Open Materials Generation with Stochastic Interpolants
Authors: Philipp Höllmer
Thomas Egg
Maya Martirossyan
Eric Fuemmeler
Zeren Shui
Amit Gupta
Pawan Prakash
Adrian Roitberg
Mingjie Liu
George Karypis
Mark Transtrum
Richard Hennig
Ellad B. Tadmor
Stefano Martiniani
Venue: ICML 2025 poster (2025)
Keywords: stochastic interpolants
generative models
inorganic crystals
materials design
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: The discovery of new materials is essential for enabling technological advancements. Computational approaches for predicting novel materials must effectively learn the manifold of stable crystal structures within an infinite design space. We introduce Open Materials Generation (OMatG), a unifying framework for the generative design and discovery of inorganic crystalline materials. OMatG employs stochastic interpolants (SI) to bridge an arbitrary base distribution to the target distribution of inorganic crystals via a broad class of tunable stochastic processes, encompassing both diffusion models and flow matching as special cases. In this work, we adapt the SI framework by integrating an equivariant graph representation of crystal structures and extending it to account for periodic boundary conditions in unit cell representations. Additionally, we couple the SI flow over spatial coordinates and lattice vectors with discrete flow matching for atomic species. We benchmark OMatG's performance on two tasks: Crystal Structure Prediction (CSP) for specified compositions, and de novo generation (DNG) aimed at discovering stable, novel, and unique structures. In our ground-up implementation of OMatG, we refine and extend both CSP and DNG metrics compared to previous works. OMatG establishes a new state of the art in generative modeling for materials discovery, outperforming purely flow-based and diffusion-based implementations. These results underscore the importance of designing flexible deep learning frameworks to accelerate progress in materials science. The OMatG code is available at https://github.com/FERMat-ML/OMatG.

Open Materials Generation with Stochastic Interpolants
Philipp Höllmer"
Thomas Egg"
Maya M. Martirossyan" 1 Eric Fuemmeler Zeren Shui 2 Amit Gupta"
Pawan Prakash3
Adrian Roitherg
Mingjie Liu George Karypis² Mark Transtrum Richard G. Hennig?
Ellad B. Tadmor
Stefano Martiniani
Abstract
1. Introduction
The discovery of new materials is essential for enabling technological advancements. Computa- tional approaches for predicting novel materials must effectively learn the manifold of stable erys- tal structures within an infinite design space. We introduce Open Materials Generation (OMarG), unifying framework for the generative design and discovery of inorganic crystalline materials. OMatG employs stochastic interpolants (SI) to bridge an arbitrary base distribution to the tar- get distribution of inorganic crystals via broad class of tunable stochastic processes, encompass- ing both diffusion models and flow matching special cases. In this work, adapt the SI frame- work by integrating sentation of crystal structures and extending it to account for periodic boundary conditions in unit cell representations. Additionally, we couple the SI flow over spatial coordinates and lattice vectors with discrete flow matching for atomic species. We benchmark OMatG's performance on two tasks: Crystal Structure Prediction (CSP) for specified compositions, and de novo generation (DNG) aimed at discovering stable, novel, and unique structures. In our ground-up implementa- tion of OMatG, we refine and extend both CSP and DNG metries compared to previous works. OMatG establishes new state of the art in gen- erative modeling for materials discovery, outper- forming purely flow-based and diffusion-based implementations. These results underscore the importance of designing flexible deep learning frameworks to accelerate progress in materials science, The OMatG code is available at https: //gittub.bb.////// com FFMM/ttMMMMMMMM.. OM.tG.
A core objective of materials science is the discovery of new synthesizable structures and compounds with the po- tential to meet critical societal demands. The development of new materials such as room-temperature superconductors (Boeri et al., 2022), high high-performance alloys with excep- tional mechanical properties (Gludovatz et al., 2014; 2016; George et al., 2019), advanced catalysts (Strmenik et al., 2016; Nakaya & Furukawa, 2023), and materials for energy storage and generation (Liu et al., 2010; Snyder & Toberer, 2008) holds the potential to drive technological revolutions.
Exploring the vast compositional and structural landscape of multicomponent materials with novel properties is essential, yet exhaustive experimental screening is infea- sible (Cantor, 2021). Quantum and classical molecular simulation offer a powerful alternative, enabling a more targeted and efficient exploration. recent decades, both experimental (Potyrailo al., 2011; Maier, 2019) and computational (Jain al., 2011 Curtarolo al., 2013) high-throughput pipelines have led to proliferation of materials databases for crystal structures (Bergerhoff et al., 1983; Mehl et al., 2017) and simulations (Blaiszik et al., 2016; Vita et al., 2023; Fuemmeler et al., 2024). These advances have already facilitated the development of more accurate machine-learned interatomic potentials (Batzner et al., 2022: Batatia et al., 2022; Chen & Ong, 2022).
equivariant graph repre-
Still, efficiently sampling the manifold of stable materi- als structures under diverse constraints such compo- sition and target propertics-remainn a major challenge. Traditional approaches to materials discovery have relied on first-principles electronic structure methods such as density functional theory (DFT)'- or more sophisticated theory, depending on the property (Booth al., 2013; Zaki et al., 2014; Isaacs & Marianetti, 2020) while power- ful and fairly accurate, are very computationally expensive. These methods include ab initio random structure searching (AIRSS) (Pickard & Needs, 2011) or genetic algorithms for structure and phase prediction (Tipton & Hennig, 2013), both of which have successfully predicted new crystal struc-
'qqual contribution 'New York University University of Min- nesota "University of Florida 'Brghham Young University. Corre- spondence to: Stefano Martiniani <sm7683 @nyu.cdu>.
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
'See Appendix for a list of acronyms used throughout this paper.

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Open Matcrials Generation with Stochastic Interpolants
tures and some of which have even been experimentally realized (Oganov et al., 2019). However, the high com- putational cost of these approaches has limited the scope and speed of material exploration, highlighting the need for cutting-edge MI. techniques to significantly accelerate the discovery of stable inorganic crystalline materials.
these equations. The flexibility of the SI framework stems from the ability tailor the choice of interpolants, and the incorporation of an additional random latent variable, further enhancing its expressivity. With their rich parameter- ization, SIs thus provide an ideal framework for optimizing generative models for materials design.
We implement the SI framework in the open- source Open Materials Generation (OMatG) package, released alongside this paper. OMatG allows to train and benchmark models for two materials generation tasks: Crystul structure predic- rion (CSP) which only learns to generate atomic positions and lattice vectors for a given composition, and de novo generation (DNG) which learns to generate both crystal structure and composition to predict novel materials. We discover that optimizing interpolation schemes for different degrees of freedom of the crystal unit cell substantially im- proves performance across diverse datasets. As result, our approach achieves a new state of the art out-ortpenonmmin both DifICSP (Jiao et al., 2023) and FlowMM (Miller et al., 2024) in CSP and DNG, as well as MatterGen (Zeni et al., 2025) in DNG-across all evaluated datasets under existing, revised, and new performance measures.
1.1. Related Works
Recent advances in machine learning techniques have gener- ated significant interest in applying data-driven approaches for inorganic materials discovery. Among these, Graph Networks for Materials Exploration (GNoME) has demon- strated remarkable success by coupling coarse sampling strategies for structure and composition with AIRSS that leverages a highly accurate machine-learned interatomic potential (MLIP) to predict material stability, leading to the identification of millions of new candidate crystal struc- tures (Merchant et al., 2023). Other frameworks have ap- proached the generation of composition and structure jointly through fully ML-based methods. Crystal Diffusion Varia- tional Autoencoder (CDVAE) leverages variational autoen- coders and graph neural network representation to sample new crystal structures from a learned latent space (Xie et al., 2022). To date, state-of-the-an performance in both crystal structure prediction for given compositions and de novo generation of novel stable materials has been achieved by diffusion models such as DifECSP (Jiao al., 2023) and MatterGen (Zeni et al., 2025), as well as conditional flow- matching frameworks such as FM (Miller et al., 2024).
2. Background
2.1. Diffusion Models
A widely used approach in generative modeling uses diffu- sion models (Sohl-Dickstein et al., 2015), which define a stochastic process that progressively transforms structured data into noise via a predefined diffusion dynamic. A model is then trained to approximate the reverse process, enabling the generation of new samples, typically by integrating corresponding SDE.
While these approaches have demonstrated that ML can push the boundaries of computational materials discovery, it remains uncertain whether score-based diffusion or flow- matching represents the definitive methodological frame- works for this problem. Furthermore, the extent to which the optimal approach depends the training data remains an open question. Thus far, each new method has typically outperformed its predecessors across datasets.
Score-based diffusion models (SBDMs) are an instantiation of diffusion models that learn a score function-the gradi- ent of the log probability density-to guide the reversal of the diffusion process via numerical integration (Song et al., 2021). SBDMs have demonstrated remarkable success in generating high-quality and novel samples across wide range of applications where the target distribution is com- plex and intractable, such as photorealistic image generation (Saharia et al., 2022) and molecular conformation prediction (Corso et al., 2023).
1.2. Our Contribution
The work we present in this paper is the first implementation and extension of the stochastic interpolants (SIs) framework (Albergo et al., 2023) for the modeling and generation of in- organic crystalline materials. SIs are a unifying framework that encompasses both flow-matching and diffusion- based methods as specific instances, while offering more gen- eral and flexible framework for generative modeling. In this context, SIs define a stochastic process that interpolates between pairs of samples from a known base distribution and a target distribution of inorganic crystals. By learning the velocity term of an ordinary differential equation (ODE) or the drift term of stochastic differential equations (SDE), new samples can be generated by numerically integrating
2.2. Conditional Flow Matching
Conditional flow matching (CFM) (Liu, 2022; Lipman et al., 2023; Albergo & Vanden-Eijnden. 2023) is a generative modeling technique that learns a flow which transports sam- ples from a base distribution at time t = 0 to a target dis- tribution at time path that describes how samples are distributed at any inter- mediate time t E [0. 1]. The velocity field associated with
1. This process defines probability
2

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Open Matcrials Generation with Stochastic Interpolants
this flow governs how individual samples evolve over time. CFM learns the velocity indirectly by constructing condi- tional vector fields that are known priori. Once trained, samples drawn from the base distribution can be evolved numerically to generate new samples from the target dis- tribution. Originally, CFM was formulated using Gaussian conditional probability paths, but Tong et al. (2024) later extended this framework to allow for arbitrary probability paths and couplings between base and target distributions. A further extension, particularly relevant to physics and chemistry, is Riemannian flow matching (RFM), which gen- cralizes CFM to Riemannian manifolds (Chen & Lipman, 2024). This allows in particular to use the flow- How-matching framework for systems with periodic boundary conditions as they appear in unit cell representations of inorganic crystals (Miller al., 2024).
(( I), wo Po- and x1 ~ Pi- For SDE-based sam- pling, an additional denoiser 2 (t) must be learned by minimizing an additional loss
(0) Re [| xt,)) 22 (t, 2]
(3)
The velocity term, along with the denoiser in the case of SDE-based sampling, enables the generation of samples from the target distribution (Albergo et al., 2023). Note that minimizing with respect to these loss functions amounts to minimizing with respect to a mean-squared error loss function (see Appendix C.2). For ODE-based sampling, (t)
in the interpolant æ(t, 20, 21, 2) is a possible choice. However, for SDE-based sampling. 7(t) > 0 is required for all t E (0, 1) (see Appendix B.1).
By appropriately selecting interpolation functions a, B, and choosing between deterministic (ODE) and stochastic (SDE) sampling schemes (see Appendix B.2), the SI frame- work not only recovers CFM and SBDM as special cases (see Appendix B.9) but also enables the design of broad class of novel generative models. The strength of OMatG's s SI implementation for materials discovery lies in its ability to tune both the interpolation and sampling schemes, as illustrated in Fig. for a pair of structures sampled from po and f1. By systematically optimizing over this large design space, we achieve superior performance for CSP and DNG tasks across datasets, as discussed in Section 5.
3. Open Materials Generation
3.1. Stochastic Interpolants
SIs provide a unifying mathematical framework for gener- ative modeling, generalizing both SBDMs and CFM (Al- bergo et al., 2023). The SI x11,, Fo, 21 bridges the base distribution po with a target distribution PI by learning time-dependent map. In this work, we focus stochastic interpolants of the form:
2z = 20, 2) = a(t)zo + 8(t)xr + (t)=.. (1)
3.2. Crystal Representation and Generation
Here, t E [0, 1] represents time and 21) are paired samples drawn from Po and Pl. respectively. The random variable z is drawn from a standard Gaussian (0, I) in- dependently of xo and 21. The functional forms of o, 3, and y are flexible, subject to few constraints (see Ap- pendix B.2). The inclusion of the latent variable 7(t)= allows sampling of an ensemble of paths around the mean interpolant x(t, = a(t)ar + (())z1, and is theorized to improve generative modeling by promoting smoother and more regular learned flows (Albergo et al., 2023).
A crystalline material is defined by its idealized repeat unit, or unit cell, which encodes its periodicity. In the OMatG representation, unit cell is described by separating the material's chemical composition given by its atomic species A E Z where N is the number of atoms in the unit cell-from its structural representation-its fractional coordinates X E [0, 1) 1)3xN with periodic boundaries and lattice vectors E R3333 During training, all three components (A, X, L) are considered simultaneously. We apply the SI framework only the continuous structural representations {X, L} with loss functions defined in Eqs (2) and (3), and use discrete flow matching (DFM) on the chemical species A (see Section 3.2.3) (Gat et al., 2024). The number of atoms N in the structure xp sampled from the base distribution po is determined by the number of atoms in the corresponding structure x1 sampled from the target distribution P1.
The time-dependent density P: of the stochastic process H, in Eq. (1) can be realized either via deterministic sam- pling through ODE (derived from a transport equation) or stochastic sampling through an SDE (derived from equation) only requiring Py ~ Po (see Ap- pendix I). This enables generative modeling by evolving samples from a known base distribution Po to the target distribution P1- For both ODE- and SDE-based sampling, the required velocity term 8 (t,, 2) is learned by minimizing the loss function
3.2.1. ATOMIC COORDINATES
L, = E2 12 wo .1 [ (1, x2
For treating fractional coordinates, we implement variety of periodic interpolants that connect the base and target data distributions (see Section 4.1). We specify the base distribu- tion for the fractional coordinates x E [0, 1) for all x E X via uniform distribution (except for the score-based dif-
(2)
- 2 2,z To,
x1, - (tt,, xx)),
where the expectation is taken independently over t ~
(0, 1), the uniform distribution between and 1, 2
3

