Paper: Hierarchical GFlownet for Crystal Structure Generation (ICLR-2024)

Reviewer comment (Novelty):
The paper lacks clarity in explaining the motivation behind the proposed method and its components, making it challenging to discern the specific advantages or necessity of employing GFlowNet in crystal material generation. The introduction of physics-informed reward functions also lacks motivation. There are totally four new reward functions introduced. While their individual effectiveness is showcased through an ablation study, without a robust theoretical underpinning or discussion compared with existing approaches, it is not convincing enough. Does any existing work try similar stuff already? I could possibly raise some random ideas of some physic-related reward functions but how to prove the authors' proposed set of functions are optimal?

Author response:
On the physical motivation of the minimum bonds information: it was obtained by querying the entire Materials Project database, and computing a bond dictionary for each structure using the function $get\_bonds()$ defined in the OgStructure (https://github.com/sheriftawfikabbas/oganesson/blob/main/oganesson/ogstructure.py). The bond distance between two atoms can fall in a range, but will not go below a specific value for any given structure, otherwise the atoms will experience a repulsive force. The large variability of materials in the Materials Project database is a basis for using a statistical minimum of available bond data for two atoms as a meaningful distance between those two atoms. Such minimum was also used in Ref. [https://www.nature.com/articles/s41524-023-00987-9] for a similar purpose.
On the physical motivation of the density parameter, we have already explained in the manuscript the reason for considering the density of the generated structure. Quoting our text from the manuscript:
“Because the bond distance preference term applies a strict penalty for structures violating the minimum distance constraint, the generative model tends to generate a structure with long distances between atoms pairs with only a few neighbor atoms. This leads the model to generate gas with low density rather than solid-state density. On the other hand, structures with very high density (i.e. larger than 10) are unlikely to be realistic as the units are crowded with atoms causing a very high formation energy.”

Reviewer comment (Evaluation):
The paper’s experimental section does not convincingly demonstrate the superiority of the proposed method. Observing Tables 1 and 2, it’s evident that the proposed methodology struggles to outperform baseline models consistently across various evaluation metrics. This modest performance raises questions about the fundamental contributions and practical viability of the proposed approach. A clearer demonstration of the proposed method's unique advantages, or a clearer justification of its design choices, is crucial. Without such improvements, the paper’s contribution remains uncertain and inadequately supported.

Author response:
The method's unique advantage is the ability to incorporate constraints into the sampling process and soft constraint to rewards function.

Reviewer comment (Reproducibility):
The presentation of the proposed method lacks important details. For instance, the paper does not mention the initialization method. To improve understanding, it would be beneficial to provide a detailed algorithm outlining the generation process.

Author response:
A1:  We provided the initialization as well as the sampling process in Algorithm 1.

Reviewer comment (Theory):
The proposed method does not consider translation invariance. In Section 4.1, fractional coordinates are directly used as inputs for the GCN model, which violates the translation invariance of 3D crystals. This issue has been well-studied in previous predictive [1] and generative [2] methods, and its omission is a notable weakness.

Author response:
A2: The atomss list only mantain a reference atom of crystallographic orbit to make sampling process light-weight. Because it is not a complete crystal structure, translation invariance is not required. At the end of the trajectory where the complete crystal structure is generated to get the formation energy, the graph representation as well as the formation energy model is SE(3) invariant.

Reviewer comment (Reproducibility):
As mentioned in Weakness 1, how to sample the initial state?

Author response:
A1:  We provided the initialization as well as the sampling process in Algorithm 1.

Reviewer comment (Reproducibility):
The procedure to sample atom positions with GFlowNets is very vaguely described. The manuscript only tangentially refers to how atom positions are obtained, despite this being a non-trivial task in crystal structure generation. Figure 1 includes a box labeled "Atom positions" without further detail, and its caption provides limited additional context. Questions remain about why the initial state has one atom at (0, 0, 0), how other atom positions are sampled, and how the total number of atoms in a crystal is generated.

Author response:
Thank you for pointing this out. The wording in the Fig. 1 is indeed misleading. It should be worded "The current state". The figure is to explain how an action of choosing an atom reference position can be translated into multiple positions with symmetry operation.

Reviewer comment (Reproducibility):
Section 4.2 mentions that atom fraction coordinates transition probabilities are parameterized by a Multivariate Gaussian distribution, but no further details are given. This leaves unanswered questions such as how many atoms are selected, whether they are sampled simultaneously or sequentially, and why a Gaussian (which can produce invalid coordinates) is chosen over more suitable alternatives like a Beta distribution.

Author response:
Each step in the trajectory samples one single atom coordinates $x_0$.  Then the sampled atom is added to the list of sampled atoms $A_{sampled}$. $A_{sampled}$ only stores the one reference atom $x_0$ of the rystallographic orbit (Eq. 1). At the end of the trajectory, we apply the symmetry operation given by the sampled space group to have the full list of atoms $A_{crystal}$.  

During the trajectory sampling, we only use the graph representation from the sampled atoms list $A_{sampled}$ as the policy network input. There are two reasons we chose this implementation design. First, with the graph representation is permutation invariant. Second, if we apply symmetry operation on the sampled atoms list $A_{sampled}$ and use the full crystal structure atom list $A_{crystal}$, the number of atoms of the crystal structure can easily reach to hundred or thousand of atoms due to high multiplicity. The computation of a crystal graph of a thousand atoms is very computationally expensive.  Creating a new crystal graph at every steps of the trajectory is not preferred.

