Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling

Kiyoung Seong; Sungsoo Ahn; Sehui Han; Changyoung Park

cs.LGFeb 23, 2026

Multimodal Crystal Flow: Any-to-Any Modality Generation for Unified Crystal Modeling

Authors: Kiyoung Seong, Sungsoo Ahn, Sehui Han, Changyoung Park

Organizations: Graduate School of AI, KAIST, Seoul, South Korea · Materials Intelligence Lab, LG AI Research, Seoul, South Korea

Abstract

Crystal modeling spans a family of conditional and unconditional generation tasks, including crystal structure prediction (CSP) and de novo generation (DNG). While recent deep generative models have shown promising performance, they remain largely task-specific, lacking a unified framework that shares crystal representations across tasks. To address this limitation, we propose Multimodal Crystal Flow (MCFlow), a unified multimodal flow model that realizes multiple crystal generation tasks as distinct inference trajectories via independent time variables for atom types and crystal structures. To enable multimodal flow in a standard transformer model, we introduce a composition- and symmetry-aware atom ordering with hierarchical permutation augmentation, injecting compositional and crystallographic priors without explicit structural templates. Experiments on the MP-20 and MPTS-52 benchmarks show that a single MCFlow model is competitive with task-specific baselines across CSP, DNG, and structure-conditioned atom type generation.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of multimodal crystal flow with any-to-any modality generation. MCFlow trains a flow model with two independent time variables corresponding to atom types ( $t$ ) and structures ( $s$ ). By selecting task-specific inference trajectories in the $(t,s)$ space, a single model performs crystal structure prediction, atom type generation, and de novo generation.

Figure 2: Composition- and symmetry-aware atom ordering with hierarchical permutation augmentation. Atoms in the primitive unit cell are lexicographically ordered by Pauling electronegativity and Wyckoff position (denoted by letter

a,b,c,\ldots

) to expose compositional and crystallographic structure. The ordering and augmentation are illustrated on a

\mathrm{Th}_{5}\mathrm{C}

crystal in the

\mathrm{R}\bar{3}\mathrm{m}

space group. Inter-orbit permutation reorders orbits sharing the same elements and Wyckoff position, while intra-orbit permutation permutes atoms within each orbit. This ordering and hierarchical permutation augmentation provides compositional and crystallographic symmetry information to sequence-based Transformers without explicitly conditioning space group or Wyckoff positions.

Model	MP-20		MPTS-52
Model	MR (%) $\uparrow$	RMSE $\downarrow$	MR (%) $\uparrow$	RMSE $\downarrow$
CDVAE (1)	33.90	0.1045	5.34	0.2106
DiffCSP (1)	51.49	0.0631	12.19	0.1786
FlowMM (1)	61.39	0.0566	17.54	0.1726
CrystalFlow (1)	62.02	0.0710	21.00	0.1613
OMatG (1)	63.75	0.0720	25.15	0.1931

Table 1: Crystal structure prediction on MP-20 and MPTS-52. MCFlow achieves the best results on MPTS-52 across all metrics and is competitive on MP-20, while achieving consistently lower RMSE. We report match rate (MR) and RMSE for single-sample (1) and multi-sample (20) generation. S/B/L denote small/base/large model sizes. Noisy guidance is applied only for single-sample generation. Bold indicates the best value within each column across all models and sampling budgets; underline indicates the best value within the single-sample group.

Model	# Params	Valid (%) $\uparrow$	Unique (%) $\uparrow$	Novel (%) $\uparrow$	Energy-based			Stability		Metastability
Model	# Params	Valid (%) $\uparrow$	Unique (%) $\uparrow$	Novel (%) $\uparrow$	$E_{f}$ (eV) $\downarrow$	$E_{\text{hull}}$ (eV) $\downarrow$	RMSD (Å) $\downarrow$	Stable (%) $\uparrow$	SUN (%) $\uparrow$	Meta (%) $\uparrow$	MSUN (%) $\uparrow$
With pre-relaxation
MatterGen	47M	95.7	95.1	70.5	$-0.70\pm 0.79$	$0.18\pm 0.18$	$0.39\pm 0.50$	2.0	0.2 (6)	33.4	15.0
PLaID++	7B	96.0	77.8	24.2	$-0.50\pm 0.44$	$0.09\pm 0.16$	$0.13\pm 0.29$	12.4	1.0 (26)	60.7	7.6
WyFormer	8B	93.4	93.0	66.4	$-0.43\pm 0.95$	$0.50\pm 0.51$	$0.81\pm 0.98$	0.5	0.1 (3)	15.7	1.9
WyFormer-DFT	8B	95.2	95.0	66.4	$-0.67\pm 0.91$	$0.27\pm 0.36$	$0.42\pm 0.60$	3.7	0.4 (9)	24.8	7.8