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Open Materials Generation with Stochastic Interpolants
a.
3.2.2. LATTICE VECTORS
Lattice vectors L are treated with a wide range of (non- periodic) stochastic interpolants (see Section 4.1 again). To construct the base distribution, we follow Miller et al. (2024) and construct an informative base distribution Po(L) by combining a uniform distribution over the lattice angles with a log-normal distribution fitted to the empirical distribution of the lattice lengths in each target dataset. This choice brings the base distribution closer to the target distribution. Unlike SBDM, which requires Gaussian base distribution, the SI framework allows such flexibility. Importantly, the model still has to learn to generate a joint, correlated distribution of lattice vectors, fractional coordinates, and atomic species.
Interpolant choice
Enc. Dec. Interpolant
Trig.
Enc. Dec.
p() (ar)
b.
d.
3.2.3. ATOMIC SPECIES
Sampling scheme ODE SDE
Tuning 7(1) V at(
The discrete nature of chemical compositions A in atomic crystals requires a specialized approach for generative mod- eling. To address this, we implement discrete flow matching (DFM) (Campbell et al., 2024). In our implementation of the DFM framework, each atomic species a E A can take values in {1. 100} U {M}} where {1 - 100} are atomic element numbers and A is a masking token used during training. The base distribution is defined as po(a) [M]* meaning that initially all N atoms masked. As sampling progresses, the identities of the atoms evolve via continuous-time Markov Chain (CTMC), and are progressively unmasked to reveal valid atomic species. At t = 1, all masked tokens are replaced. To learn this process, we define a conditional flow pta (a)lai) that lin- early interpolates in time from the fully masked state du toward a1 and thus yields the composition a, of the inter- polated structure Ft. Based on these conditional flows, neural network is trained to approximate the denoising dis- tribution p (a1 (zz), which yields the probability for the composition a given the entire structure Ig. by minimizing a cross-entropy loss
0.1 0.2
Figure 1. Visualization of the tunable components of the SI frame- work for bridging samples zp (gray particles) and 2i (purple parti- cles). Interpolation paths are shown only for one pair of highlighted particles. (a) The choice of the interpolant changes the path of the time-dependent interpolation trajectory. (b) During inference, the learned velocity term 5 (t, v) and denoiser 2 (f, .) generate new samples via ODE or SDE integration, here for a linear interpolant with v (0 - t). (c) The inclusion of a latent variable >(t)= changes the interpolation path. For SDE-based sampling, 7(1) > 0 is required. (d) The function 7(4, a) /at(1 L)
depends on a that also influences the interpolation path.
fusion interpolant that, following the approach of Jiao al. (2023), uses wrapped normal distribution po (z)) which be- comes a uniform distribution in the limit of large variance). To satisfy periodic boundary conditions on the paths defined by the interpolants, we extend the SI framework to the sur- face of a four-dimensional torus in this paper. Reminiscent of RFM (Chen& & Lipman. 2024), the linear interpolant on the torus traverses a path equivalent the shortest-pale -aath geodesic which is always well-defined. ? Other interpolants, however, are more complex. In order to uniquely define them, we always define the interpolation with respect to the shortest-path geodesic. That is, for interpolation between Zo and Φ1 with a periodic boundary at and 1, we first unwrap 2n to the periodic image æ' which has the shortest possible distance from æo. Following this, the interpolation between 2o and x is computed given choice of interpolant, and the traversed path is wrapped back into the boundary from 0 to 1. This approach is illustrated in Appendix B.5.
=
[
(4)
In doing this, we are able to directly construct the marginal rate matrix (2 i) for the CTMC that dictates the rate of a, at time t jumping to a different state during generation (see Appendix B.6). It is important to note that the learned probability path is function of the entire atomic configu- ration {A, X, .}} which is necessary for the prediction of chemical composition from structure.
3.3. Joint Generation with Stochastic Interpolants
For both CSP and DNG tasks, we seek to generate samples from a joint distribution over multiple coordinates. For DNG. this joint distribution P1 encompasses all elements of a crystal unit cell. For CSP we similarly model the joint
TThe only exception being when two points are precisely half the box length apart. However, this case is not relevant for the given hase distribution.

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Open Matcrials Generation with Stochastic Interpolants
distribution, P1, but with atom types fixed compositions sampled from the target dataset. For both tasks, the total loss function is formulated as weighted sum of the individual loss functions for each variable (see Appendix C.2), and their relative weights are optimized (see Appendix C.3). We illustrate both types of models and their structure generation process in Fig. 2a.
interval [o, 1]. Different noise schedules of the variance- preserving and variance-exploding SBDMs can likewise be encoded in different variants of the VP and VE SBD inter- polants (see Appendix B.9). For the results presented in this paper, we only consider a constant noise schedule for the VP SBD interpolant, and a geometric noise schedule for the VE SBD interpolant. The SBD interpolants assume that pi is Gaussian distribution, and unlike the previous interpolants it involves explicit latent variable; instead the a())ao term takes on this role. The incorporation of VP and VE SBD interpolants enables OMatG to reproduce similar conditions to those in DiffCSP and MatterGen.
Additionally, for DNG, we consider a two-step process in which composition is learned separately from structure, seen in Fig. 2b. In this approach, we first train chem- ical formula prediction (CFP) model (see Appendix C.I) to generate compositions optimized for SMACT stability (Davies et al., 2019), similarity in the distribution of N- arity of known structures, as well as uniqueness and novelty. The predicted compositions are then used as input for a pretrained CSP model, which generates the corresponding atomic configurations.
The trajectory of the encoder-decoder interpolant between times = Twwitch and = resembles that of the SBD interpolants between times t and = 1. For the ex- ample of using the encoder- interpolant only for the coordinates, however, we emphasize that the Gaussian- distributed coordinates at t = Tswiihh are conditioned on other coordinates that are partially interpolated at this point. Conversely, for SBD interpolation. the Gaussian distributed coordinates at = 0 are only conditioned on other random variables since, at this point, all elements of To are randomly distributed.
4. Methodology
4.1. Choice of Interpolant
In training OMaGG, we optimize the choice of the interpo- lating function that is used during training for the lattice vectors (without periodic boundary conditions) and the frac- tional coordinates (with periodic boundary conditions). We consider four interpolants of the form defined in Eq. (1), each shaping the interpolation trajectory differently (see also Appendices B.2 and B.9 for further details).
To investigate how different interpolants affect generative performance, we consider all interpolants outlined above for both the atomic positions X and the lattice vectors L. We noted that learning accurate velocities and denoisers for the atomic positions, X is more challenging than for the other degrees of freedom. Accordingly, we optimize all hypprparameters- for L L-separaelly for each interpolant applied to X. This results in a set of experiments specific to the positional interpolants, where the best performing lattice interpolant may vary.
The linear interpolant defines a constant velocity trajectory from To to 21- When combined with an ODE sampling scheme and 7 O, this reproduces the specific instantia- tion of CFM implemented in FlowMM (see Appendix B.9). However, combining the linear interpolant with an SDE sampling scheme or nonzero already introduces key dif- ferences. The inclusion of the latent variable can promote smoother learned flows (Albergo et al., 2023), while stochas- tic sampling alters the generative dynamics compared to the deterministic formulation in FlowMM. The trigonometric interpolant prescribes trajectories with more curvature than the linear interpolant. The encoder-decoder interpolant first evolves samples from Pu at t = 0 to follow an interme- diate Gaussian distribution at a switch time Twiichh before mapping them to samples from the target distribution P1 at = 1. This approach has been found to interpolate more smoothly between distributions, potentially mitigating the formation of spurious features in the probability path at intermediate times (Albergo et al., 2023). Lastly, we consider variance-preseen preserving score-based diffusion (VP SBD) and variance-exploding seore-based diffusion (VE SBD) interpolants. When paired with an SDE sampling scheme, these interpolants are mathematically equivalent to the corresponding SBDM, but on the continuous time
the choice of interpolant
4.2. Equivariant Representation of Crystal Structures
Imposing inductive biases on the latent representation of the crystal structure can promote data efficiency and improve learning. The CSPNet architecture (Jiao et al., 2023), orig- inally adopted in DiffCSP, is an equivariant graph neural network (EGNN) (Satorras al., 2021) that produces permutation- and rotation-equiivarian,, as well as translation- invariant representation of the crystal structures.
In the current OMatG implementation, we employ CSPNet as an encoder that is trained from scratch. The CSPNet ar- chitecture encodes atomic types using learnable atomie em- beddings and represents fractional coordinates through sinu- soidal positional encodings (see Appendix C.1). These fea- tures are processed through six layers of paassage-aassing after which the encoder produces the velocity 6 6((t. 2) of both the lattice and the fractional coordinates, as well as potentially predicting the denoiser 2t(t, 2²). For DNG, the
5

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Open Materials Generation with Stochastic Interpolants
b.
(gg Ny, Lad Pi
(Aua Nosas loge?
(Ause Noser logss?
CSP
CSP
CFP
DNG
DNG
1=0
Generation
Figure 2. Illustration of crystal structure prediction (CSP) and de novo generation (DNG) tasks, (a) For CSP, the species A fixed with known compositions from = 0. From this, we predict X and L from randomly sampled initial values. For DNG, we predict (A, X, jointly. Our implementation of discrete flow matching (DFM) initializes A sequence of masked particles that unmasked through series of discrete jumps to reveal physically reasonable composition. (b) Two avenues for performing DNG of materials. The first uses two steps: chemical formula prediction (CFP) model predicts compositions and then uses a CSP model find accompanying stable structures. The second trains a DNG model over cell, species, and fractional coordinates jointly as shown in (a).
network must also predict logefje (a (z)). The resulting outputs inherently preserve the permutation, rotational, and translational symmetries embedded in CSPNet.
these models. In nearly all cases, we were able to gener- ate structures using the DiffCSP, FlowMM, and MatterGen source codes whose metrics closely matched the previously reported metrics their respective manuscripts. For Diff- CSP and MatterGen, we relied on published checkpoints while we retrained FlowMM from scratch. The observed differences be attributed to the of of SMACT composition rules (Davies et al., 2019) and to natural fluctuations during generation and model retraining,
The output of CSPNet is invariant with respect to transla- tions of the fractional coordinates in the input. Thus, one should, in principle, use representation of the fractional coordinates that does not contain any information about translations. While this is straightforward in Euclidean space by removing the mean of the coordinates of the given structure, this cannot be done with periodic boundary con- ditions where the mean not uniquely defined. We follow Miller et al. (2024) and instead remove the well-defined center-of-mass motion when computing the ground-truth velocity (t,
version
Since the focus of this work to assess our model's ability to learn unconstrained and unconditioned flows, we do not compare against commetry-constrainee generation meth- ods (AI4Science et al., 2023; Cao et al., 2024; Zhu al., 2024; Kazeev al., 2024; Jiao et al., 2024). Symmetry constraints can be incorporated in future extensions of the flexible OMatG framework.
(t, No, 21, 2) in Eq. (2).
Alternative EGNNs such as NequlP (Baizner et al., 2022), M3GNet (Chen & Ong, 2022), or MACE (Batatia et al., 2022) which have been widely used for the development of MIIPs can also serve as plug-and-play encoders within OMatG's SI framework. Integrating different architectures a direction that we plan to explore in future iterations of the framework.
5. Experiments
5.1. Performance Metries
Crystal structure prediction. We assess the performance of OMatG's and competing models using a variety of stan- dard (introduced in (Xie et al., 2022; Zeni et al., 2025)), refined, and contributed benchmarks. For the CSP task, we generate structure for every composition in the test dataset. We then attempt to match every generated struc- ture with the corresponding test structure using Pymatgen's
4.3. Comparison other Frameworks
We compare results DiffCSP and FlowMM models for both CSP and DNG. For DNG, we additionally consider the MatterGen-MP model that was trained the same MP- 20 dataset as OMatG's s DNG models. We detail in Section 5 how we improve the extant benchmarks used in the field and therefore recompute all CSP and DNG benchmarks for
"The SMACT Python library updated its default oxidation states with the release of version 3.0.
6

