Open Materials Generation with Stochastic Interpolants

We appreciate the reviewer’s thorough engagement with our manuscript and thank them for their insightful feedback.

Ablation studies, hyperparameters, and model performance

Regarding the performance of different stochastic interpolants, we direct the reviewer to the CSP ablation study tables provided in response to Reviewer w22y in which we highlight the best-performing models for each unique combination of positional interpolant, sampling scheme, and latent variable.

For the perov-5 ablation study, we show across the board that all interpolants but the linear one can beat the state-of-the-art match rate set previously by DiffCSP/FlowMM’s models, all with larger RMSEs (our linear interpolant performs comparably to both models). The increased RMSE is partly why the match rate increases: for SIs outside the linear interpolants, we find that particles generally find the correct local chemical configurations to flow towards, but are not able to end up in the precise symmetric sites. By contrast, the linear interpolants have the lowest RMSE because the particles flow to more symmetric positions, but the local environments are not correct due to species mismatch. We note that the trigonometric interpolant is not unique in its ability to have a high match rate. We suspect that the encoder’s ability to learn relevant representations for species may pose a limitation, which noised or non-geodesic interpolating paths can overcome. We can add this revised discussion to the main text.

We also show that for MP-20, the trigonometric interpolant with an SDE can also outperform on match rate compared to previously published models.

Comparing the perov-5 and MP-20 datasets to understand the effect of unit cell size would not be effective since these datasets have vastly different atomic, species, and unit cell distributions. The comparison between MP-20 and MPTS-52 would be more pertinent as they are more similar: they are both taken from the MP database, and differ by the max. number of atoms (20 vs. 52). Their match rates are reported in the original manuscript.

Concerning the number of trained models: Many models were partially trained and compared in the process of hyperparameter tuning: on average 27 models (perov-5) and 32 models (MP-20) for each choice of positional interpolant, sample scheme, and latent variable. We will add ranges for the hyperparameters to the appendix.

Comparison to FlowMM

The notable differences between our models and FlowMM are (a) discrete flow matching on species for OMG vs. analog bits for FlowMM; (b) the cell representation; (c) FlowMM’s use of a slightly modified CSPNet encoder while OMG utilizes CSPNet out of the box. We direct the reviewer to our CSP results, which show improvement over the FlowMM’s model we train without any species learning. Thus the handling of species is not sufficient to fully explain the differences in model performance.

Clarifications

SBD base distribution

We agree with the reviewer that the wrapped normal distribution with a large variance as used in DiffCSP can be approximated by a uniform distribution. We made the referenced note in line 181 to reflect our implementation and to highlight the connection to one-sided interpolants in the SI framework (that require a normal base distribution). We will update the corresponding sentence to clarify this.

Geodesic and periodic interpolants

The geodesic is indeed the same as the linear interpolant wrapped back into the box. The reason for computing the geodesic first for all other interpolants is because there are multiple ways to connect two points on a torus (e.g., in a periodic box one can connect two points with or without crossing the box boundaries). Using the geodesic as the “starting point” for computing the interpolating path allows for them to be uniquely defined. We also point to our response to Reviewer gm5E in section “PBCs.”

Subtraction of COM motion

The removal of the center-of-mass motion (as similarly implemented by FlowMM) in the loss function is basically analogous to choosing translationally invariant representations of the unit cells (see the discussion in Appendix D of Miller et al., 2024; arXiv:2406.04713). This allows to train of the translationally invariant CSPNet model in a consistent manner. Phrased differently, CSPNet cannot predict any COM motion which is why this part has to be removed in the ground-truth velocity for a consistent training target.

Species filtering

Our models are indeed being trained on the full MP-20 dataset (with all atom types) and the filtering of atoms is only done during relaxation with MatterSim. We will highlight this more clearly in the revised manuscript.

We thank the reviewer for their feedback and address their concerns below.

DFT relaxation

We agree that DFT relaxations offer a more rigorous evaluation of structure stability. As such, we are currently running DFT calculations for a large batch of generated structures. To assess consistency between the MLIP (MatterSim) and DFT results, we analyze 10 random subsets of 100 structures each (from the ~800 structures for which DFT relaxations are currently complete). For each subset, we compute the metastable S.U.N. (M.S.U.N.) rate based on the MLIP and DFT relaxed structures, respectively:

As the table shows, we observe close agreement between the MLIP and DFT, with DFT M.S.U.N. rates consistently tracking the MLIP rates while showing slightly more conservative values.