Table 2: De novo generation evaluated on LeMat-GenBench ( Betala et al., 2025 ) with MP-20 training. MCFlow achieves competitive or superior performance across most metrics both with and without MLIP pre-relaxation. Reported values are averaged over an ensemble of three MLIPs (MACE-MP, UMA, Orb-v3) with LeMat-Bulk as the reference set, with all percentages computed over the 2,500 submitted structures following the LeMat-GenBench convention. Standard deviations are computed across the 2,500 generated structures, averaged over the three MLIPs. SUN and MSUN values are reported as percentage (count). Results are organized into two groups: models with and without MLIP pre-relaxation; MCFlow uses Orb-v3 for pre-relaxation. Bold and underline indicate the best and second-best results within each group.

Figure 3: Effect of the number of integration steps on performance. MCFlow reaches competitive crystal structure prediction (CSP) match rate and de novo generation (DNG) validity with far fewer integration steps than FlowMM and DiffCSP.

Figure 4: Distributions of crystallographic symmetry properties. MCFlow better captures crystallographic symmetry distributions than FlowMM and ADiT. From left to right, panels show the distributions of space groups (P1, Fm

\bar{3}

m, C2/m, P6 3 /mmc, I4/mmm, Pnma, R

\bar{3}

m, Cmcm, Pm

\bar{3}

m, P4/mmm, P

\bar{1}

\ldots

), Wyckoff multiplicity (1, 2, 3, 4, 6), and Wyckoff dimensionality (0, 1, 2, 3). Space groups and Wyckoff positions are determined using the SpaceGroupAnalyzer module in pymatgen with distance of 0.1 Å.

Figure 5: Distributions of relaxed structures. MCFlow samples are closer to local energy minima than FlowMM and ADiT, with smaller relaxation RMSE, fewer convergence steps, and lower

E_{\text{hull}}

. From left to right, plots show distributions of convergence steps, RMSE between initial and relaxed structures, and

E_{\text{hull}}

along CHGNet ( Deng et al., 2023 ) geometry optimization trajectories. The convergence rates are 99.73/60.79/87.65/99.54% for MCFlow/FlowMM/ADiT/ADiT Joint.

Model	MP-20		MPTS-52
Model	Comp (%) $\uparrow$	MR (%) $\uparrow$	Comp (%) $\uparrow$	MR (%) $\uparrow$
CGCNN	87.06	4.71	82.08	3.50
FlowMM (1)	83.39	2.40	76.12	0.88
MCFlow-S (1)	88.97	1.57	83.14	1.01
MCFlow-B (1)	90.17	3.12	83.56	1.20
MCFlow-L (1)	90.07	3.32	83.67	1.24

Table 3: Atom type generation on MP-20 and MPTS-52. MCFlow shows higher compositional validity (Comp) and MR than the baselines when sampling multiple candidates per structure. Comp follows Xie et al. (2022) ; details in Appendix G .

Rank	AcGaO 3 (mask Ac)				Zr 4 Fe 4 Si 4 (mask Fe)				Mn 2 VGa (mask V)				Ca 2 In 4 O 8 (mask Ca)
Rank	Element	Freq (%)	MP ID	Split	Element	Freq (%)	MP ID	Split	Element	Freq (%)	MP ID	Split	Element	Freq (%)	MP ID	Split
1	Ac	23	1183053	Test	Co	40	1102565	Test	Mn	23	999550	—	Mn	32	35162	Train
2	Yb	22	1187561	—	Cu	32	22522	Train	V	16	21235	Test	Ca	19	22766	Test
3	Sr	10		—	Ni	19	1103272	Val	Nb	15	1185972	Train	Cd	12	19803	Val