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Open Matcrials Generation with Stochastic Interpolants
Table 1. Results from crystal structure prediction. Match rate and RMSE of matched structures without (left) and with (right) filtering for structural and compositional validity are reported for all models. For OMalG's models, the choice of positional interpolant, latent variable component y, and sampling scheme are noted. For all SDE sampling schemes the inclusion y is assumed and not noted; for SBD interpolants is not relevant. Further details and complete results for perov-5 and MP-20 can be found in Appendix C.3.
perov-5
MP-20
MPTS-52
Alex-MP-20
Method
Match (%) f
RMSE 1
Match (%) f
RMSE 1
Match (%)
RMSE 1
Match (%) f
RMSE!
DiRCSP
53.08 51, 11.94
0.0774 000775
57.82/ 52.51
0.0627 0.0610)
15.79 14.29
0,1533 0.1489
FlowMM
53.63 51.8
0.1025 000994
66.22 / 59.98
0.0661 / 0.0629
22.29 / 20.28
0,1541 / 0..4486
OMaiG
Linear (ODE) wo 7
51,86 / 50.62
0.0757 / 0,0760
69.83/ / 63.75
(.0741/0,077)
27.38/ / 25.15
0.1970 0.1191
72.02/ 64.23
0.0683 / 0.0677
Linear (SDB) q
74.16 72.87
0.3307 00.315
68.20 61.88
0.1632 0.1611
23.95 21.70
0.2402 0.2353
61.07 54.44
0.1870 / 0.1860
Trig (SDE) w/
73.37 / 71.60
0,3610 / 0,3614
68,90 / 62.65
0,1249 / 0,12
24,51 / 22.26
0,1867 1804
72.50 22.50/64.77 64.71
0,1261 / 00.1
Ene-Dec (ODE) ?
68.08 64.60
0,4005 0.4003
55,15 0.1306 49:445
0.1260
14.65 13.53
0.2543 0.2500
0,0957 0.0993
VP SBD (ODE)
83.06/ 81.27
0.3753 0,3755
45.57 / 39.48
0.1880 / 0.1775
9.66 8,36
0.3088 / 0.3041
46.23 39.96
0,1718 / 0.1618
VE SBD (ODE)
60.18/ / 52.97
0.2510 0,23337
63,79 57.82
0.0809 / 0,0780
21.42/ 19.57
0,1740 0.1102
67.79 / 60.25
0.0674 / 0,0649
Table Results from de novo generation of 10000 structures with models trained on the MP-20 dataset. The integration steps for OMatG is chosen based on best overall performance. For OMatG's model, the choice of positional interpolant, latent variable component ? and sampling scheme are noted. Best scores in each category are bolded."
Validity (% 1)
Coverage (% f)
Property (U)
Method
Integration steps
Structural
Composition
Combined
Recall
Precision
wdist (p)
wdist (Nary)
wdist ((CN)
DifCSP
1000
99.91
82.68
82.65
99.67
99.63
0.3133
0.3193
0,3053
FlowMM
1000
92.26
83.11
76.94
99.34
99.02
1.0712
0.1130
0,4405
MatterGen-MP MP
1000
99.93
83,89
83.89
96.62
99.90
0,2741
0,1632
0.4155
OMatG
Linear (SDE) w/1
710
99.04
83.40
83.40
99.47
98.81
0.2583
0.0418
0.4066
Trig (ODE) w/7
680
95.05
82.84
82.84
99.33
94.75
0.0607
0.0172
0.1650
Ene-Dec (ODE) w/y
97.25
84.19
99.62
99.61
0.1155
0.0553
0.0465
VP SBD (SDE)
870
93.38
80.66
80.66
98.95
92.76
0.1865
0.0768
0,1637
CFP + CSP [Linear (ODE) w/ v]
130+210
97.95
79.68
78.21
99.67
99.50
0,5614
0,2008
0.6256
"We do not bold any values in the structural validity category as the CDVAE model reports the state of the art with 100% structural validity. For the Wasserstein distances of the density and Nary distributions, we only bold values lower than 0.075 and 0.079 respectively, these were the values reported by FlowMM for their model with 500 integration steps (not included in this table).
structuraMatcher module (Ong et al., 2013) with tol- erances (stol= = 0.5, ltol = 0.3, angletol = 10). We finally report the match rate and the average root-mean square displacement (RMSE) between the test structures and matched generated structures. Here, the RMSEs com- puted by Pymatgen are normalized by (V/N)1/3, where V is the (matched) volume and N is the number of atoms. During hyperparameter optimization, we only attempt to maximize the match rate (see Appendix C.3).
De novo generation. For the DNG task, metrics include validity (structural and compositional), coverage (recall and precision), and Wasserstein distances between distributions of properties including density p. number of unique ele- ments N (i.e., an Nary material), and average coordination number by structure (CN).. We newly introduce the average coordination number benchmark due to the difficulty of gen- erating sy symmetric structures; a structure's 's average coordina- tion number is a useful fingerprint, and higher-coordinated structures tend to be more symmetric.
Previously reported match rates filtered the matched gener- ated structures by their structural and compositional validity (see Appendix D.2). We note, however, that the datasets themselves contain invalid structures-fo for example, the MP- 20 test dataset has ~ 10% compositionally invalid structures. Thus, we argue that the removal of these invalid structures for computation of match rate and RMSE is not reasonable for assessing learning performance; we do, however, pro- vide both match rates (with and without validation filtering).
The previous DNG metrics are used in conjunction dur- ing optimization of hyperparameters (see Appendix C.3). For the best models, we then structurally relax the gener- ated structures in order to calculate the stability and the S.U.N. (stable, unique, and novel) rates using MatterGen's code base (MatterGen, 2025). The S.U.N. rate is defined as the percentage of generated structures that are stable with respect to reference convex hull (within 0.1 e eV/atom), are not found within the reference set (novel), and are not dupli-
7

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Open Matcrials Generation with Stochastic Interpolants
cated within the generated set itself (unique). The machine- learned interatomic potential MatterSim (Yang et al., 2024)
Table 3. Stahility (defined as S 0.1 e/atoo above hull), unique- ness, and novelty results from de novo generation on the MP-20 dataset computed for the same models as in Tab. 2. All evalua- tions are calculated with respect to 1000 relaxed structures (see Appendix D.4), utilizing the MatterGen code base (MatterGen, 2025) and the included reference Alex-MP-20 dataset. The average RMSD is between the generated and the relaxed structures, and the average energy above hull is reported in units of e eVatom.
is utilized for initial structural relaxation, which is subse- quently followed by a more computationally expensive DFT relaxation. Full workflow details are given in Appendix D.4. In addition to the stability-based metrics, we also report the root mean squared displacement (RMSD) between the gen- erated and relaxed structures (unnormalized, in units of A).
Method
(Py/ ())
RMSD (A.
Novelty Rate (%.
Stability Rate (56.
S.U.N.
5.2. Benchmarks and Datasets
above hull
Rate (56, 1)
DiSCSP
0.1751
0.3861
70.04
43.43
15.95
We use the following datasets to benchmark the models: perov-5 (Castelli et al., 2012), dataset of perovskites with 18 928 samples with five atoms per unit cell in which only lattice lengths and atomic types change; MPP-20 (Jain et al., 2013; Xie et al., 2022) from the Materials Project that con- tains 45 231 structures with a maximum of N = 20 atoms per unit cell, and MPTS-52 (Baird et al., 2024) which is a chronological data split of the Materials Project with 40 476 structures with up to N = 52 atoms per unit cell and is typically the most difficult to learn. We use the same 60- 20-20 splits as Xie et al. (2022); Jiao et al. (2023); Miller et al. (2024). Additionally, we consider the Alex-MP-20 dataset (Zeni et al.. 2025), where we used an 80-10-10 split constructed from MatterGen's 's 90-10 split, in which we re- moved 10% of the training data to create a test dataset. This dataset contains 675 204 structures with 20 or fewer atoms per unit cell from the Alexandria (Schmidt et al., 2022a;b) and MP-20 datasets. We do not include the carbon- 24 dataset (Pickard, 2020) in our results, as the current match rate metric is ill-defined for this dataset; because all ele- ments are carbon, it is not clear how many generated struc- tures are unique and producing a structure that matches one in the reference dataset is trivial.* 4
MM
0.1917
0.6316
69.13
41.20
11.73
MatterCien-MP
0.1772
0.1038
72.40
44.79
20.30
OMat
Linear (SDF) w/ ?
0.1808
0.6357
73.11
46.18
22.48
Trig (ODE) w/
0.1670
0.6877
66.45
$2.81
21.10
Enc Ene-Dec (ODE) w/
0.1482
18.77
vP SBD (SDE)
0.2120
0.7851
76.25
40.83
20.73
CHP CSP
0.2302
79.08
20.17
vious models' match rate for CSP on the MP-20 and MPTS- 52 datasets with the linear (both ODE and SDE sampling scheme) and the trigonometric positional interpolants. Fi- nally, OMatG establishes the first performance baseline for the CSP task on the Alex-MP-28 dataset.
De novo generation. For DNG models, we show the va- lidity, coverage, and property metrics of the models in Tab. 2 and the stability, uniqueness, and novelty results in Tab. 3. Further ablation results across OMatG variants can be found in Tab. 10 in the Appendix, and qualitative plots of various distributions are compared in Appendix D.5, OMatG achieves state-of of-the-art performance over DiffCSP, FlowMM, and MatterGen-MP for multiple positional in- terpolants thanks to the broader design space brought by the SIs. Figure compares the distributions of the average energies above the hull for generated structures, exhibit- ing OMatG's superior performance for the generation of stable structures. OMalG consistently produces lower en- ergy structures compared to previous models, and they are also generated close to their relaxed configuration. This, together with high novelty rates, begets improved S.U.N. rates. OMatG also outperforms FlowMM in settings where large language models are used as base distributions (Sriram et al., 2024), with results shown in Appendix E.
5.3. Results
Crystal structure prediction. We report the CSP perfor- mance of DiffCSP, FlowMM, and six OMatG models on the four benchmark datasets in Tab. 1. Further ablation re- sults across OMatG variants can be found in Tables 6 and 8 in the Appendix. OMatG significantly outperforms previ- ous approaches with respect to match rate on all datasets. We highlight the strong match rates on the perov-5 dataset achieved using the VP SBD and trigonometric positional interpolants with ODE sampling schemes, as shown in Ta- ble and Table 6, that greatly surpass (by a factor of 1.6) the match rates of previous models. We note that the relative performance for ODE SDE sampling schemes depends on the positional interpolant. OMalG also outperforms pre-
5.4. Discussion
Crystal structure prediction. We generally observe during hyperparameter optimization that the main learning chal- lenge lies in the accurate prediction of the atomic coordi- nates, which tended to have a higher relative weight in cal- culating the full loss function in Eq. (32) (see Tabs 7 and 9 in the Appendix). However, the two best perfforrfo mod- els for the perov-5 dataset instead exhibited the opposite- lending the most weight to learning of the cell vectors.
"Previous papers (Xie et al., 2022; Jiao al., 2023; Miller et al., 2024) report mathh rate for carbon-24. but they do not compare each generated structure to the entirety of the reference dataset; their results suggest the match tolerance is larger than the differences between the carbon- 24 structures themselves.
8

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Open Materials Generation with Stochastic Interpolants
not need to be learned, making this dataset a useful bench- mark for evaluating positional interpolant performance. It is possible that the superior performance of the VP SBD and trigonometric interpolants arise from their ability to gener- ate more circuitous flow trajectories compared to the strictly geodesic paths imposed by the linear interpolant- akin to the reasoning behind using latent variables to enhance learn- ing in SIs (Albergo et al., 2023).
DiffCSP
10-
FlowMM
MatterGen-MP
OMatG (Enc-Dec)
Density
6-
De novo generation. Thorough hyperparameter optimiza- tion enables us to note trends among the best-performing DNG models. In Tabs 10 and 11 in the Appendix. we show the performance metrics and hyperparameters for each model by choice of positional interpolant, sampling scheme, and (t) in the latent variable. We observe that several of our best performing models (with respect to S.U.N, and RMSD) possess lower levels of 'species noise' 1j which sets the probability that an atom will change its identity if already in an unmasked state (see Appendix B.6). Addi- tionally, we find that linear and trigonometric interpolants favor an element- permutation as a data-dependent cou- pling during training (see Appendix B.8), while the encoder- decoder and SBD interpolants prefer not use this coupling. Finally, we note that VP SBD models require similar mag- nitude of velocity annealing during generation for positions and lattices (see Appendix B.7). This is in stark contrast to all other models, where a significantly larger velocity annealing parameter is required for generating the positions.
-0.2
0.0 0.2 0.4
0.6
0.3
Energy Above Hull (eV/Atom)
Figure 3. Histogram of the computed energies above the convex hull for structures generated by FlowMM, DifICSP, MatterGen-MP. and OMatG (Ene-Dec Dee interpolant). The OMatG model consistently produces lower energy structures compared to competing models. See Appendix D.4 for caleulation details.
Our CSP results indicate tradeoff between match rate and is only computed if matched. We find that as the number of generated structures matching known com- positions in the test dataset increases, the structural fidelity- quantified by the accompanying RMSE-alss tends to in- crease. This tradeoff most strongly influences our results on the perov-5 dataset where all positional interpolants but the linear one can beat the previous state-of-the-ar match rate. Here, particles generally find the correct local chem- ical configurations to flow towards during generation, but are not able to end up in the precise symmetric sites. The ODE-based linear interpolant without a latent variable, in contrast, has the lowest RMSE because the particles flow to more symmetric positions, but the local environments are not correct due to species mismatch. We quantify this effect in Fig. 7 in the Appendix.
6. Conclusion
We adapt stochastic interpolants (SIs) for material genera- tion tasks and propose Open Materials Generation (OMatG), a material-generation framework that unifies score- based diffusion and conditional flow-matching approaches under the umbrella of SIs. By incorporating an equivariant graph representation of crystal structures and explicitly handling periodic boundary conditions, OMatG jointly models spatial coordinates, lattice vectors, and discrete atomic species in cohesive flow-based pipeline. Our extensive experiments on crystal structure prediction and de novo generation tasks demonstrate that OMatG sets a new state of the art in gener- ative modeling for inorganic materials discovery, yielding more stable, novel, and unique structures than either pure diffusion or pure conditional flow-matching counterparts.
For the CSP task on the perov-5 dataset, we highlighted the particularly strong performance of the VP SBD and trigonometric interpolants in achieving a high match rate. Unlike other datasets, perov-5 has a fixed number of N 5 atoms per unit cell and fixed (cubic) cell with varying side lengths and similar fractional positions- a combination which should not expose the model to a large variety of unit cell choices during interpolation or generation. By contrast, in other datasets, no singular representation of the periodic repeat unit is imposed on flows, meaning the model cannot learn the invariance (or even equivariance) to the choice of periodic repeat unit.² This likely contributes to the difficulty of unconstrained based models in generating highly symmetric structures. Thus, the perov-5 dataset presents a unique case where the invariance to unit cell choice does
We underscore the importance of flexible ML frameworks like OMatG, which can adapt to different types of materials datasets by optimizing the generative model accordingly. Our work represents a key step forward in applications of machine- learning methods to materials discovery. Look- ing forward, we plan to extend the flexibility of OMatG to additional interpolating functions, improve on the evalua- tion metrics and datasets, and investigate how different SIs influence the discovery of suitable materials.
Usiing Niggli reduction during learning to enforce unique choice of unit cell structures from our datasets is not sufficient for enforcing this invariance during generation structures.
9