To further validate consistency, we also compared the energy above the convex hull between MLIP and DFT-relaxed structures. We find strong agreement, with a linear regression producing $R^2 = 0.986$, indicating that the MLIP (MatterSim) serves as a reliable surrogate for DFT.

The complete results will be included in the revised version of the paper.

Interpolant choice and structural diversity

We thank the reviewer for raising the important question of how the interpolant choice affects diversity in generated materials. We refer the reviewer to our response to Reviewer w22y for additional ablation studies for CSP (to be added to appendix), and to Reviewer K1be in section “Ablation studies, hyperparameters, and model performance” for a more detailed discussion of how interpolant choice affects performance (to be added to main text).

For the DNG task on the MP-20 dataset, we have also obtained $N$-ary distributions (i.e., number of unique elements per structure) and element-wise distributions of average coordination number across all positional interpolants used in OMG. We find that the best OMG models show superior agreement between the test set and the generated structures for these elemental distributions, and thus conclude that OMG can closely reproduce the elemental and structural diversity present in the data. In particular, the OMG-Linear, OMG-EncDec, and OMG-CFP+CSP positional interpolants show best agreement for the $N$-ary distributions, and all OMG models show superior performance on the element-wise distributions of average coordination number where DiffCSP and FlowMM’s models show significant under-coordination of atomic environments. We will include these results as new figures in the appendix.

Evaluation before and after relaxation

As the reviewer correctly notes, evaluation can be performed either on generated structures as-is or after relaxation with DFT or an MLIP. Our evaluation is split accordingly:

• Table 2 (main text) reports DNG performance before relaxation, focusing on coverage (recall, precision), property distributions (e.g., density, average coordination number, $N$-ary count), and validity metrics (structural, compositional, and combined). These metrics are used to assess how well the model captures the target data distribution and should reflect the quality of generation prior to any refinement.
• Table 3 (main text) reports DNG results after structural relaxation using the MatterSim MLIP, evaluating stability, novelty and uniqueness, as well as RMSD between initial and relaxed structures. These metrics assess the model’s utility for materials discovery. The RMSD values directly quantify structural discrepancy between generation and relaxation. We discovered a transcription error in the initially reported values and provide the corrected RMSDs below:
  • DiffCSP: 1.295
  • FlowMM: 0.651
  • OMG-Linear: 0.294
  • OMG-Trig: 0.763
  • OMG-EncDec: 0.390
  • OMG-SBD: 0.759
  • OMG-CFP+CSP: 0.488

These corrected values support OMG’s ability to generate structures that are not only diverse and realistic but also close to relaxed local minima, especially for the linear and encoder-decoder interpolants.

We thank the reviewer for raising questions about our paper, and address them topically below.

Comparison with FlowLLM

We agree that FlowLLM is an important material generation method representing the most recent trends. It uses an LLM to sample structures and FlowMM to refine them. We note that this is an orthogonal feature to our method which uses the general SI framework, and that FlowLLM’s approach can be incorporated into OMG. For a fair comparison, we evaluate both FlowMM (FlowLLM) and OMG (OMG-LLM) on the LLM dataset released by the FlowLLM authors (see https://github.com/facebookresearch/flowmm) and show results below.

Refining the same structures generated by an LLM, OMG’s linear interpolant outperforms FlowMM in almost all DNG metrics. We appreciate the reviewer’s suggestion as this comparison–to be included in the revised paper–further demonstrates the advantage and flexibility of OMG.

Computational cost

We compare the cost of training and integrating OMG on the MP-20 dataset and show low computational costs for OMG’s ODE scheme for both training and inference. The SDE scheme is more expensive but competitive. For these experiments, we use an Nvidia RTX8000 GPU with a batch size of 512 and 1000 integration steps. These results will be included in the appendix.

CSP

DNG

Ablation studies

Regarding the performance across positional interpolants, we provide ablation studies for perov-5 and MP-20 on the CSP task, broken down by choice of positional interpolant, sampling method, and latent variable $\gamma$ (to be added to the appendix). We note different trends for the perov-5 dataset, which has cubic unit cells and similar positions, and the MP-20 dataset, which exhibits more structural and chemical variation. We direct the reviewer to our discussion in response to Reviewer K1be in section “Ablation studies, hyperparameters, and model performance”.

Perov-5 CSP

MP-20 CSP

Limited baselines

We reiterate that symmetry-aware methods would not be an apples-to-apples comparison to our method, and thus are not utilized in benchmarks. However, they are discussed in the manuscript and can be incorporated into future iterations of OMG.

We thank the reviewer for the thoughtful and constructive comments. Below we address the main concerns and the three questions raised.