Table 4: Atom type substitutions across four MP-20 test-set materials. MCFlow recovers chemically meaningful substitutions in different applications: the top-3 predicted elements mostly correspond to known materials sharing the same space group as the original structure. For each case, we mask one element type and draw 100 samples using inference-time guidance built on atom type generation. The four parent structures, all drawn from the MP-20 test set, are AcGaO 3 (mp-1183053, cubic Pm

\bar{3}

m perovskite oxide), Zr 4 Fe 4 Si 4 (mp-1102452, orthorhombic Pnma silicide intermetallic for hydrogen storage), Mn 2 VGa (mp-21235, cubic Fm

\bar{3}

m full Heusler spintronic half-metal), and Ca 2 In 4 O 8 (mp-22766, cubic Fd

\bar{3}

m spinel photocatalyst). A blank in the MP ID column means no matching Materials Project entry exists; a “—” in the Split column means the entry is not in MP-20.

Model	MP-20				MPTS-52
Model	MR	RMSE	Struct	Comp	MR	RMSE	Struct	Comp
MCFlow	63.14	0.0611	99.44	89.85	26.46	0.1572	98.34	85.57
$-\,$ Inter	60.41	0.0772	99.48	89.64	22.80	0.1927	98.12	85.20
$-\,$ Intra	58.54	0.1023	99.35	89.30	18.80	0.2231	97.70	84.83
$-\,$ PosEmb	54.83	0.0979	99.48	85.30	16.44	0.2105	98.20	75.20

Table 5: Ablation results on MP-20 and MPTS-52. Removing the proposed hierarchical permutation augmentation and atom ordering leads to consistent performance degradation across tasks. Rows denote cumulative ablations from the full MCFlow model by removing inter-orbit permutation (

-\,

Inter), intra-orbit permutation (

-\,

Intra), and positional embedding (

-\,

PosEmb). MR and RMSE are reported for CSP, while Struct and Comp denote structural and compositional validity for DNG; details in Appendix G .

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Description
$C$	Infinite periodic crystal
$N$	Number of atoms in the primitive unit cell
$A$	Atom types sequence ( $[a_{i}]_{i=1}^{N}\in\mathcal{A}^{1\times N}$ )
$F$	Fractional coordinates ( $[f_{i}]_{i=1}^{N}\in[0,1)^{3\times N}$ )
$X$	Cartesian coordinates ( $X=LF$ )
$L$	Lattice matrix

Appendix

Table A.1 : Summary of Notations

Modality ( $m$ )	Base dist. $p_{0}$	Cond. target	Parameterization $u^{m}_{\theta}$	Bregman div. $D_{m}(u,v)$
Atom type ( $A$ )	$\delta_{M}$	$\displaystyle\frac{\delta_{A_{0}}(A_{t})\delta_{A_{1}}}{1-t}$	$\displaystyle\frac{\delta_{A_{0}}(A_{t})\text{Cat}(\text{softmax}(A_{\theta}))}{1-t}$	$\displaystyle\sum_{j}u_{j}\log\!\left(\frac{u_{j}}{v_{j}}\right)-\sum_{j}u_{j}+\sum_{j}v_{j}$
Frac. coord. ( $F$ )	$\mathcal{U}[0,1)^{3\times N}$	$\displaystyle\frac{\log_{F_{s}}(F_{1})}{1-s}$	$\displaystyle\frac{\log_{F_{s}}\left(\text{proj}(F_{\theta})\right)}{1-s}$	$\\|u-v\\|^{2}$
Length ( $L^{l}$ )	$\text{LogNorm}(\mu,\sigma)$	$\displaystyle\frac{L^{l}_{1}-L^{l}_{s}}{1-s}$	$\displaystyle\frac{L^{l}_{\theta}-L^{l}_{s}}{1-s}$	$\\|u-v\\|^{2}$
Angle ( $L^{a}$ )	$\displaystyle\mathcal{U}\left[60^{\circ},120^{\circ}\right]^{1\times 3}$	$\displaystyle\frac{L^{a}_{1}-L^{a}_{s}}{1-s}$	$\displaystyle\frac{L^{a}_{\theta}-L^{a}_{s}}{1-s}$	$\\|u-v\\|^{2}$