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Open Materials Generation with Stochastic Interpolants
Acknowledgements
High Dimensions, January 2025. URL http:/t://aaxit org/ aaaabb//// /2501 1. 00988.
The authors thank Shenglong Wang at NYU IT High Per- formance Computing and Gregory Wolfe for their resource- fulness and valuable support. The authors acknowledge funding from NSF Grant OAC-2311632 P. H. and S. M. also acknowledge support from the Simons Center for Com- putational Physical Chemistry (Simons Foundation grant 839534, MT). The authors gratefully acknowledge use of the research computing resources of the Empire AI Con- sortium, Inc, with support from the State of New York, the Simons Foundation, and the Secunda Family Founda- tion. Moreover, the authors gratefully acknowledge the additional computational resources and consultation sup- port that have contributed to the research results reported in this publication, provided by: IT High Performance Computing at New York University; the Minnesota Su- percomputing Institute (http:: /www. . mssi. umn edu) the University of Minnesota; UFIT Research Computing (http:/ /www://wwww//o/////////////// rc. ufl. edu) and the NVIDIA Al Tech- nology Center at the University of Florida in part through the Al and Complex Computational Research Award; Drexel University through NSF Grant OAC-2320600.
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Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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2311. 17916.
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Open Materials Generation with Stochastic Interpolants
A. Acronyms
Table 4 provides a list of acronyms used throughout this paper for reference.
Table 4. Acronym definitions.
Acronym
Full Name
AIRSS
Ab initio Random Structure Searching
CDVAE
Crystal Diffusion Variational Autoencoder
CFP
Chemical Formula Prediction
CFM
Conditional Flow Matching
CTMC
Continuous- Time Markov Chain
CSP
Crystal Structure Prediction
DFT
Density Functional Theory
DFM
Discrete Flow Matching
DNG
De Novo Generation
EGNN
Equivariant Graph Neural Network
GNoME
Graph Networks for Materials Exploration
MLIP
Machine-Learned Interatomic Potential
NequlP
Neural Equivariant Interatomic Potentials
ODE
Ordinary Differential Equation
OMatG
Open Materials Generation Riemannian Flow Matching
RFM
SBD
Score-Based Diffusion
SBDM
Score-Based Diffusion Model
SDE
Stochastic Differential Equation
Stochastic Interpolant
SMACT
Semiconducting Materials from Analogy and Chemical Theory Stable, Unique, and Novel
SUN
B. Implementation Details of Stochastic Interpolants
B.1. OMatG Framework
Figures 4 and 5 summarize the training and the integration pipeline of the OMatG framework, respectively. Depending on the specific task, there several stochastic interpolants at once. For CSP, one stochastic interpolant considers lattice vectors L, and another one considers fractional coordinates X. The model output of CSPNet (see Appendix C.1) depends on the full structural representation (A, X, L} and time I, where A are the atomic species. For the DNG task, we additionally use discrete flow matching for the atomic species A (Campbell et al., 2024).
During the numerical integration in the CSP task, X and L are integrated jointly while A is fixed. For DNG, A is evolved according to discrete flow matching (Campbell al., 2024) (see Appendix B.6). For the SDE sampling scheme Fig. 5, one chooses a time-dependent noise =(t) that only appears during integration and not during training (see Appendix B.4). Also, (t) has to be unequal zero in order to prevent the divergence in 1/)(t). However, since ytt) necessarily vanishes times t 0 and 1 (see Appendix B.2), one should choose time-varying E(t) that vanishes near these endpoints (see Appendix B.4) (Albergo et al., 2023).
B.2. Interpolant Choice
In this work, are concerned with spatially linear interpolants of the form specified in Eq. (1). The following conditions must be met (Albergo et al., 2023):
(00) 3(1)= = a(1) 3(0) = (0) = 0, >()) > 0 At E (0, 1).
(5)
Under these constraints, the form of the SI is relatively flexible, and many different interpolants can be defined. Also, the base distribution can be arbitrary in CFM (Liu, 2022; Albergo & Vanden-Fijnden, 2023; Tong et al., 2024). In this work,
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Open Materials Generation with Stochastic Interpolants
(b),
Update batch loss C+ O₁
MSE Loss C2 belween (bes bf) and (=,
CSPNet:
(t,
(b,, =)
(t,
Catasea Ferove MP 20. MPTSO Alex MP)
Draw batch
For structure #1 nV Pr batch
Time (4 1) and initial structure zhy ~V Pu
Stochastic interpolants:
21 (t, zng 411 2), by 0aT
After computation batch loss C
step on C over CSPNet parameters
Figure 4. Training pipeline of the OMatG framework: batch of structures is drawn from dataset with target distribution P1- Every structure 21 is connected with a structure za from the base distribution with stochastic interpolants that yield the interpolated structure F, x=t, 20. 21, 2) and drift b. d.r, time 20 1). possibly using random variable (0, I). The model CSPNet predicts 6e = 6 (t, x2) and 2! = z(t, re) and its parameters are minimized based on the MSE losses in Eqs (2) and (3) [see also Eq. (32)).
Numerical integration
d X, Xt)
SDE: dX, Xi)dt
<0 (t, (e(W)aWW 70
CSPNet: (t, m))
For time 0,...,1
Initial structure
(t, 21)
Final structure #1 ~
Figure 5. Numerical integration pipeline of the OMalG framework: An initial structure æo from the base distribution is numerically integrated following either an ODE an SDE based model predictions by and 2! For an SDE, one can choose noise e(t) during integration.
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Open Materials Generation with Stochastic Interpolants
Table 5. SI parameters from Alhergo (2023).
Stochastic Interpolant
((t)
B(t)
7(1)
linear
-
v all -
Arbitrary po
trig
Cos (t
sin (らt)
v aat - t)
enc-dec
cos 2 ( (o.
cos (
sin sin?(Tt)
Gaussian Po VP SBD
v1 -
we mostly rely on interpolants originally defined in Albergo et al. (2023) and listed in Tab. 5. For the VP SBD interpolant, we allow for different widths on of the Gaussian base distribution
In Appendix B.9, we introduce additional VP and VE SBD interpolants beyond the one in Tab, 5 that are implemented in OMatG. The encoder-decodee (enc-dec) interpolant as defined in Tab. 5 evolves samples from the base distribution Po to follow an intermediate Gaussian distribution with variance the switch time Twitch 0.5, before mapping them to sample from P1- This can be generalized to arbitrary variances
and switch times Twicch (0, 1):
x(1 (s
ax(t) cos?
I (t),
(Tswitch Tswichh /PP + (1 (G-10 (w
x(t Tswicctt)
(tt) cos²
1 I(Tana.:
(6)
(Tswitch Tswicch)PP + (t TT-Tswwnee swincch) x11 - T Towichhl)P (Tswitch tssiicht)
(t) V sin?
Tswucht)P
where 2 1/2. We consider the cases E {1/2,1] 1) and note that the general interpolant in Eq. (6) reduces to the interpolant in Tab. 5 for = 1, = 1, and Twitch = 0.5.
B.3. Antithetic Sampling
As shown by Albergo et al. (2023), the loss function can become unstable around t and t = 1 for certain choices of (t)). To account for this, we implement antithetic sampling. This requires simultaneously computing the loss at both x+ and x: where
x: (t, 3 2) = at) + 3(t)21 + (t)2,
(7)
x (t, Fo. 3 2) att) -7))2
(8)
Both losses computed using the same value of and subsequently averaged.
B.4. Diffusion Coefficient
An important inference-time parameter for models integrated with an SDE is the choice of e which plays the role of diffusion coefficient. Albergo al. (2023) note that the presence of - (t) in the drift term seen in Fig. 5 can pose numerical instability at the endpoints integrating from some nonzero time t 2 0 to S 1 in order to avoid the singularity. Alternatively, one can design form for el) such that vanishes at these endpoints. In OMatG, we opt for the latter approach and consider a diffusion coefficient, Evanish(t) (t), which vanishes at the endpoints
0 and = during integration. For the choice cconst (t) c, they consider
Cvanish(t))
(1 + +ee (1+e-
Here, e dictates the magnitude of the diffusion, j sets the times at which the midpoints between c(t) = 0 and c(t) = c are reached, and a controls the rate this increase from e(t) = 0 to <(t) = c. The only constraints on these parameters are that c 2 0. j > O, and a > Importantly, these parameters should be chosen such that they are near zero at the endpoints.
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Open Materials Generation with Stochastic Interpolants
B.5. Interpolation with Periodic Boundary Conditions
We adopt a task-specific formulation for handling periodic boundary conditions with SIs tailored to flat tori, which are the relevant manifolds for fractional coordinates in erystal generation. We do not attempt generalize stochastic interpolants (SIs) to arbitrary manifolds in Riemannian flow matching (Chen & Lipman, 2024).
As in FlowMM (Miller et al., 2024), in order to uniquely define the interpolating paths, we rely on shortest geodesic interpolation paths between pairs of fractional coordinates from zn and 21, ensuring that interpolants are well-defined and differentiable. As noted in Section 3.2.1, this shortest geodesic path is computed by first unwrapping one of the coordinates (say 21) into its periodic image z', such that it is the closest image 2g- We then compute, for example, the linear 1) la as if in Euclidean space and finally wrap the interpolated path back onto the torus. (The geodesic is the same as the linear interpolant wrapped back into the box.)
interpolant 20, 21)
We perform this procedure for all choices of interpolants. The reason for unwrapping according to the closest image first- for all interpolants- is because there are multiple ways to connect two points on a torus (e.g., in periodic box one can connect two points with or without crossing the box boundaries). All periodic stochastic interpolants are then defined this way, by computing ætt,, z0, x11, 2) = a(t)xr + 8(1)z1 + > in the unwrapped (Euclidean) space and wrapping back onto the torus. We emphasize that this procedure is important not only for the choice of interpolant, but also for the addition of the latent variable (t)= which also moves the interpolation trajectory away from the geodesic. Our process yields exactly the same shortest- geodesic in FlowMM if using the linear interpolant, and thus recovers its corresponding conditional flow-matching loss. We depict our implementation of periodic stochastic interpolants in Fig. 6. We also demonstrate in Fig. 6c that averaging over the latent variable z recovers the deterministic base interpolant path, as required by the SI framework.
b.
a.
C.
Figure 6. Extending interpolants to incorporate periodic boundary conditions. (a-b) The path for based diffusion interpolant calculated by first computing the shortest-path geodesic (blue) between the initial (green dot) and final positions (red dot). Next, the path of the interpolant moving the final position outside the bounding box is computed (green). and finally the path is wrapped back into the hounding box to produce the interpolant trajectory (orange). (c) The effect of adding a latent variable to any interpolant must be handled similarly calculating the path of non-linear interpolant. For linear interpolant with nonzero y, we show samples of possible paths (blue) and their averaged path (orange) which collapses onto the path of the linear interpolant.
B.6. DFM Details
DFM allows for generative modeling of discrete sequences of tokens while respecting the discrete nature of the design space. As discussed, parameterized neural network pf (x1 (1ze) is learned, which attempts to predict the final sequence from the sequence at time f. Borrowing from Campbell et al. (2024), we choose a conditional rate matrix Rd (Ix, ((zi)-giving the rate 2, jumping to a different state given 2 which generates the conditional flow Pt (x, (x1) of the form:
(r i ReLU (Dwru
(Oaru)illa)) (ilm1) O.Prn (xpe(())(a1)) S P:p (2)(01)
(10)
where S is the number of possible tokens a sequence element can take on. This conditional rate matrix can be modified by including a term that introduces stochasticity the form of a detailed balance rate matrix RPB by writing Rÿ = R, RRRD
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Open Materials Generation with Stochastic Interpolants
Here, (Campbell et al., 2024):
ny + I- o6o
(i,j(,)) =
(1I)
where M is the masking token. The parameter 1) E TR+ represents the level of stochasticity that only appears during generation.
During generation, our objective is to compute (xx, based on the learned distribution P/ (21( Formally, we have
R2 (z2, i) Beg (e (m.)
[R] (1,, 1(2r)
(12)
In practice, Campbell et al. (2024) show that need not compute full expectation, but rather, simply draw xı pi (1) (azz), evaluate the conditional rate matrix (a,, ilz1), and perform an update of at to Fe ae with discrete time step At directly from this by sampling at according to
Pt = 8121, + (2
(z,, itt
(13)
B.7. Velocity Annealing
Velocity annealing- amnealing-rescalingg the learned velocity field during generation to increase velocity over time as 6 (t, x) + (1 + st) 69 x)) with as an hyperparameter during integration has been empirically shown to improve performance in number of studies that apply CFM to physical systems (Yim et al., 2023; Bose et al., 2024; Miller et al., 2024). For instance, Miller et al. (2024) demonstrated that applying velocity annealing significantly improves performance in CSP and DNG benchmarks for materials. Motivated by these findings, we include velocity annealing in OMatG as a tunable hyperparameter, while emphasizing that this technique lacks a formal theoretical justification within the mathematical frameworks underlying flow models and stochastic interpolants.
B.8. Data- Data-Depenneet Coupling
SIs have been used with data-dependent couplings (Albergo et al., 2024), where a coupling function v( 21) enables biasing of zy based on the sampled æ1- In OMatG, we incorporate an optional data-dependent coupling that enforces an ordering (i.e,, a permutation on the order of atomic elements within a structure) that produces the minimum fractional- coordinate distance between each particle pair (a) xi) from structures Zo E Pu and I1 P1- We find that the inclusion of this data-dependent coupling is optimal during hyperparameter tuning depending on the type of model: CSP models typically performed better without this coupling, but DNG models (see Tab. 11) can benefit in certain cases from minimizing traveled distance via permutation of elements.
Formally, our coupling is conditional on the sampled (mo, x1) and is defined as
arg min p
Σ d(p(z)).a1).
(14)
Here, :(:.)) a distance metric which we define on a periodic manifold in fractional-coordinate space (i.e., four- dimensional torus) and p some permutation function that permutes the discrete indices i. Under this coupling, we still sample (xo, independently but then bias the sampled zo travel the minimum permutational distance necessary to reach the target structure.
B.9. SI unifies CFM and SBDM
The SI framework implemented in OMatG unifies the frameworks of CFM, as implemented in FlowMM (Miller et al., 2024), and SBDM, as implemented in DiffCSP (Jiao et al., 2023) and MatterGen (Zeni et al., 2025). FlowMM is naturally subsumed by OMatG. For the choice of ODE-based sampling, the velocity term 1 (1,xx) is learned by minimizing the loss function in Eq. (2). By using (t) = 0 in the linear interpolant ætt, Zo. x1) ) = (1 - t)20 + tæ (see Appendix B.2), Eq. (2) becomes identical to the FlowMM loss (see Eq. (15) in Miller et al. (2024)). Furthermore, the treatment of periodic boundary conditions for the linear interpolant (see Appendix B.5) leads to the same geodesic paths as in FlowMM, and the center-of-mass motion of the ground-truti velocity is removed similarly in both frameworks.
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Open Materials Generation with Stochastic Interpolants
The connection between SIs and SBDM requires the discussion of both variance-preservingg (VP) and variance-explodingg (VE) cases. In the VP case (Sohl-Dickstein et al., 2015; Ho et al., 2020), sequence of N noise increments with variance B perturbs data yo as
yi V
Bi Vc-1 V 2-21, i= 1.,....,
(15)
where 2, ~ (OO,f). A + oo, this converges to the SDE
dy. -
2-0=--110 (s)y dt + ((s)
(16)
where w, is the standard Wiener process (Song et al., 2021). Under this stochastic process, the data distribution time is transformed into a Gaussian base distribution - 30. This differs from the time convention in SI where samples from the base distribution at time f are transported to samples from the data distribution f 1. With corresponding
change of variables st) log(t), Aranguri et al. (2025) show within the SI framework that y/ is equal in law to the
one-sided interpolant
z(t, Lt, x1) V1 - 7t)) To (())a1
(17)
where
1 log(t) B(u) du
rtt) = exp
2
(18)
In accordance with SBDM, the base distribution po is Gaussian, that is, zo (N Here, introduced the width dp of the base distribution as a tunable hyperparameter.
The SDE in Eq. (16) and the aariance-schedule B(s) are often considered on the time interval [o, 1], which implies that r()) from Eq. (18) is only used in / E [e,e, 1] (Aranguri et al., 2025). OMatG implements three such schedules. The linear schedule (Ho et al., 2020; Aranguri et al., 2025) is (sin = Smin + 5Bmmax Bminn) where 8min and 8max are chosen
empirically. This implies
8min log(!) 1 [Smmax Sminn) (1)
(1) exp
(19)
This function well-hehaved for the entire time range / E [0, 1] and implemented as such in OMatG.
For the cosine schedule, which is used DiffCSP for the lattice vectors, the noise variance in Eq. (15) is given by 3. ā with
f(i) f(0)'
π i/N + +d) 2 + d
f(i)
cos2
(20)
where d a small constant offset (Nichol & Dhariwal, 2021). As N + co, one gets for s E [0, 1]
cos2 2
std I+d
goos (s) = ds
log
π
tan
TT +
(21)
1+d
21+ d
cos
l+d
For t E [1/e, 1], this leads to
π + log(t) + 2d
008 (t)
csc
sin
(22)
2 2d
and for t E 00,1)e), we use (t) = 0.
The schedule const (t) = on E [0. 1] corresponds to a constant schedule gconst(s) = 2 and yields the SBD interpolant of Appendix B.2 derived in Albergo et al. (2023). This is the VP SBD interpolant considered throughout this paper.
In the VE case (Song& & Ermon, 2019), a sequence of N noise increments with variance a perturbs data yo
y/ 9 + oot 02_1
(23)
As N- + co, this converges to the SDE (Song et al., 2021)
(s))
dy
dw.-
(24)
ds
20