Novelty and contribution

We acknowledge that the novelty of our work could have been more clearly emphasized. To clarify:

• While DiffCSP and MatterGen specifically implement score-based diffusion with SDE-based sampling, and FlowMM uses a fixed linear interpolant with ODEs, our approach in OMG builds on the broader stochastic interpolants (SI) framework which enables both ODE- and SDE-based generation and a much wider range of interpolants. To our knowledge, this flexibility has not previously been explored for crystal generation. By systematically studying this much broader design space, we demonstrate state-of-the-art performance across the CSP and DNG tasks. We also set the first CSP baseline for the Alex-MP dataset by reporting OMG's performance on it.
• We refine the match-rate metric for CSP by eliminating unnecessary filtering present in prior work (e.g., CDVAE, DiffCSP and FlowMM), and we introduce the average coordination number metric for DNG to better evaluate the similarity of generated and test structures.
• From a methodology perspective, OMG is the first work to incorporate periodic boundary conditions (PBCs) into the SI framework. As noted by the reviewer, the use of discrete flow matching (DFM) for atomic species generation in DNG is also novel.
• We introduce the minimum permutation distance option as a data-dependent coupling during training that permutes atoms within structures to minimize the per-atom displacement during interpolation.
• Our proposed framework is highly flexible and extensible. It can be easily adapted for LLM-enhanced material generation (see response to Reviewer w22y for OMG-LLM / FlowLLM results).

Claim of unification

We thank the reviewer for raising this point and agree it would strengthen the paper to make the unification claim more explicit. We will add a dedicated section in the appendix cataloging how prior approaches can be recovered within the SI framework:

• Conditional Flow Matching (CFM) as implemented in FlowMM is naturally subsumed by SI. When using ODE-based sampling [considering only the loss in Eq. (2)] with the linear interpolant $x(t, x_0, x_1) = (1 - t) x_0 + t x_1$, Eq. (2) becomes identical to the FlowMM loss [see Eq. (15) in Miller et al., 2024; arXiv:2406.04713].
• Score-based diffusion models (SBDMs) are recovered via specific stochastic interpolants, both in their variance-preserving (VP) and variance-exploding (VE) forms as they appear in DiffCSP and MatterGen (see Aranguri et al., 2025; arXiv:2501.00988). The SBD interpolant $x(t, x_0, x_1) = \sqrt{1 - t^2} x_0 + t x_1$ (derived in Section 5.1 of Albergo et al., 2023; arXiv:2303.08797) recovers the VP variant of SBDMs. In this work we only explicitly employ this SBD interpolant, as we only implemented the spatially linear interpolants outlined in Section 4 of (Albergo et al., 2023). Nevertheless, we will report the appropriate choices of interpolants to recover both VP and VE SBDMs in the revised discussion.

PBCs

We agree that the discussion of periodic boundary conditions (PBCs) should be extended:

• We do not attempt to generalize stochastic interpolants (SIs) to arbitrary manifolds (as in Riemannian flow matching, or RFM). Instead, we adopt a task-specific formulation tailored to flat tori, which are the relevant manifolds for fractional coordinates in crystal generation.
• As in FlowMM, in order to uniquely define the interpolating paths, we rely on shortest geodesic interpolation paths between pairs of fractional coordinates $x_0$ and $x_1$, ensuring that interpolants are well-defined and differentiable. As briefly discussed in Section 3.2.1, this shortest geodesic path can be computed by first unwrapping one of the coordinates (say $x_1$) into its periodic image $x_1^{\prime}$, such that it is the closest image to $x_0$. We then compute the linear interpolant $x(t, x_0, x_1^{\prime}) = (1 - t) x_0 + t x_1^{\prime}$ as if in Euclidean space, and finally wrap the result back onto the torus. This yields exactly the same shortest-path geodesic as in FlowMM, and thus recovers its corresponding CFM loss.
• All periodic stochastic interpolants are then defined similarly by computing $x(t, x_0, x_1^{\prime}, z) = \alpha(t) x_0 + \beta(t) x_1^{\prime} + \gamma(t) z$ in the unwrapped (Euclidean) space and wrapping back onto the torus. In Appendix A.3, we show that averaging over the latent variable $\gamma(t) z$ recovers the deterministic base interpolant path, as required by the SI framework.

We will revise and expand our discussion in Section 3.2.1 and Appendix A.3 to elaborate on this approach, clarify its compatibility with SI, and contrast it more explicitly with RFM.

Computational overheads

We agree this is an important point. We refer to our response to Reviewer w22y.