Reviewer comment (Novelty):
The paper fails to adequately address the complexities of sampling in a continuous state space, which is challenging due to geometric and physical constraints in crystals. The only related statement—"As the original GFlowNets only work on the discrete space, we adapt it to work on the continuous space..."—may also be misleading, as it could give the impression that the adaptation is novel to this paper, despite referencing prior work.

Author response:
We also change the claim to "As the original GFlowNets only work on the discrete space, we follow the recent work on continuous GFlownet (Lahlou et al., 2023)  to work on the continuous space of the atoms’ coordinates and lattice parameters."

It is true that sampling from Gaussian distribution may result in the negative fraction coordinate. Therefore, we cap the sampled action to $[0,1]$ to ensure the validity of the action.

Reviewer comment (Reproducibility):
Other important method components are described vaguely, including the selection of lattice parameters, the evaluation criteria for structure matching, the criteria for diversity in evaluation, and the definition of a mode in Section 5.2.

Author response:
We have provided more information regarding sampling lattice parameters, criteria for stability experiments, and definition of a mode as following and added to Appendix section.

Reviewer comment (Reproducibility):
The reward function, an essential part of GFlowNets, is insufficiently described. Section 4.3 explains sub-rewards encoding soft constraints, but the main component—the formation energy predicted by a model—is specified without identifying the model in the main text. Only in the supplementary material is M3GNet mentioned, without reference, and with vague cutoff details such as "Any prediction lower than the cut-off is set to 10 eV/atom" but without specifying the cutoff value or its frequency of occurrence.

Author response:
We add the reference of the prediction model in Sec. 4.3 to A.2.2 . We set the cut-off to $-10$ eV/atom as any structure having lower formation energy than that is more likely to be invalid. It is very difficult to tell exactly how often it occurs as the prediction model is a black box function. We can not tell exactly which type of structure causes this. In our own experience, it rarely happens, less than 10 times out of 100000 sampled crystal structures.

Reviewer comment (Presentation):
The lattice parameters given in Figure 1 (a = 4, b = 6, c = 4) do not seem to match the drawing in the figure.

Author response:
We have adjusted the Figure to match the number.

Reviewer comment (Novelty):
The description of "more meaningful actions" in the hierarchy is unclear. It is not explained why such actions are "closely related" to the reward function or what “close” means. The described hierarchical decomposition appears to be an intrinsic feature of GFlowNets rather than a novel contribution.

Author response:
The term "more meaningful actions" means that the actions are designed to have a more direct connection the target properties. The connection in the design is provided by prior knowledge, for example, perovskite structure class with conductivity. As discussed in the third paragraph in the Introduction section, the crystal structure class can  associate with the properties that we are optimizing. However, to generate the crystal structure, we need to sample the lattice parameter and the atoms' coordinates. Instead of only relying on very complex interactions between atoms using only coordinates to optimize properties, we can choose simpler actions but can directly associate with target properties. In this paper, we limit our experiment to only space group. In future applications, we can extend to more sophisticated higher-level tasks such as choosing composition to associate with crystal structure neutral charge.

Reviewer comment (Writing):
The statement "In the hierarchy, lower states represent discrete concepts constrained by the higher states that represent more abstract concepts" is unclear, as "discrete concepts" is not defined. A formal definition or greater clarification is needed.

Author response:
The term "discrete concepts" means that the actions or states are directly involved in the sampling process and can not be omitted. For example in the crystal structure generation task, we cannot have a structure without lattice parameters or atoms' type and coordinates. Meanwhile, the action of choosing a spacegroup is optional but can greatly assist the sampling process. Another example is in the control task. The action of moving the robot's joint is necessary and directly linked to the robot's movement. On the other hand, the action "move left", "move right", or "move to the door" is more at an abstract level.

Reviewer comment (Writing):
The paper contains numerous typos and errors in the methods description, including:
- "a triplet (L, A, X) of lattice parameters L, atom list L, and atom coordinates X"
- "$\tau = s0, \ldots, s_n, \tau \in \tau$ where $\tau$ is the trajectory set" (lowercase τ repeated for different meanings)
- "for all the state s"
- Mislabeling the "flow of the trajectory" when defining forward probability
- Inconsistent use of lowercase o and uppercase O for atom positions
- Mentioning "flow transitions" instead of "probability transitions" in Section 4.2
- Equations in Section 4.2 are generally hard to interpret and may cause confusion rather than clarity.

Author response:
Thank you for pointing out these errors. We have corrected those accordingly.

Reviewer comment (Evaluation):
Evaluation details are lacking, making it difficult to draw insights. Although results suggest CHGFlowNet achieves better metrics than a simpler GFlowNet or PGCGM, the reported average formation energies are positive, making them practically irrelevant for stability assessment. Furthermore, comparisons to PGCGM results raise doubts due to inconsistencies with previously reported distributions and the lack of detailed methodology.

Author response:
PGCGM performs the post-generation step to refine the crystal structure. To make it fair, in all methods, we only evaluate the generated structure without any further refinement or post-processing. As we pointed out in the 1st paragraph of Sec 5.1, PGCGM faces the common problem of sampling from data-induced distribution without any
refinement, which is low structure validity and atoms are very close to each other thus leading to high formation energy.

Reviewer comment (Novelty):
There exists concurrent work with significant overlap to this paper: https://arxiv.org/abs/2310.04925

Author response:
Thank you for introducing us to the concurrent work. Our work is different as we sample the complete crystal structure while other work only samples crystal structure parameters without atom coordinates. The formation energy is only provided by a newly trained prediction function without any coordinates.