---

Open Materials Generation with Stochastic Interpolants
The corresponding one-sided interpolant in the SI framework is given by
Tt = vo2 - - (0)
(0) z0 + 21,
(25)
where again 20 ~V(0, o) (Aranguri et al., 2025). As in DiffCSP for the fractional coordinates, the schedule a(s) on 8 E [0, 1] is typically given by ((s) = min ( (m min) which we implement OMatG. The parameters min and max are optimizeable hyperparameters. The reported match rate and RMSE of the CSP task for the VE SBD positional interpolant for the MP-20 dataset in Tab. are close to the ones DifICSP. This highlights that OMatG is able to reproduce similar conditions to those in DiffCSP.
1010. Comparison OMatG-Linner to FlowMM
subset of OMatG models, specifically those which use linear interpolants for both the fractional coordinates and lattice vectors, map closely onto the conditional Now-matching model FlowMM (Miller al., 2024). The notable differences between OMatG-Linear models and FlowMM
follows: (1) Discrete flow matching on species for OMatG analog bits for FlowMM. (2) Lattice matrix representation for OMatG vs. lattice parameter representation (lengths and angles) for FlowMM. (3) Original CSPNet encoder for OMatG slightly modified CSPNet for FlowMM.
OMatG's CSP results improve upon FlowMM's, Since the CSP task does not utilize any species learning, the different handling of species not sufficient to fully explain the differences in model performance for CSP. For DNG models, the handling of species is also a relevant difference between OMatG-Linear and FlowMM.
C. Model Architecture
C.1. Graph Neural Network
We implement message-passing graph neural network (GNN) with CSPNet as introduced in Jiao et al. (2023):
h On (a')
(26)
(e) 4m (h 1 I, SimusoidalEEmbeddinng. xz))
(27)
N
m Σ
2j
(28)
==1
n()0 = hi(a-1) + spn (h mi
(29)
bx = Px (iimax))
(30)
by 10
( 8)
(31)
n
Here, node embeddings hi,) of node j at layer s are initialized as a function of atom types, a. Embeddings are then updated by message passing scheme through series of graph convolution layers. Messages are computed with parameterized neural network, Pm. from neighboring node embeddings well as information about the lattice, L, and distance between the fractional coordinates All necessary drift and denoiser terms are computed from single layer MLPs applied to the final node embeddings.
For the CFP model that should only predict compositions, we simply remove the input of the lattice I and the fractional coordinates x from the computation of the message in Eq. (27). This ensures that the output (arla) of CSPNet for the composition does not depend on lattice vectors or fractional coordinates, while preserving permutational equivariance.
21

---

Open Materials Generation with Stochastic Interpolants
C.2. Loss Function
With Eqs (2), (3), and (4), we can construct a loss function for the modeling of our joint distribution of interest for the DNG task,
((0) = -1,,2,00.11
20, 20. 21, 21,2) 2) bg xt,) + Az,= x,)) (t, -2) + λ1, [1] xx)]² -1, zo, 00, 2) (1, xx)] + d1 [1, - (1, (1)
An, [6 (t, (t,
(32)
+ Aa pog (espi((a1aaa) [2)1)))]
For the CSP task, the last line is left out. The À terms correspond to the relative weights of each term in the loss function. These weighting factors are hyperparameters that are included in our hyperparameter sweep. The respective terms for the fractional coordinates and lattice vectors corresponding to Eqs (2) and (3) are equivalent to mean-squared error (MSE) loss function as, for instance,
Cb
MSF (I) = Er. [bb (t, xz) ) 0,r(t, zo, 21 2)1²]
(33)
for the velocity term. They only differ by constant term that does not influence gradients. We do not include that constant term because the possible divergence of (t) near and can artificially inflate the absolute value of the loss, even when antithetic sampling upplied (see Section B.3).
C.3. Hyperparameter Optimization
For every choice of the positional interpolant, sampling scheme, and latent variable y, an independent hyperparameter optimization was performed using the Ray Tune package (Liaw et al., 2018) in conjunction with the HyperOpt Python library (Bergstra al., 2013) for Bayesian optimization. The tuned hyperparameters include both those relevant during training-the relative loss weights Å, the choice of stochastic interpolant for the lattice vectors, the parameters for chosen (t) (if necessary), the sampling scheme, the usage of data-dependent coupling, the batch size, and the learning rate- and during inference-the number of integration steps, the choice of the noises s(l) and v,, and the magnitude of the velocity annealing parameter& s for both lattice vectors and atomic coordinates. Hyperparameters are sampled according to the distributions below
Number of integration timesteps ~ Uniform (100,, 1000).
Batch size Choice(32, 64, 128, 256, 512, 1024).
Min. permutational distance data coupling ~ Choice False).
Relative weigths Nx,t, Ax,,, dn LogUniform(0.1, 2000.0).
Relative weight λτ_ LogUniform(O... 100.0)
Niggli reduction of cell during training Choice(True, False).
DFM Stochastictity ~ Uniform(O, 50.0).
Learning rate ~ LogUniform(10-1 5,10-2).
Weight decay LogUniformm11 10-
Velocity annealing coefficient (both for x and I) ~ Uniform(O.0, 15.0).
Diffusion coefficient parameter (both for x and Uniform (0.1, 10.0).
Diffusion coefficient parameter y (both for x and I) ~ Uniform(0.0s 0.3).
Diffusion coefficient parameter a (both for x and 1) Uniform(0.005, 0.05).
*Relative loss weights for are only swept over for DNG. Otherwise only weight parameters for and are optimized. Relative loss weights for the denoiser are only included when SDE integration is used.
22

---

Open Materials Generation with Stochastic Interpolants
Parameter switch of encoder-decoder interpolant Uniform(O01,
Parameter of encoder-decoder interpolant ~ Choice(O.5, 1.0).
Parameter of +(t) functions LogUnfform
Standard deviation og of Gaussian Py for SBD interpolants ~ LogUniform(0.01, 10.0).
Parameter min of the (s) schedule of the VE SBD interpolant Uniform(0.000) 0.01).
Parameter max of the schedule of the VE SBD interpolant Uniform 1.0).
The relative loss weight for the velocity of the lattice vectors is fixed: d1 = 1. By ensuring that the sum of all relevant loss weights is one,
transform the relative weights À to the weights in Eq. (32).
For the CSP models, the hyperparameter optimization attempts to maximize the match rate. For the DNG models, we combine the metrics in Tab. 2 to a single evaluation metric evalpNG that is supposed to be minimized by the hyperparameter optimization:
evalpNG avg combined validity,
avg wdist(p), (p), waiss(NNrr),, wdist (CN)))
(34)
avg coverage recall, coverage precision
Here, the function avg returns the average of its arguments.
We perform hyperparameter optimization for the DNG task only for the MP-20 dataset. For the CSP task, we optimize hyperparameters for the perov-5 and MP-20 datasets. For the CSP task on the MPTS-52 and Alex-MP-20 datasets, we simply transfer the hyperparameters of the optimized MP-20 models. We provide hyperparameter-tuner models with the relevant performance metrics and hyperparameters for perov-5 CSP in Tabs 6 and 7, MP-20 CSP Tabs 8 and 9, and MP-20 DNG Tabs 10 and 11.
Many models were partially trained and compared the process of hyperparameter tuning: on average models (perov-5) and 32 models (MP-20) for each choice of positional interpolant, sample scheme, and latent variable.
23

---

Open Materials Generation with Stochastic Interpolants
Table Study perov-S dataset comparing CSP performance metrics for choice of positional interpolant, sample scheme, and (1) in the latent variable (or width of the Gaussian base distribution po for the SBD interpolants, and parameters Omin and Omax of the ((s)
schedule the VE SBD interpolant).
Positional
Positional
Positional
Match rate
RMSE
interpolant
sampling scheme
7(1)
(%. Full Valid)
(Full Valid)
Linear
ODE
None: 7
51.86% 50.62%
0.0757 / 0.0066
Linear
ODE
LatentSqrt: 7 V0.034
72.21% 62.54%
0.3510/0.3444
Linear
SDE
LatentSqrt: 7 = v0.028t (1 -
74. 16% / 72.87%
0,3307 0.3315
Trigonometrie
ODE
None: y
81.51 / 52.36%
0.3674 0.3628
Trigonometric
ODE
LatentSqrt: 7 v /0.011
80.85% / 79.55%
0,3864 / 0,3873
Trigonometric
SDE
LatentSqrt: v0.063
73.37% 71.60 %
0.3610 0.3614
Encoder-Decone Decoder
ODE
Enc-Dec: 7 = V0.66 sin
=1t-0.800
68.08%
64.60%
0,4005 / 0.4003
(0.30 (0.80-0.800--11---01)
Encoder-Decoder
SDE
Enc- Dec: /8.45 sin
zic-0.611)
78.28%
76.80%
0.3616 0.3620
(0.61 -0.11))
0.616]
VP Scure-Based Diffusion
ODE
dg 0.28
83.06% / 81.27%
0,3753 / 0.3755
VP Score-Based Diffusion
SDE
(7g
0.13
76.54%
64.46%
0.3529 / 0.3402
VE Kcore-Based Diffusion
ODE
dg
8.96; miin
0.0078, man
0.5165
60.18% 52.97%
0,2510 0.2337
Table Study penov-5 dataset CSP comparing hyperparameters for each choice positional interpolant, sample scheme, and (t)
(as reported in Tab. 6).
Pos. interpolant,
Cell interpolant
Annealing param.
Integration
Min. dist.
Niggli
Sampling scheme. 7
Sampling scheme. ((t)
(Pos./ Cell)
steps
permutation
Linear, ODE, None
Linear, ODE, 7
14.11 2.90
820
False
False
0.9729/0.0271 /-/-
Linear, ODE, LatentSqrt
Linear, ODE,
0.008 12.19
820
Truc
True
0.9724 /.0276//
Linear, SDE, LatentSgrt
Linear, ODE,
8.20/ 1.46
910
True
True
0.0024/0.0051 0.9925 /-
/0.013 t)
Trig. ODE, None
Linear, ODE,
14.99 14.97
880
True
False
/..007//
0.021 f
Trig. ODE, LatentSqrt
Linear, ODE, 7
9.68 / 2.42
110
False
False
0.1130 0.1130/0.88700 0,8870 /-/-
Trig, SDE, LatentSqrt
Linear, ODF,
3.43 0.03
900
True
True
0.6868 0.0643 0.2489
0.051 (1
Ene-Dec, ODE, Ene-Dec
Linear, ODE, 7
0..94/0.318
460
True
True
0.8563 0..63/0014377 /-/-
Ene-Dee, SDE, Ene-Dec
Linear, ODE.
14.55/0.075
930
True
False
0.2828 0.0004 / 0.7168 /-
(0.154((1-0)
VP SBD, ODE
SBD. SDE, 0.61
12.79 2.69
130
True
True
0.0035 /.035/ / /
VP SBD, SDE
Trig, SDE,
11.54 11.53
350
True
False
0.2898 0.3259 / 0818
== 00.02 (1-t)
VE SBD, ODE
Trig, SDE,
0.003/ 14.93
14.93
380
False
False
0.9800 0..00//0.010///0000000
7 = 0.024r
24