Appendix

Table B.1 : Configurations for Multimodal Flow Matching. We leverage the flexibility of the flow matching framework to design components for each modality. The Base dist. column specifies the prior distribution

p_{0}

, selected to match the support of each modality. The Cond. target column defines the conditional target vector fields or rate vectors derived from data pairs, utilizing linear interpolation for Euclidean space, geodesic paths for the torus, and independent mixture paths for discrete atom types. The Parameterization column describes how to approximate the target velocity field from the clean data prediction (

F_{\theta},L^{l}_{\theta},L^{a}_{\theta}

) or the target categorical posterior distribution via atom type logit prediction

A_{\theta}

. Finally, the Bregman div. column lists the specific divergence metric minimized during training to match the parameterized and target velocity field or rate vector. Since the torus is flat and locally Euclidean, the Bregman divergence on fractional coordinates is the squared Euclidean distance in the tangent space.

A_{t}\sim\mathrm{Cat}((1-t)\delta_{A_{0}}+t\delta_{A_{1}}),\quad F_{s}\leftarrow\exp_{F_{0}}(s\log_{F_{0}}(F_{1})),\quad L^{l}_{s}\leftarrow(1-s)L^{l}_{0}+sL^{l}_{1},\quad L^{a}_{s}\leftarrow(1-s)L^{a}_{0}+sL^{a}_{1}.

Appendix

Algorithm 1 MCFlow preprocessing and training

Figure D.1 : Permutation space reduction. Logarithmic scale comparison of full permutation space

N!

and reduced space

\prod_{i}|\mathcal{W}^{i}|!\prod_{j}|O^{i}_{j}|!

across unit cell sizes

N=1

20

Model	NG	CSP				ATG
		MP-20		MPTS-52		MP-20		MPTS-52
		MR (%) $\uparrow$	RMSE $\downarrow$	MR (%) $\uparrow$	RMSE $\downarrow$	Comp (%) $\uparrow$	MR (%) $\uparrow$	Comp (%) $\uparrow$	MR (%) $\uparrow$
MCFlow-S	✗	54.40	0.1339	22.58	0.1880	88.89	1.53	83.19	0.98
MCFlow-S	✓	56.10	0.1269	22.88	0.1845	88.97	1.57	83.14	1.01
MCFlow-B	✗	62.11	0.0624	25.32	0.1640	90.05	2.69	84.29	1.09
MCFlow-B	✓	63.14	0.0611	26.46	0.1572	90.17	3.12	83.56	1.20

Appendix

Table E.1 : Ablation study of noisy guidance on CSP and atom type generation (ATG). Noisy guidance improves structural fidelity in CSP and yields competitive performance in ATG across datasets. Single-sample generation per composition or structure. Bold indicates the best performance within each model size group.

Figure E.1 : Sensitivity analysis of noisy guidance. Single-sample CSP performance on MP-20 depending on the atom type noise level

\sigma

and guidance scale

\omega

Model scale	Small (S)	Base (B)	Large (L)
# Parameters	34M	134M	464M
Hidden dimension	384	768	1024
Transformer layers	12	12	24
Attention heads	6	12	16

Appendix

Table F.1 : MCFlow model configurations.

Hyperparameter	Value
Optimizer	AdamW
Learning rate	$1\times 10^{-4}$
Weight decay	0.0
Batch size	256
Max epochs	10,000
Precision	BF16 mixed precision

Appendix

Table F.2 : MCFlow training configurations.

Hyperparameter	Value
Integration steps	500
Guidance scale $\omega$	2.0 (single-sample)
Noise level $\sigma$	0.1 (single-sample)

Appendix

Table F.3 : MCFlow inference configurations.

