SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\nabla_ {\boldsymbol {F} _ {t} ^ {\prime}} E \left(\boldsymbol {Q} \boldsymbol {L} _ {t}, \boldsymbol {F} _ {t} ^ {\prime}, \boldsymbol {A} _ {t}, t\right) | _ {\boldsymbol {F} _ {t} ^ {\prime} = w \left(\boldsymbol {F} _ {t} + \boldsymbol {t 1} ^ {\top}\right)} = \nabla_ {\boldsymbol {F} _ {t}} E \left(\boldsymbol {L} _ {t}, \boldsymbol {F} _ {t}, \boldsymbol {A} _ {t}, t\right), \tag {29}
$$
$$
\nabla_ {\boldsymbol {A} _ {t}} E (\boldsymbol {Q} \boldsymbol {L} _ {t}, w (\boldsymbol {F} _ {t} + \boldsymbol {t 1} ^ {\top}, \boldsymbol {A} _ {t}, t) = \nabla_ {\boldsymbol {A} _ {t}} E (\boldsymbol {L} _ {t}, \boldsymbol {F} _ {t}, \boldsymbol {A} _ {t}, t), \tag {30}
$$
which implies that the gradient to  $L_{t}$  is O(3) equivariant, and the gradient to  $\pmb{F}_{t}$  and  $\pmb{A}_{t}$  is periodic E(3) invariant. Such symmetries maintain that the introduction of energy guidance does not violate the periodic E(3) invariance of the marginal distribution.
The detailed algorithm for energy-guided sampling is provided in Algorithm 3. We find that  $s = 50$  works well on the three datasets. We evaluate the performance of the energy-guided model with the same metrics as Xie et al. [9]. We sample 100 structures from the testing set for optimization. For each structure, we apply  $T = 1,000$  and  $T' = 100,200,\dots,1,000$ , leading to 10 optimized structures. We use the same independent property predictor as in Xie et al. [9] to select the best one from the 10 structures. We calculate the success rate (SR) as the percentage of the 100 optimized structures achieving 5, 10, and 15 percentiles of the property distribution. We select the formation energy per atom as the target property, and provide the results on Perov-5, Carbon-24 and MP-20 in Table 12, showcasing the notable superiority of DiffCSP over the baseline methods.
Aside from the Carbon-24 dataset, the composition is flexible in the above experiments. We also attempt to follow the CSP setting and optimize the crystal structures for lower energy based on the fixed composition. We visualize eight cases in Figure 10 and calculate the energies of the structures before and after optimization by VASP [68]. Results show that our method is capable to search novel structures with lower energies compared to existing ones.
Original Structures
Figure 10: Visualization of 8 pairs of structures before and after optimization.
Table 13: GPU hours for yielding 20 candidates over the testing set.
Perov-5	Carbon-24	MP-20	MPTS-52
P-cG-SchNet	2.1	3.0	10.0	22.5
CDVAE	21.9	9.8	95.2	178.0
DiffCSP	1.5	3.2	18.4	34.0
Table 14: Results on MP-20 with different inference steps.
Steps	Match rate (%)↑	RMSD↓
DiffCSP	1,000	51.49	0.0631
5,000	52.95	0.0541
CDVAE	1,000	30.71	0.1288
5,000	33.90	0.1045
H Computational Cost for Inference
We provide the GPU hours (GeForce RTX 3090) for different generative methods to predict 20 candidates on the 4 datasets. Table 13 demonstrates the diffusion-based models (CDVAE and our DiffCSP) are slower than P-cG-SchNet. Yet, the computation overhead of our method is still acceptable given its clearly better performance than P-cG-SchNet. Additionally, our DiffCSP is much faster than CDVAE across all datasets mainly due to the fewer generation steps. CDVAE requires 5,000 steps for each generation, whereas our approach only requires 1,000 steps. We further compare the performance of CDVAE and DiffCSP with 1,000 and 5,000 generation steps on MP-20 in Table 14. Our findings indicate that both models exhibit improved performance with an increased number of steps. Notably, DiffCSP with 1,000 steps outperforms CDVAE with 5,000 steps.
I Error Bars
We provide a single run to generate 20 candidates in Table 1. We apply two more inferences of each generative method on Perov-5 and MP-20. Table 15 shows similar results as Table 1.
Table 15: Results on Perov-5 and MP-20 with error bars.
# of samples	Perov-5	MP-20
Match rate (%)↑	RMSE↓	Match rate (%)↑	RMSE↓
P-cG-Schnet [53]	1	47.34±0.63	0.4170±0.0006	15.59±0.41	0.3747±0.0020
20	97.92±0.02	0.3464±0.0004	32.70±0.12	0.3020±0.0002
CDVAE [9]	1	45.31±0.49	0.1123±0.0026	33.93±0.15	0.1069±0.0018
20	88.20±0.26	0.0473±0.0007	67.20±0.23	0.1012±0.0016
DiffCSP	1	52.35±0.26	0.0778±0.0030	51.89±0.30	0.0611±0.0015
20	98.58±0.02	0.0129±0.0003	77.85±0.23	0.0493±0.0011
J More Visualizations
In this section, we first present additional visualizations of the predicted structures from DiffCSP and other generative methods in Figure 11. In line with Figure 3, our DiffCSP provides more accurate predictions compared with the baseline methods. Figure 12 illustrates 16 generated structures on Perov-5, Carbon-24 and MP-20. The visualization shows the capability of DiffCSP to generate diverse
structures. We further visualize the generation process of 5 structures from MP-20 in Figure 13. We find that the generated structure  $\mathcal{M}_0$  is periodically translated from the ground truth structure, indicating that the marginal distribution  $p(\mathcal{M}_0)$  follows the desired periodic translation invariance. We provide the detailed generation process in the Supplementary Materials.
Figure 11: Additional visualizations of the predicted structures from different methods. We translate the same atom to the origin for better visualization and comparison.
Figure 12: Visualization of the generated structures by DiffCSP.
Figure 13: Visualization of the generation process on MP-20. The column "Translated" means translating the same atom in the generated structure  $\mathcal{M}_0$  to the origin as the ground truth for better comparison.