---

Open Materials Generation with Stochastic Interpolants
Table 8. Study for the MP-20 dataset comparing CSP performance metrics for choice of positional interpolant, sample scheme, and 7(1) the latent variable (or width on of the Gaussian base distribution po for the SBD interpolants, and parameters miin and maax of the ((s)
schedule of the VE SBD interpolant).
Positional
Positional
Positional
Match rate
RMSE
interpolant
sampling scheme
(()
(9, Full Valid)
(Full / Valid)
Linear
ODE
None: 7
69.83 / 63.75%
0.0741 0.0720
Linear
ODE
LatentSqrt: 7 v0.258 (2588(() ()
55.60% / 50.04%
0.1531 0.1494
Linear
SDE
LatentSqrt: 7 < /0.063 (1
t)
68.20% 61.88%
0.1632 0.1611
Trigonometrie
ODE
None: y =
65.30% / 58.94%
0.1184 / 0.1149
Trigonometric
ODE
LatentSqrt: 7 /0.033/ ()
66.19% / 59.81%
0.1002 0.0968
Trigonometric
SDE
LatentSqrt: 7 / /0.049 t)
68.90%
62.65%
0.1249 0.1235
Encoder-Decoder
ODE
Enc-Dec: 7 1.99
-It- 654)
55.15% / 49.45%
0.1306 / 0.1260
[0:65-0.650)
0.855
Encoder-Decoder
SDE
Ene-Dec: 7 00.04 sin
m(t 020
57.69% / 52.44%
0.1 0.1160 / 0.1125
[0.42
4242
VP Score-Based Diffusson
ODE
0g 0.22
45.57% / 39.48%
0,1880 / 0.1775
VP Score-Based Diffusion
SDE
Øg 2.29
42.29% / 38.08%
0.2124 / 0.2088
VE Score-Based Diffusion
ODE
8n 9.77: Gmin 0.0047, dmax
0.9967
63.79% 57.82%
0.0809 0.0780
Table 9. Study for the MP-20 dataset comparing CSP hyperparameters for choice of positional interpolant, sample scheme, and (t) (as reported in Tab. 8).
interpolant,
Cell interpolant
Annealing param.
Integration
dist.
Niggli
Sampling scheme, 7
Sampling scheme, 7())
(Pos. / Cell)
steps
permutation
Linear, ODE, None
Linear, ODE, 7
10,18 1.82
210
False
False
0,9994 0.0006 /-/-
Linear, ODE, LatentSqrt
Trig. ODE,
7.76 4.12
690
Folse
True 0.9976 00.02///000//1---
/ (1
Linear, SDE, LatentSqrt
Linear, SDE,
11.58/ 5.08
310
False
False
0,0073 00073/0.0000 /.0064 0,000442/0.954/0.03.3
10.132 (1 -
Trig. ODE, None
Enc Dee, SDE,
12.34 3.66
170
False
False
0.9967/0.0023//
7= v5.27 sim?
0.1194
Trig. ODE, LatentSqrt
Linear, SDE,
13.54 /2338
780
False
True
0.9930//0.117///00.000
/0.017
Trig. SDE, LatemSyrt
Trig. ODE,
11.48 0.43
740
True
True
0.2468 0.0301 0.7231
Enc Dec, ODE, Enc Dec
Trig. SDE,
12.29 / 4,30
820
False
True
0.6892 /
0.2219 (1 - t)
Enc-Dec, SDE, Enc-Dec
Linear, ODE,
3.78 / 1.14
710
False
True
0.6143 0,4433//0003//377000
v 44961 [(
VP SBD, ODE
Linear, ODE, 7 =
6.61 2.45
890
True
True
0.9598 0.0440 /-/-
VP SBD, SDE
Linear, ODE.
6.46 / 0.67
600
True
True
0.6060 / 0.01 0,3828 /-
/ (1 f)
VE SBD, ODE
Linear. SDE,
8.28 / 0.43
660
False
False
0.9813 /00///00000///0000
7 v 00.017 -
25

---

Open Materials Generation with Stochastic Interpolants
Table 10. Study for the MP-20 dataset comparing DNG performance metrics for choice of positional interpolant, sample scheme, and (tt)
in the latent variable (or width du of the Gaussian base distribution pu for the SBD interpolants, and parameters mmin and mmx of the ((s)
schedule of the VE SBD interpolant). S.U.N. rates are computed according to the MatterSim potential.
Positional
Positional
Positional
S.U.N.
RMSD
interpolant
sampling scheme
7())
Rate
Linear
ODE
None: 7 0
18.59%
0.2939
Linear
ODE
LatentSqrt: ? V1.450t t)
9.95%
1,6660
Linear
SDE
LatentSqrt: 7 V -t)
22.07%
0.6148
Trigonometric
ODE
None: 7 0
19.63%
0.8289
Trigonometric
ODE
LatentSqrt: 7 /0.027 /
19. 19.96%
0.6570
Trigonometric
SDE
LatentSqrt: 7 /
17.60%
0.7763
Secoder-Decoder
ODE
Enc-Dec sin (Tt)
17.59%
0.3899
Encoder-Decoder
SDE
Enc-Dec: ? v0.10
TLt-0.736
16.27%
1.1795
0.733. (
VP Score-Based Diffusion
ODE
0.23
17.30%
1.1376
VP Score-Based Diffusion
SDE
7.14
22. 20.10%
0.7631
VE Score-Based Diffusion
ODE
0.45; amin = 0.0021, max 0.8319
20.38%
0.6644
Table 11. Study for the MP-20 dataset comparing DNG hyperparameters for choice of positional interpolant, sample scheme, and 7(t) (as reported in Tab. 10).
Pos. interpolant,
interpolant
Anncaling param.
Integration
Specics
Niggli
Sampling scheme,
Sumpling scheme. (())
(Pos. / Cell)
steps
permutation
moise
Lineur, ODE, None
Lineur. ODE,
13.62 1,07
150
True
False
7.08
0.9775 0.00060/////600 /00000000////0111
Linear. ODE, LatentSgt
Eno-Dec, SDE,
14.83 5.91
130
True
False
23.87
0.7683 000009900000000/002202000
145
10.1--1 (1
Lineur, SDE, LateniSqrt
Lineur. ODE,
6.33 1.07
710
True
False
0.19
0.1309 0.009/00.06///000 /-
Trig. ODE, None
Trig, ODE,
0.59/
860
True
False
32.69
0.3502 0.3302/0.002//- 0000000002/////06677
(1
Trig. ODE, LatentSqri
Linear, SDE,
7.79 0.30
680
True
True
27.25
0.2322 / 1..00/55..0.333//....
0.8818 []]
Trig. SDE, LatentSgit
Trig, ODE,
12,80 /4,3
760
True
False
13.15
0.6304 111.11822..073/-/..11
y 0.316
Enc-Do Dee, ODE, Ene-Dec
Lincar, ODE, =
10.27 0.00
840
False False
False
0.85
0.7268 0.0084/-/--02.0
Ene-Dee, SDE, Ene-Dee Lincan, ODE,
7,87 / 3.92
610
False
19.78
0.2143 /0,1960.188111 /.11611-/004341
7= y 1.651
VP SBD,
Trig, ODE,
2.30 /
False
20.27
0.005/00447/--/005500
7= v (1.79
VP SBD, SDR
Trig, SDF,
9.06 11.77
870
False
False
8.52
0.5184/0.0044 10.0000/0..111111
7 31.00
VE SBD, ODE
Linear. SDE,
/..92/0..6
330
False
Trac
5.87
0.2209 /.0000/ //0.00///00..7/ 0,0900
(1
26

---

Open Materials Generation with Stochastic Interpolants
D. Evaluation Metrics
In this section, we provide details and discussion of the various metrics we use to evaluate CSP and DNG models (see Section 5.1).
D.1. Match Rate and RMSE
The tradeoff between match rate and RMSE most strongly influences the perov-5 dataset. We show in Fig. how different positional interpolants for the atomic coordinates (trigonometric vs. linear with ODE sampling schemes) learn generate matched structures differently. For the linear case. the change in matching tolerance (via the Itol parameter of Pymatgen's Struct wetictureaatcio makes little difference. For the trigonometric interpolant, it makes far more significant difference and leads to much higher match rate, suggesting that the trigonometric interpolant learns structures more reliably but less accurately.
Itol=0.2
Itol=0.3
All
Valid only
Valid only
OMatG (Trig)
Density
Density
300
130
655
015
930
635
05 35
05
RMSE (dimensionless)
RMSE (dimensionless) 80.85 mr valid 79.55
mr all 20.73 mr valid 20.13
All
All Valid only
Valid only
OMatG (Linear)
Density
Density
RMSE (dimensionless)
RMSE (dimensionless) mr 51.86
all 46.58
valid 45.60
mr valid 50.75
Figure 7. We show here the effect of making matching difficult by decreasing the length tolerance used by Pymatgen's St reett We plot the density of the normalized RMSE distributions from CSP models trained on the perov-5 dataset (Linear and Trigonometric positional interpolants with ODE sampling schemes). We note that the curves for all generated structures and only valid generated structures overlap significantly.
D.2. Validity Metrics
The structural validity of generated structures is defined according the bond lengths present in the structure all lengths must be >0.5 Á to be considered valid. The compositional validity is defined according to the SMACT software package (Davies et al., 2019). We note that the default oxidation states have been updated with the release of SMACT version 3.0 which changed the DNG compositional validity rates by several percent. This also impacts the CSP match rate when filtered by valid structures. As such, all values for all models were recomputed with the most up-to-date version (3.0) of the SMACT software.
27

---

Open Materials Generation with Stochastic Interpolants
D.3. Coverage and Property Statisties
As in (Xie al., 2022), we evaluate coverage recall and precision (reported rates) by measuring the percentage of crystals in the test set and in the generated samples that match each other within defined fingerprint distance threshold. For structural matching, we use CrystalNN, and for compositional matching, we report Wasserstein distances between property distributions of the generated and reference datasets. The considered properties are the mass density (p), the number of unique elements (Nary), and the average coordination number for each element in the unit cell ((CN)).
Magpic fingerprints. Additionally,
D.4. Calculation of S.U.N. Rates
Evaluation of DNG structures was performed using scripts provided by the developers of MatterGen (MatterGen, 2025). A total of 10,000 structures were generated from each of OMatG, DiffCSP, FlowMM, and MatterGen-MP. These struc- tures were then filtered to remove any that contained elements not supported by the MatterSim potential (version MatterSim-vil O 0-1M) (Yang et al., 2024) or the reference convex hull. These included heavy elements with atomic numbers >89, radioactive elements, and the noble gases (specifically: 'Ac', 'U', 'Th', 'Ne', "Te', 'Kr', 'Pu', 'Np'. 'Xe', PPm', 'He', 'Pa')." " Stability and novelty were computed with respect to the default dataset provided by MatterGen which contains 845 997 structures from the MP-20 (Jain et al., 2013; Xie et al., 2022) and Alexandria (Schmidt et al., 2022b;a) datasets. This provides a more challenging reference for computing novelty as each model was trained only on the 27 000 structures from the MP-20 training set.
The MatterSim potential was utilized for the first structural relaxation, requiring far less compute resources compared to full DFT. Results derived from the MatterSim- Matleeei relaxed structures are shown in Tab. 12. Following the MatterSim relaxtions, 1000 structures were relaxed with DFT. All DFT relaxations utilized MPGGADoubl eR aeeelaattatit flows from the Atomate2 (Ganose et al., 2025) package to produce MP20-come cP20-compatible data.
Comparing the results from Tab. 12 to Tab. 3, we find that overall there is reasonable agreement between the metrics computed at the machine learning potential and DFT level. Relative performance ordering between methods remains fairly consistent, allowing for qualitative trends to be made at the much cheaper ML potential level. Nevertheless, for a full quantitative understanding, DFT is essential.
Table Stability (defined as - 0.1 eV/ttom above hull), uniqueness, and novelty results from de novo generation on MP-20 dataset computed for the same models in Tab. 2. All evaluations are performed with the MatterGen code base (MatterGen, 2025) with respect the included reference Alex-MP-20 dataset. The average RMSD between generated and the relaxed structures, and the average energy above hull is reported in units of eV/atom. All structures were relaxed with the MatterSim potential. Note that the results in Tab. 3 relied on a subsequent DFT relaxation.
(E)/N (L)
RMSD (A,
Novelty Rate (%, 1)
Stability Rate (%,
S.U.N. Rate (%, f)
Method
above hull
DiffCSP
0.1984
0.367
72.73
43.04
19.00
FlowMM
0.2509
0.651
72.76
37.47
13.86
MatterGen-MP
0.1724
0.142
72.17
47.07
22.66
OMaiG
Linear
0.1823
0.615
72.00
45,00
22.07
Trig
0.1857
0.657
65.35
51.40
19.96
Enc-Dec
0.1699
0.390
54.97
58.56
17.59
SBD
0.2189
0.763
75.80
42.60
22.10
CFP + CSP
0.2340
0.488
75.85
42.21
20.50
D.5. Stability and Structural Analysis of Generated Structures
In Fig. 8, we show the distribution of computed energies above the convex hull across various OMatG models, showing best stability of generated structures for linear, encoder-decoder, trigonometric, and VP SBD positional interpolants.
These elements were not removed from any datasets during training.
28