Model	Params	Batch	Train (1 Epoch)	Infer (1 Step)
DiffCSP	12.3M	256	26.0s	111.1ms
FlowMM	28.3M	256	52.11s	240.7ms
MCFlow-S	33.7M	256	8.54s	23.9ms
MCFlow-B	134.0M	256	22.27s	68.7ms
MCFlow-L	464.6M	128	77.68s	113.5ms

Appendix

Table H.1 : Computational cost comparison. MCFlow-B achieves faster per-step training and inference than the baselines despite more parameters, because Transformer dense operations are more hardware-efficient than E(3)-equivariant GNN message passing. Combined with faster metric saturation ( Figure 3 ), MCFlow achieves lower total inference cost. We report wall-clock training time per epoch and inference time per integration step, measured on a single NVIDIA RTX 3090.

Split	Model	METRe (%) $\uparrow$	RMSE $\downarrow$	cRMSE $\downarrow$
Polymorph	DiffCSP	53.1	0.084	0.279
	FlowMM	65.2	0.079	0.226
	OMatG	70.5	0.056	0.187
	MCFlow-B	70.7	0.068	0.195
Original	DiffCSP	58.8	0.064	0.244
	FlowMM	67.0	0.067	0.210

Appendix

Table H.2 : Polymorph-aware CSP evaluation using METRe ( Martirossyan et al., 2025 ) . MCFlow achieves the highest METRe under polymorph and original splits, indicating robustness to evaluation methodology. METRe (Match Everyone to REference) computes the fraction of reference structures that find at least one matching generated structure; this corrects the standard match rate, whose one-to-one pairing underestimates models that correctly generate multiple stable polymorphs per composition.

Model	MP-20		MPTS-52
Model	MR (%) $\uparrow$	RMSE $\downarrow$	MR (%) $\uparrow$	RMSE $\downarrow$
DiffCSP++	80.27	0.0295	46.29	0.0896
SGFM	82.74	0.0288	51.79	0.0827
MCFlow-S	79.21	0.0238	73.91	0.0415
MCFlow-B	85.31	0.0210	77.61	0.0361
MCFlow-L	85.96	0.0201	74.63	0.0386

Appendix

Table H.3 : Template-conditioned crystal structure prediction on MP-20 and MPTS-52. MCFlow outperforms the template-conditioned baselines DiffCSP++ and SGFM on both datasets. Evaluation under structural templates extracted from the target structure; for MCFlow, MR and RMSE are computed on generated samples matching the target template.

Ordering	CSP		DNG
Ordering	MR (%) $\uparrow$	RMSE $\downarrow$	Struct (%) $\uparrow$	Comp (%) $\uparrow$
Atomic Number	59.81	0.0951	99.30	88.09
Rand. Perm.	54.25	0.1184	99.45	86.12
Electronegativity	63.14	0.0611	99.44	89.85

Appendix

Table H.4 : Impact of atom ordering and permutation strategies on CSP and DNG. Electronegativity ordering outperforms atomic number ordering due to the direct correspondence with cation-anion roles exploited by the DiT. Full

N!

random permutation substantially degrades performance, as the permutation space overwhelms learning capacity. All variants use MCFlow-B on MP-20.

Figure H.1 : Scaling behavior of MCFlow across model sizes. Larger MCFlow variants consistently improve training loss, DNG validity, and CSP match rate. Training loss, DNG valid rate and CSP match rate over training epochs for small, base, and large models (left). Correlation between model size and training loss, validity, and CSP match rate at epoch 4000 (right).

Model	MP-20		MPTS-52
Model	MR (%) $\uparrow$	RMSE $\downarrow$	MR (%) $\uparrow$	RMSE $\downarrow$
DiffCSP	52.51 / 57.82	0.0600 / 0.0627	14.29 / 15.79	0.1489 / 0.1533
FlowMM	59.98 / 66.22	0.0629 / 0.0661	20.28 / 22.29	0.1486 / 0.1541
OMatG	63.75 / 69.83	0.0720 / 0.0741	25.15 / 27.38	0.1931 / 0.1970
MCFlow-S	56.10 / 61.78	0.1269 / 0.1321	22.88 / 24.74	0.1846 / 0.1920
MCFlow-B	63.14 / 69.23	0.0611 / 0.0663	26.46 / 28.77	0.1577 / 0.1610

Appendix

Table H.5 : Filtered and unfiltered CSP match rates on MP-20 and MPTS-52. MCFlow-L consistently outperforms all baselines under both evaluation schemes. The unfiltered MR is included to capture physically real structures (

\sim

10% in MP-20) that fail SMACT validity rules. Performance is reported as filtered / unfiltered MR using single-sample generation with noisy guidance. OMatG corresponds to the LinearODE variant.

