a reduced number of denoising steps. When using 20 denoising steps, which is the main setting Torsional Diffusion reported in the paper, MCF achieves better performance than Torsional Diffusion on the challenging GEOM-DRUGS dataset. Also, as shown in the Tab. 9 MCF surpasses Torsional Diffusion with 50 sampling steps. Indeed, Torsional Diffusion show advantage when the number of sampling steps is very small (3 or 5 steps). We attribute this to the fact that Torsional Diffusion uses a pre-processing step in which a cheimformatics method predicts local substructure in 3D spaces before applying their model. So even before inference the cheimformatics prediction provides a strong prior. In addition, we would like to mention there are lines of works concerning distillation of diffusion models [3] so that it can effectively sample in even one single step. Such techniques can be applied to MCF and further increase the sampling efficiency.

Reviewer comment (Experiments):
The training settings for the proposed method use 8*A100 with a batch size of 64 (or one A100 with a batch size of 8). In contrast, Torsional Diffusion is trained on a singular A6000 with a batch size of 32. Consequently, the computational expense for each denoising phase for the proposed method is at least four times that of Torsional Diffusion. Considering Torsional Diffusion necessitates only 20 denoising steps, whereas MCF demands 1000, the proposed method, with a computational cost over two hundred times more, should ideally yield superior results, especially with larger molecules.

Author response:
MCF takes 0.5 GPU/s and 100 CPU/s to sample one conformer (much lower than CREST). As GPUs are widespread computing platforms in scientific computing, we believe GPU/s is a practical metric for assessing the inference time computational demands.  With MCF now greatly outperforming Torsional Diffusion in our updated results we provide an approach that is very efficient when running inference on GPU. When running inference on CPU, MCF provides better results with an increased runtime. It should also be noted that current runtime benchmark is conducted with batch size 1. In practice, one can stack multiple molecules as a batch and feed into MCF, which further improves the efficiency. In addition, there’s a few approaches one can use to distill MCF to use a very small number of denoising steps during inference [3,4,5] which will further decrease runtime.

Reviewer comment (Evaluation):
A pivotal metric for the conformer generation task is runtime. Even though some early deep-learning research might not have emphasized runtime, recent studies, such as GeoMol [Figure.7] and Torsional Diffusion [Table.2], regard it as a critical experimental metric. Historically, cheminformatics methods recognized runtime as a vital metric even before the advent of deep learning, as documented in [Conformation Generation: The State of the Art].

Author response:
MCF takes 0.5 GPU/s and 100 CPU/s to sample one conformer (much lower than CREST). As GPUs are widespread computing platforms in scientific computing, we believe GPU/s is a practical metric for assessing the inference time computational demands.  With MCF now greatly outperforming Torsional Diffusion in our updated results we provide an approach that is very efficient when running inference on GPU. When running inference on CPU, MCF provides better results with an increased runtime. It should also be noted that current runtime benchmark is conducted with batch size 1. In practice, one can stack multiple molecules as a batch and feed into MCF, which further improves the efficiency. In addition, there’s a few approaches one can use to distill MCF to use a very small number of denoising steps during inference [3,4,5] which will further decrease runtime.

Reviewer comment (Experiments):
CREST, a method based on in metadynamics, is used to generate GEOM-Drugs dataset. The computational cost of CREST in GEOM-DRUGS is about 90 CPU core-hours per drug-like molecule, as detailed in [Torsional Diffusion Section 2, Page 2]. In contrast, Torsional Diffusion requires roughly 5-CPU core seconds for a conformer, while MCF might need between 15 to 20 CPU core-minutes (considering the computational cost is over 200 times). Given that the average number (N) of conformers per molecule is about 100, MCF might need a minimum of 60 CPU core-hours to generate conformers (2N) for a single molecule. This speed is comparable to the dataset's generation rate (90 CPU core-hours per molecule). Hence, the proposed method's significance is low, especially when weighed against its high computational demands.

Author response:
In the response to previous questions, we have demonstrated efficient sampling technique, i.e., DDIM [2], can be directly applied to MCF while maintaining high conformer quality. There are also other efficient sampling methods like consistency model [4] that can be adapted as well. Besides, recent studies have proposed distillation methods for diffusion model that can achieve single-step diffusion with high quality [3,5]. There techniques can seamlessly be leveraged for MCF to increase sampling efficiency. We have also included more discussions about possible strategies to further reduce the computational cost in Appendix A.1.

Reviewer comment (Theory):
if we have n atoms in one molecule, then the conformation space is \mathbb{R}^{3n} instead of \mathbb{R}^3. in the paper authors indicate f: G -> \mathbb{R}^3. I am confused.

Author response:
Here, the notation represents a field (or implicit function) that maps any node from a graph to 3D Euclidean space. Namely it means mapping an atom (node) in a molecular graph to its 3D coordinates in the Euclidean space instead of mapping a molecular graph to a 3D conformer.

Reviewer comment (Theory):
section 4.1. Score field network: the author cite and use perceiverIO net and explain why they use that. this is good. the author should explain further for the benefits of the readers, what line 8 of algorithm 1 entails. \epsilon_q ~ N(0,I), line 8 want \epsilon_\theta to map to N(0,I). what is the physical significance in the context of molecule conformations?

Author response:
The loss in line 8 of algorithm 1 is adapted from DDPM loss. $\epsilon_q$ are sampled Gaussian noises that are injected to corrupt the output space of conformer field (i.e, 3D atomic coordinates). We train the score network $\epsilon_\theta$ to predict the Gaussian noise added to the coordinates. Therefore, the gradient is not intended to push $\epsilon_\theta$ to a standard Gaussian, but rather predict the sampled noise $\epsilon_q$ given the corrupted context and query pairs at current timestep $t$. So that in inference, MCF can generate realistic molecular conformers starting from random Gaussian noise.