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Open Materials Generation with Stochastic Interpolants
Linear
10
Enc-Dec
Trig
SBD
Density
CFP CSP
2-
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Figure 8. Histogram of the computed energies above the convex hull for structures generated by the five OMarG DNG models highlighted in the main text (see Table 3). We show that all positional interpolants are effective at generating structures close to the convex hull, with the VP SBD interpolant and CFP CSP method performing slightly worse than other interpolants.
By evaluating distribution of Nary structures (Fig. 9), the distribution average coordination numbers (both by structure in Fig. 10 and by species in Fig. 11), well as distribution of crystal systems (Fig. 12) which are related a structure's 's Bra- vais lattice, provide qualitative analysis for model performance. Space groups (and thus crystal systems) were determined using the spgl lib software (Togo et al., 2024) and choosing the most common space group identification with exponentially (geometrically) decreasing tolerance. We find that OMatG models have superior performance on the Nary metric across the board, with DiffCSP performing the most poorly. Specific OMatG models (with positional Encoder-Decoder interpolant and CFP+CSP with Linear interpolant) and DiffCSP showed the best performance in matching the distribution of average coordi- nation number for each structure, particularly for high-coordinated structures. The average coordination number for species were best-matched by OMatG models across the board (with the exception of the OMaiG-VPSBD model which tended overpredict the coordination environments), and broadly underpredicted by DiffCSP, FlowMM, and MatterGen-Mi Finally, the best matching results for distribution across crystal systems for the OMatG CFP+CSP model, with all other inter- polants (and DifCCSP and FlowMM showing propensity for generating low-symmetry (triclinic and monoclinic) crystal structures. Overall, we find that OMatG models closely reproduce the elemental and structural diversity present in the data.
29

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Open Materials Generation with Stochastic Interpolants
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Figure 9. Qualitative performance of the distribution of Nary crystals for (a) Non- OMa- models and (b) OMatG models across structural benchmarks computed on generated structures and test set structures from the MP-20 datasct. Atomic elements are listed increasing atomic number from left right.
30

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Open Materials Generation with Stochastic Interpolants
(a)
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Figure 10. Qualitative performance of the distribution of average coordination number by structure for (a) Non Oor-O models and (b) OMatG models across structural benchmarks computed on generated structures and test set structures from the MP-20 dataset.
31

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Open Materials Generation with Stochastic Interpolants
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Figure 11. Qualitative performance of the distribution of average coordination number by species (listed left to right order of atomic number) for (a) Non-OMarG models and (b) OMatG models across structural benchmarks computed generated structures and test set structures from the MP-20 dataset.
32

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Open Materials Generation with Stochastic Interpolants
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Figure 12. Qualitative performance of the distribution of crystal system by structure for Non-OMatG models and (b) OMatG models across structural benchmarks computed on generated structures and test set structures from the MP-20 dataset.
33

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Open Materials Generation with Stochastic Interpolants
E. Large Language Models as Base Distributions
FlowLLM (Sriram et al., 2024) combines large language models (LLMs) with the conditional flo-mmacching framework FlowMM (Miller et al., 2024) to design novel crystalline materials in the DNG task. A fine-tuned LLM serves as the base distribution and samples initial structures; FlowMM then refines the fractional coordinates and lattice parameters as in the CSP task. This idea can be similarly applied to OMatG, which allows the use of LLMs within the general SI framework for materials generation.
We extend OMalG to OMaiG-LLM by allowing for -LM-generated structures as the initial structures. We evaluate both FlowLLM and OMatG-LLM, on the LLM dataset released by FlowLLM. Specifically, we use the training (containing 40 000 structures) and validation sets (6000 structures) from https: /tttttbbbbbbbgggg com/ /aw./eoooooorerrrcccc//aaa fl owwm and the LLM-ggenrated initial structures (10000 structures) from https: : / / /ggtttb....///g// com/ htt////arbboooeerrre..aaaaaa//ee tesstall-eettt 11m as the test set. These initial structures are generated by fine-tuned Llama-70B model (Gruver et al., 2024). As shown in Tab. 13, DMatG-LLM' linear and trigonometric interpolants outperform FlowLLM in almost all DNG metrics. Since the Wasserstein distance with respect to the Narity distributions and the compositional validity only depend on the atomic species generated by the LLM, these two metrics are necessarily equal for FlowLLM and OMatG-LLM. Note that the original FlowLLM (Sriram et al., 2024) is trained on 3M LLM-generated structures while our experiments are conducted on the 40K structures open- spen-sourced by the authors. The performance of FlowLLM in our experiments thus differs from the scores reported in (Sriram al., 2024).
Table 13. LlowLLM''s and OMaG-LLM''s performance (with linear and the trigonometric interpolants) when using the same fine-tuned LLM (Gruver et al., 2024) as the base distribution. The best performance for each metric is in bold.
Validity (% t)
Coverage (% t)
Property (L)
S.U.N. Rate (% f)
Method
Structural
Composition
Combined
Recall Precision wdist (p)
wdist (Wary)
wdist ((CN)
FlowLLM
96.27
86.40
83,55
97.98
96.55
0.9922
0.5427
0.5936
10.28
OMaIG-LLM
Linear
97.86
86.40
84.85
99.16
98.40
0,9100
0.5427
0.8600
12.61
Trigonometric (ODE)
95.70
86.40
83.25
98.57
98.24
0.7410
0.5427
0.6165
11.14
Trigonometric (SDE)
97.78
86.40
84.72
97.41
99.12
3.6214
0.5427
0.4448
11.86
E. Computational Costs
In Tabs 14 and 15, we present the computational costs for both the CSP and DNG tasks. We compare the cost of training and integrating OMatG on the MP-20 dataset and show low computational costs for OMatG'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.
Table 14, Computational costs for DiffCSP, FlowMM, OMarG (ODE) and OMaGG (SDE) models trained on the CSP task.
Task
OMatG (ODB)
FlowMM
OMatG (SDE)
DPfCCSP
Training (s/ epoch)
56.8 + 0.75
70.35 + 1.38
89.0 + 1.41
21.89 + 0.31
Sampling (s / batch)
313.67 + 9.29
424.125 + 11.78
479.5 + 13.5
338.11 11.93
Table 15, Computational costs for DiffCSP, FlowMM, OMatG (ODE) and OMatG (SDE) models trained on the DNG task.
Task
OMaiG (ODE)
FlowMM
OMaiG (SDE)
DifCCSP
Training (s epoch)
75.26 + 2.08
73.32 + 0.47
102.65 1.87
21.85 + 0.36
Sampling (s/ / batch)
473. + 13.20
469.93 + 6.12
617.2 = 18.2
322.63 + 10.28
34