Model	Structural Validity (%) $\uparrow$	Compositional Validity (%) $\uparrow$
DiffCSP	67.47	55.80
SymmCD	87.11	78.18
DiffCSP++	99.87	77.52
SGEquiDiff	97.79	79.83
MCFlow-S	96.73	84.62
MCFlow-B	98.34	85.57

Appendix

Table H.6 : De novo generation results on the MPTS-52 benchmark. MCFlow achieves the highest compositional validity and competitive structural validity on MPTS-52. The MPTS-52 dataset contains 100.00% structurally valid and 88.24% compositionally valid structures. Bold and underline indicate the best and second-best results, respectively.

Method	MR (%) $\uparrow$	RMSE $\downarrow$
CSP-only	61.92	0.0638
MCFlow-B w/o NG	62.11	0.0624
MCFlow-B	63.14	0.0611

Appendix

Table H.7 : CSP-only vs. unified training on MP-20. CSP-only (

t=1

during training) and unified training yield nearly identical MR, indicating no destructive interference. The unified model additionally supports DNG, ATG, and masked ATG without retraining, and noisy guidance (NG) provides a further gain only available through decoupled time training.

Method	MR (%) $\uparrow$	RMSE $\downarrow$
DiffCSP DNG	0 3.76	0.3396
FlowMM DNG	27.01	0.1947
MCFlow-B coupled ( $t{=}s$ )	45.58	0.1499
MCFlow-B	63.14	0.0611

Appendix

Table H.8 : Decoupled time axes ablation and comparison with adapted DNG baselines on MP-20 CSP. Decoupled time axes are essential: coupling

t{=}s

substantially degrades CSP performance. DNG-trained DiffCSP and FlowMM, adapted to CSP by overwriting composition with ground-truth values at every integration step, also achieve substantially lower MR due to their coupled noise schedules.

Figure I.1 : Crystal structure prediction. MCFlow explores multiple crystal structures with different space groups conditioned on a single composition, illustrated for the alloy AlFe 3 (top-left, target crystal).

Figure I.2 : Atom type generation. MCFlow captures multimodal compositional distributions conditioned on a structure, generating diverse compositions for a given perovskite structure of Ca 4 Ni 4 O 12 (top-left, target crystal).

Figure I.3 : De novo generation. MCFlow jointly generates diverse atom types and crystal structures.

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Crystal structure prediction (CSP) is fundamental to computational materials discovery. Generative models including CDVAE, DiffCSP, FlowMM, and CrystalFlow learn stable-crystal distributions directly, but diffusion and flow-matching inference requires tens to thousands of sequential network evaluations per candidate. We introduce uFlowCSP, a MeanFlow-based CSP model that learns the average, rather than instantaneous, probability-flow velocity. It generates a complete structure in one to five evaluations, delivering 5x-58x faster inference with equal or better performance. A chemistry- and symmetry-aware Transformer uses canonical atom ordering, global composition, and per-token chemistry embeddings. A coarse crystal-system token is used only during training; it provides additive gains, particularly improving space-group agreement despite being absent at inference, which remains formula-only. On MP-20 with 20 candidates per target, one step matches CrystalFlow (78.38% vs. 78.34%) with 100x fewer evaluations and about 10x lower wall-clock time. Five steps reach 83.64%, exceeding CrystalFlow (78.34% at 2,000 evaluations) and DiffCSP (77.93% at about 20,000), while using 20x fewer evaluations. uFlowCSP generates 10,000 structures in 0.39-1.31 minutes, versus 6.5 for CrystalFlow and 76.1 for DiffCSP. Under CSPBench's energy-ranked top-five structure-and-space-group criterion, five-step uFlowCSP reaches 72%/72%/65% structure, space-group, and consensus match rates. CrystalFlow reaches 78%/73%/68% at 100 steps but falls to 49%/32%/31% at five. Thus, uFlowCSP improves accuracy per network evaluation, not merely peak accuracy.

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