Reviewer comment (Theory):
the authors claimed that the strength of their method is that there is no need to know some domain knowledge such as torsion angles of molecules. in my opinion this is a serious weakness. would the predictions generate physically non-viable confirmations if it disregard some basic physics and chemistry?

Author response:
We updated Tab. 2 with results from an updated MCF model, which demonstrates superior performance over all previous approaches on the challenging GEOM-DRUGS dataset. Also, in updated Tab. 8, MCF achieves competitive performance on ensemble property prediction. We note that although MCF does not enforce explicit inductive biases (eg. modeling torsional angles) it generates physically/chemically realistic samples. In addition other machine learning domains like computer vision [1,2] and natural language processing [3,4] that attention-based transformers and its variants can generalize well to various downstream applications when pre-trained on sufficiently large datasets. This indicates that if the domain knowledge is properly included in sufficient training data (e.g., the molecular conformers with low energies investigated in this work), the model can implicitly learn inductive biases from data without the need to explicitly enforcing them into the architecture. Besides, it should be noted that integrating domain knowledge also comes at a cost. For example, in Torsional Diffusion, one needs to use rule-based method to find the rotatable bonds which can fail in some cases especially for large complex molecules. It also relies on cheminformatic methods to predict local substructures which may neglect long-range interactions between substructures and restrict the conformer quality by the local methods. On the other hand, MCF provides a framework that directly operates on Euclidean coordinates which is easier to scale up both in terms of number of parameters as well as larger molecular systems.

Reviewer comment (Theory):
molecules are rotational invariant. that means, one can rotate the conformation by any angles in 3D and the conformation remains a valid conformation. is the diffusion model able to generated rotational invariant distribution of conformation?

Author response:
In this work, we highlight the question of whether in-/equi-variant neural networks are required for models that generate conformers. In particular, we note that what matters in conformer generation is the intrinsic geometry of the conformer and not the rigid coordinate system (as reflected by the metrics used in to evaluate approaches, which involve an alignment step, thus disregarding information about rigid SO(3) transformations). Using MCF, one can always sample conformers from the model and then apply a post-hoc SO(3) transformation if needed.

Reviewer comment (Reproducibility):
page 7, paragraph 1, "We generate 2K confirmers for a molecule with K ground truth conformers", how does the author do this? it is important to have 'correct' ground truth. if the authors use MD or MCMC with proper settings, then the ground truth can be correct. what are the settings? explicit water? mean field? what are the molecular interactions?

Author response:
Following the previous works [5,9], we use test data from standard GEOM-QM9 and GEOM-DRUGS benchmarks as ground truth. GEOM datasets applies semi-empirical DFT to calculate the molecular conformers. Each molecule can contain different number of conformers. In testing, we use all the K conformers for a molecule and sample 2K conformers to calculates the coverage and AMR metrics as [5,9] so that we can collect numbers for apple-to-apple comparison. Also, when trained with other data from MD or MCMC, our method can be applied to model the distributions as well.

Reviewer comment (Theory):
does the ground truth distribution obeys statistical physics law? NVT? grand canonical ensemble? micro-canonical ensemble?

Author response:
We follow the setting in GeoMol to randomly pick at most 10 or 20 conformers for each molecule in training on GEOM-QM9 and GEOM-DRUGS, respectively. Therefore, the ground truth training data doesn’t exactly obey statistical physics laws. Note that this is done in order to provide a fair comparison with previous work on conformer generation.

Reviewer comment (Novelty):
The novelty is limited from a technological perspective. I see little difference between the diffusion probabilistic field and the proposed molecular conformer field.

Author response:
In this work we generalize the problem setting in DPF to deal with functions on graphs (as opposed to functions on Euclidean spaces). Our proposed approach is conceptually simple to implement while being very powerful in its application. **Note that MCF sets the current state-of-the-art for conformer generation.** We believe that approaches that are both simple to implement, scalable and powerful in their empirical results tend to have the longest lasting impact in the field (ref the bitter lesson by Rich Sutton http://www.incompleteideas.net/IncIdeas/BitterLesson.html) .

Reviewer comment (Theory):
Using eigenvectors of the position of the graph in the field seems dangerous for the following two reasons. (1) This modeling does not take key information, such as atomic numbers, into consideration. (2) The eigenvector based position cannot effectively indicate distance in the graph without the corresponding eigenvalues.

Author response:
In MCF, we use atomic features which are concatenated with the graph Laplacians in both query and context pairs, including atomic numbers, aromatic, degree, hybridization, formal charge, etc. Therefore MCF has included essential atomic information that models the molecules. We have added an extra section A.2.2 in Appendix to clarify the atomic features used in this work. Though we didn’t include eigenvalues as the input to MCF, the eigenvectors are ranked based on eigenvalues. This can provide the extra information regarding the distance and connectivity between atoms. Finally, in practice we see that even without eigenvalues MCF sets the new state-of-the-art for conformer generation.

Reviewer comment (Theory):
Can this proposed method distinguish two different 2D molecular graphs that have the same eigenvectors but different eigenvalues?

Author response:
As mentioned in response to the previous question, eigenvectors are ranked based on eigenvalues. Therefore,  MCF can still distinguish graphs with same eigenvectors but different eigenvalues. In addition, we use atomic features as input to MCF which solves this type of problem in practice.