Reviews:
[{"review_id": "vdIeZ2QgpX", "reviewer": "Reviewer_gm5E", "summary": "This paper extends flow-based inorganic crystalline structure prediction (CSP) to the stochastic interpolants (SI) framework. The authors use an equivariant graph representation (CSPNet) and wrapped interpolants to account for periodic boundary conditions of atomic coordinates and discrete flow-matching (DFM) to generate atomic species for De Novo Generation (DNG). By placing CSP into the SI framework, they are are able to show empirical performance gains over prior methods (DiffCSP, FlowMM, MatterGen) by ablating the over the additional tuning knobs offered by SI.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thoughtful and constructive comments. Below we address the main concerns and the three questions raised.\n## Novelty and contribution\nWe acknowledge that the novelty of our work could have been more clearly emphasized. To clarify:\n- 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.\n- 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.\n- 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.\n- 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.\n- 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).\n## Claim of unification\nWe 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:\n- **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].\n- **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.\n## PBCs\nWe agree that the discussion of periodic boundary conditions (PBCs) should be extended:\n- 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.\n- 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.\n- 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.\n\n**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. \n## Computational overheads\nWe agree this is an important point. **We refer to our response to Reviewer w22y.**", "claims_and_evidence": "This paper claims that\n-The SI framework is a unifying formulation that generalizes both flow matching and diffusion-based generative models.\n-The method is flexible and tunable through the choice of interpolants and noise scheduling, contributing to better empirical results. Notably, they achieve state-of-the-art performance on both CSP and DNG tasks\n\nEvidence:\nThe paper definitely supports its performance claims with comprehensive experimental comparisons across several datasets. Although the SI framework generally comes with more tuning knobs “out of the box”, conceptually it is mostly using the formulation of the previous flow-matching and diffusion works for material generation. With the right reparameterizations, the SI paths can be realized by flow matching probability paths.  Additionally, I think it's hard to argue that SI unifies FlowMM since it is based on Riemannian flow matching and intrinsically handles the PBC conditions (although I think in the case of flat tori, the geodesics proposed here for SI seem consistent).", "methods_and_evaluation_criteria": "The SI framework are formulated pragmatically and well-motivated by CSP and DNG tasks. The evaluation criteria such as matching rate, RMSD and coverage are all reasonable metrics.", "theoretical_claims": "It is suggested that SI generalizes both diffusion and flow-matching frameworks, but the equivalence to previous frameworks isnt explicitly noted anywhere. The authors mention the use of periodic interpolants to account for PCBs which seems to be compatible with the original SI framework, but it seems the paper mostly gives intuitive arguments for this, citing Albergo 2023 and Jiao 2023.", "experimental_designs_or_analyses": "The experimental design and analysis appears thorough. The authors compare against DiffCSP and FlowMM, reproducing their results accurately for several benchmarking. They also provide an informative ablations regarding hyperparameter tuning, interpolant choice.\nIt could be informative to also display computational cost associated with training and inference for OMG compared to existing methods.", "relation_to_broader_scientific_literature": "I find it pretty clear to follow OMG’s relationship to previous works DiffCSP and FlowMM in the setting of CSP and DNG tasks.", "essential_references_not_discussed": "I’m not aware of additional material generation references.", "other_strengths_and_weaknesses": "Strengths:\n-OMG delineates how to perform CSP and DNG with the stochastic interpolants framework and provides informative ablations with SOTA results.\n\n-The use of DFM for atomic species generation and DNG is novel to my knowledge.\n\n-The open-source implementation contributes to reproducibility and benchmarking for future works.\n\nWeaknesses:\n\n-The claims of unification of other material generation approaches under SI is not completely supported, although SI does offer some additional tuning knobs over basic CFM probability paths. \n\n-The ideas here are mostly extensions of previous formulations and not incredibly novel.", "other_comments_or_suggestions": "None", "questions_for_authors": "1. To support this claim of “unification” can we show explicitly how each previous framework is realized by SI? It could be conceptually and practically interesting to catalog the previous approaches. This would make it clear how one would propose future extensions of those methods under the SI framework. \n\n2. Could the periodic interpolants and how they are supported by the SI framework be formulated in more detail? I'm not very convinced by the section 4.1 and appendix regarding the well-definedness of the paths and their geodesics on flat tori. This would help us better contextualize this framework with what FlowMM is doing with Riemannian flow-matching and propose extensions.\n\n3. Would it be possible to compare any computational overheads for training and sampling across SI and the related methods? This would further help us understand tradeoffs and address any issues regarding scalability.\n\nThese points would make me feel more confident about the contributions of this work.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hQcXWY308S", "reviewer": "Reviewer_ENYj", "summary": "This paper introduces a framework called OMG that applies stochastic interpolants to generate inorganic crystalline materials. The authors adapt the stochastic interpolants framework to handle periodic boundary conditions for crystal structures and integrate discrete flow matching for atomic species. Their approach provides flexibility in choosing interpolation schemes and sampling methods, outperforming existing methods on CSP and de novo generation tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback and address their concerns below.\n\n## DFT relaxation\nWe 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:\n| MLIP M.S.U.N. | DFT M.S.U.N. |\n|-------------|-----------|\n| 0.16        | 0.09      |\n| 0.13        | 0.12      |\n| 0.06        | 0.06      |\n| 0.12        | 0.10      |\n| 0.14        | 0.12      |\n| 0.13        | 0.12      |\n| 0.16        | 0.15      |\n| 0.10        | 0.09      |\n| 0.13        | 0.13      |\n| 0.11        | 0.08      |\n\nAs 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.\n\nTo 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.\n\n**The complete results will be included in the revised version of the paper.**\n\n## Interpolant choice and structural diversity\n\nWe 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**). \n\nFor 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**. \n## Evaluation before and after relaxation\nAs 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:\n- 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.\n- 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:\n  - DiffCSP: 1.295\n  - FlowMM: 0.651\n  - OMG-Linear: **0.294**\n  - OMG-Trig: 0.763\n  - OMG-EncDec: **0.390**\n  - OMG-SBD: 0.759\n  - OMG-CFP+CSP: **0.488**\n\nThese 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.", "claims_and_evidence": "The main claim that OMG achieves state-of-the-art performance on CSP and DNG tasks is supported by comprehensive benchmarking across multiple datasets. The authors demonstrate performance improvements over DiffCSP and FlowMM, and show comparable results with MatterGen. The authors also claims for the flexibility of their approach. I believe the claims can be supported by the ablation studies on how different interpolant choices optimize performance for different datasets and tasks.", "methods_and_evaluation_criteria": "The stability evaluation using MatterSim provides a computationally efficient alternative to DFT relaxations. However, I would still suggest DFT calculation for accurate and fair comparison on crystal structure stability evaluation.", "theoretical_claims": "The integration with discrete flow matching for atomic species are novel.", "experimental_designs_or_analyses": "Experiments are comprehensive.", "relation_to_broader_scientific_literature": "The authors place their work appropriately within the context of both materials generation and generative modeling literature. They acknowledge the state-of-the-art in both fields and clearly articulate how their approach bridges these domains.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Weaknesses:\n1. I suggest some more theoretical analysis explaining why specific interpolant choices work preferentially on each dataset.\n2. While stability rate is reported, the approach is not validated by experimental synthesis of novel materials. DFT calculation or CHGNet should provide a more comprehensive comparison to other methods on quality and stability of the generated structures.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. Have you explored how the choice of interpolants affects the diversity of generated structures beyond the standard property distribution metrics, for example elemental distribution? \n2. Are you evaluating structures without relaxation? Have you analyzed the discrepancy of your generated structures before and after relaxation?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Gnm3wVkzzA", "reviewer": "Reviewer_w22y", "summary": "The paper introduced Open Materials Generation (OMG), a framework that leverages stochastic interpolants in generative models for inorganic crystalline materials. The method is built on existing architecture in the literature (CSPNet), which is based on an equivariant graph neural network (EGNN). The authors addressed two materials tasks: Crystal structure prediction for fixed compositions and de novo generation. The method has been evaluated on two materials datasets: perov-5 and MP-20.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for raising questions about our paper, and address them topically below.\n\n## Comparison with FlowLLM\nWe 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. \n\n|| LLM model size | Cov. precision | Cov. recall | wdist $\\rho$ | wdist $\\langle CN \\rangle$ | Structural validity |\n| --- | --- | --- | --- | --- | --- | --- | \nFlowLLM | 70B | 96.55 | 97.98 | 0.9922 | 0.5936 | 96.27 |\nOMG-LLM | 70B | 98.40 | 99.16 | 0.9100 | 0.8600 | 97.86 |\n\nRefining 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.  \n\n## Computational cost\nWe 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.**\n### CSP\n|   Task   | OMG (ODE) | FlowMM | OMG (SDE) | DiffCSP |\n|-----------|-------------|--------|-----------|---------|\n| Training (s / epoch) |     $56.8 \\pm 0.75$     |    $70.35 \\pm 1.38$   |     $89.0 \\pm 1.41$     |    $21.89 \\pm 0.31$    |\n| Sampling (s / batch) |     $313.67 \\pm 9.29 $     |    $424.125 \\pm 11.78$   |     $479.5 \\pm 13.5$     |    $338.11 \\pm 11.93$    |\n\n### DNG\n|   Task   | OMG (ODE) | FlowMM | OMG (SDE) | DiffCSP |\n|-----------|-------------|--------|-----------|---------|\n| Training (s / epoch) |     $75.26 \\pm 2.08$     |    $73.32 \\pm 0.47$   |     $102.65 \\pm 1.87$     |    $21.85 \\pm 0.36$    |\n| Sampling (s / batch) |     $473.14 \\pm 13.20$     |    $469.93 \\pm 6.12$   |     $617.2 \\pm 18.2$     |    $322.63 \\pm 10.28$    |\n\n## Ablation studies\nRegarding 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”.\n\n**Perov-5 CSP**\n| Pos. Interpolant | Pos. sampling | Pos. gamma | Match rate (%, Valid only) | RMSE (Valid only) | \n| --- | --- | --- | --- | --- | \n| Linear | ODE | $\\gamma(t)=0$ | 50.62 | **0.0760** |\n| Linear | ODE | $\\gamma(t)=\\sqrt{0.034t(1-t)}$ | **62.54** | 0.3444 |\n| Linear | SDE | $\\gamma(t)=\\sqrt{0.028t(1-t)}$ | **72.87** | 0.3315 |\n| Trig | ODE | $\\gamma(t)=0$ | **52.36** | 0.3628 |\n| Trig | ODE | $\\gamma(t)=\\sqrt{0.011t(1-t)}$ | **79.55** | 0.3873 |\n| Trig | SDE | $\\gamma(t)=\\sqrt{0.063t(1-t)}$ | **71.60** | 0.3614 | \n| Enc-Dec | ODE | $\\gamma(t)=\\sqrt{0.66} \\sin^2(\\pi(t-0.80t) / ((0.80-0.80t) + (t - 0.80t)))$ | **64.60** | 0.4003 |\n| Enc-Dec | SDE | $\\gamma(t)=\\sqrt{8.45} \\sin^2(\\pi(t-0.61t) / ((0.61-0.61t) + (t - 0.61t)))$ | **76.80** |0.3620 | \n| SBD | ODE | $\\sigma = 0.28$ | **81.27** | 0.3755 |\n| SBD | SDE | $\\sigma = 0.13$ | **64.46** | 0.3402 | \n\n**MP-20 CSP**\n| Pos. Interpolant | Pos. sampling | Pos. gamma | Match rate (%, Valid only) | RMSE (Valid only) | \n| --- | --- | --- | --- | --- | \n| Linear | ODE | $\\gamma(t)=0$ | **63.75** | 0.0720 |\n| Linear | ODE | $\\gamma(t)=\\sqrt{0.257t(1-t)}$ | 50.04 | 0.1494 |\n| Linear | SDE | $\\gamma(t)=\\sqrt{0.063t(1-t)}$ | **61.88** | 0.1611 |\n| Trig | ODE | $\\gamma(t)=0$ | 58.94 | 0.1149 |\n| Trig | ODE | $\\gamma(t)=\\sqrt{0.033t(1-t)}$ | 59.15 | 0.0998 |\n| Trig | SDE | $\\gamma(t)=\\sqrt{0.049t(1-t)}$ | **61.39** | 0.1321 | \n| Enc-Dec | ODE | $\\gamma(t) = \\sqrt{1.99} * \\sin^2(\\pi(t - 0.65t) / ((0.65 - 0.65t) + (t - 0.65t)))$ | 49.45 | 0.1260 |\n| Enc-Dec | SDE | $\\gamma(t) = \\sqrt{0.04} * \\sin^2(\\pi(t - 0.42t)^{0.5} / ((0.42 - 0.42t)^{0.5} + (t - 0.42t)^{0.5}))$ | 52.44 | 0.1125 | \n| SBD | ODE | $\\sigma=0.22$ | 37.39 | 0.1890 |\n| SBD | SDE | $\\sigma=2.29$ | 38.08 | 0.2088 | \n\n## Limited baselines\nWe 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.", "claims_and_evidence": "The authors claimed to achieve state-of-the-art but I think there is a reference missing (see the Essential References section).", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "No.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "The method is built on the existing approaches in the literature like CSPNet architecture and stochastic interpolants.", "essential_references_not_discussed": "The following reference is missing in the comparison and the method used the same benchmark:\n\n- FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions. Sriram et al., 2024.", "other_strengths_and_weaknesses": "**Strengths**:\n\n- Integration of stochastic interpolants: I think the idea of extending stochastic interpolants to generate materials is novel.\n- Experimental results: The paper showed strong empirical results of the proposed method on materials datasets. \n\n**Weaknesses**: \n\n- Computational requirements: The paper did not mention the training or inference costs of the proposed method, or how it compares to previous approaches.  \n- Limited baselines: As mentioned by the authors, the paper did not compare with symmetry-aware models (like Crystal-GFN or  WyCryst) which narrows the scope of the proposed method.", "other_comments_or_suggestions": "No.", "questions_for_authors": "Can you explain more about the choice and limitations of each interpolant, and how this affects the performance across different datasets?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aJYUOu8R7s", "reviewer": "Reviewer_K1be", "summary": "The paper presents an extension of stochastic interpolants for the modelling of crystalline materials. Stochastic interpolants are a general framework that encompasses diffusion models and flow matching as specific instances. As the fractional coordinates live on a torus, they adapt the interpolants to respect the circular nature of the space. They use stochastic interpolants also to model the unit cell parameters (lengths and angles), while they rely on discrete flow matching in the case of atom types. They present results for two different tasks: crystal structure prediction (CSP) and de novo generation (DNG).\n\n\n### After rebuttal period\nI think that the author's rebuttal clarified the minor concerns I had and the additional results make the paper even stronger.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer’s thorough engagement with our manuscript and thank them for their insightful feedback. \n\n## Ablation studies, hyperparameters, and model performance\n\nRegarding 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.\n\nFor 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.**\n\nWe also show that for MP-20, the trigonometric interpolant with an SDE can also outperform on match rate compared to previously published models. \n\nComparing 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.\n\nConcerning 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.**\n\n## Comparison to FlowMM\n\nThe 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.\n\n## Clarifications\n\n### SBD base distribution \n\nWe 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.**\n\n### Geodesic and periodic interpolants \n\nThe 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.”\n\n### Subtraction of COM motion \n\nThe 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.\n\n### Species filtering\n\nOur 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.**", "claims_and_evidence": "The claims that the paper presents are that stochastic interpolants can be used for crystalline material generation and that they outperform all the other deep generative model approaches in the literature. While the first claim is well supported, the second one is a bit tricky as although they are better from the experiments table than the other methods, this is not an apple-to-apple comparison. Therefore, I think that more analysis is required to support this claim (see weaknesses)", "methods_and_evaluation_criteria": "The benchmark and evaluation metrics are the ones that are usually considered in this context of generative models for materials. They also evaluate DNG samples using a foundation model, i.e. Mattergen (FlowMM used a different one but they recomputed results for that baseline using Mattergen). Although they are not evaluating with DFT as done in FlowMM, I think that this evaluation is enough to compare the different methods.", "theoretical_claims": "-", "experimental_designs_or_analyses": "The experimental setup follows closely the ones of CDVAE and DiffCSP, therefore these are the classic experiments considered in papers for material generation.", "relation_to_broader_scientific_literature": "The paper extends Stochastic interpolants in the context of crystalline material generation. It presents a discussion of the main deep generative approaches used in the context of crystalline material generation.", "essential_references_not_discussed": "/", "other_strengths_and_weaknesses": "The paper is well-written and easy to follow. Also, the evaluation of the unrelaxed DNG samples in terms of average energy above the hull, stability, uniqueness, and novelty using MatterGen as a foundation model strengthens the results. The main weakness is that comparison with previous models is not exactly apple-to-apple and therefore it is difficult to understand which ingredients are making the approach better. I think the paper will benefit from more ablation studies. For example, in the DNG task, do the improvements come from the use of stochastic interpolants for fractional coordinates and lattice parameters or mostly from the discrete flow matching approach for the atom types? Indeed, FLowMM uses analog bits and continuous flow matching. In addition to that, they propose different interpolants (both stochastic and deterministic), but present results only for the best approach, making it difficult to get the full picture of the design space.", "other_comments_or_suggestions": "/", "questions_for_authors": "I have a few questions regarding some parts of the paper that I personally think would be nice to be discussed in the paper:\n\n- The results on perov-5 make me a bit confused. I appreciate the discussion in the appendix on the sensitivity of the trigonometric interpolants, but at the same time, it is impressive how the tolerance affects the match rate for that specific interpolant in contrast to the linear one. Also by looking at the distribution over the RMSD it seems that the error is x3 bigger than the linear interpolant. It would also be interesting to get the full picture of how the trigonometric interpolant is performing in the other dataset. Is there something that can be learned from the experiment on perov-5 to use on the other dataset to get approximately the same improvement? Also, it seems that the trigonometric interpolant is the only one (from our experiments) that works better with $\\gamma(t)$, is there a reason why in this case the model benefits more from this stochasticity at training time? It would be helpful to see the results of all the interpolants on perov-5 and MP-20 to see how also the increase in the number of atoms affects the performance.\n\n- I am a bit confused when on line 181, you say that the base distribution for the score-based diffusion interpolants needs to be a wrapped normal distribution as in DiffCSP. In DiffCSP the wrapped normal is used to define the transition kernel $p_{t|0}(x_t|x_0)$, but the diffusion converges to a uniform distribution also in that case. I would be happy if you could comment on that sentence a bit more.\n\n- It seems that there are plenty of hyperparameters to tune to get the final model. How many models have you trained for each dataset? I think it would be helpful to mention in the appendix all the ranges considered for every hyperparameter. \n\n- In Figure 5 in the appendix, I am a bit confused as to why you need to first compute a geodesic before computing the interpolant. Can you elaborate more on this? Also, a geodesic on a torus is also an interpolant, does that correspond to the wrapped linear interpolant (with wrapped I mean the way you are computing interpolations as described in section 3.2.1) in your case?\n\n- I know that removing the Euclidean mean from the target is done in FlowMM too, but is it enough to get a consistent training target for the network?\n\n- Just to clarify: are you training your model on the usual MP20 dataset used also in DiffCSP and just doing the filtering you mention in the appendix (page 17) after sampling?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents the method referred to as _Open Materials Generation (OMG) with Stochastic Interpolants (SI)_. The method is designed to tackle the problem of generating crystal structures and the paper offers an evaluation on _de novo_ crystal structure generation and crystal structure prediction from compositions. 

The reviews are overall positive, with a consensus in the recommendation scores of _Weak accept_. The reviewers seem enthusiastic about the positive results and do not critique the claims of achieving state of the art performance. My own assessment and critique is that the metrics used in the evaluation are a poor indication of the usefulness in practical downstream applications in materials discovery. Nonetheless, this is an unfortunate trend in this area rather than a unique flaw of this paper.

The main criticisms are 1) that the comparisons with other methods are not apples-to-apples comparisons and 2) that the claims of unifying FlowMM and diffusion models are not fully supported. In my view, the rebuttal responses are not solid in addressing these comments and I would strongly encourage the authors to improve these aspects in a subsequent update of the paper.

Despite these weaknesses and the lack of more enthusiastic comments by the reviewers, my position is to not recommend the rejection of papers that have sufficient scientific quality, are sufficiently well-written and that could be interesting to part of the ICML audience. This paper should pass that bar.
Decision: Accept (poster)