EP-Flow: Disordered Crystal Structure Prediction without Site-Level Annotations

Qiuliang Liu; Liming Wu; Qi Li; Zhonglong Peng; Chang Chen; Xiaolong Chen; Wenbing Huang; Shifeng Jin

cs.LGOct 1, 2026

EP-Flow: Disordered Crystal Structure Prediction without Site-Level Annotations

Authors: Qiuliang Liu, Liming Wu, Qi Li, Zhonglong Peng, Chang Chen, Xiaolong Chen, Wenbing Huang, Shifeng Jin

Organizations: Institute of Physics, Chinese Academy of Sciences, Beijing, China · University of Chinese Academy of Sciences, Beijing, China · Gaoling School of Artificial Intelligence, Renmin University of China

Abstract

Generative models have made rapid progress in ordered crystal structure prediction, yet many functional materials are intrinsically disordered, with substitutional mixing, vacancies, or interstitial species controlling their properties. Existing crystal generators either assume deterministic site occupations or require site-level disorder annotations, which are often unavailable when the chemical formula is the primary input. We formulate disordered crystal structure prediction through an Occupancy Distribution Matrix (ODM), a continuous site-by-species representation that unifies ordered crystals, solid solutions, vacancy disorder, and interstitial occupancy. A valid ODM must satisfy coupled site-wise occupancy, mass-conservation, and non-negativity constraints, placing each sample on a formula-dependent transportation polytope. We propose Entropic Polytope Flow (EP-Flow), a marginal-constrained flow matching framework that canonicalizes heterogeneous polytopes into a shared double-centered space, learns a marginal-preserving flow, and recovers feasible occupancies through a Sinkhorn inverse map. By jointly generating occupancies, fractional coordinates, and lattice parameters, EP-Flow achieves state-of-the-art performance on formula-conditioned disordered CSP benchmarks derived from COD and MPDS, substantially outperforming adapted ordered-crystal generators. Analyses further show that EP-Flow recovers sparse and chemically meaningful local disorder patterns rather than merely matching global composition statistics.

Figures & tables

Figure 1: Formula transportation polytopes.

Figure 2: Overview of EP-Flow for disordered crystal generation. (a) Disordered crystal represented by $({\bm{A}},{\bm{F}},{\bm{l}})$ . (b) Occupancy matrix canonicalized from formula polytope $\mathcal{P}(\bm{1}_{N},{\bm{c}})$ to double-centered space $\mathcal{M}_{0}(N,M)$ via log-domain double centering, which recovers back by Sinkhorn projection. (c) Joint flow matching over $({\bm{A}}^{\prime},{\bm{F}},{\bm{l}})$ conditioned on ${\bm{c}}$ , with projection back to ${\bm{A}}$ for the final structure.

Dataset	Model	Match Rate ( $\uparrow$ )	RMSD ( $\downarrow$ )	OccMAE ( $\downarrow$ )	Validity ( $\uparrow$ )
Sub-20	Ada-FlowMM	0.65%	0.0146	0.0002	87.40%
	Ada-OMatG	10.21%	0.0298	0.0147	90.40%
	Ada-DMFlow	21.61%	0.0580	0.0189	90.60%
	EP-Flow	45.39%	0.0393	0.0111	95.10%
Sub-50	Ada-FlowMM	0.67%	0.0728	0.0092	86.90%
	Ada-OMatG	3.28%	0.0459	0.0218	90.90%

Table 1: Disordered CSP Performance on the COD dataset. When match rate is zero (i.e., all failed), RMSD and OccMAE are undefined (N/A). The best results are marked in bold. “Ada” denotes the adapted baselines.

Dataset	Model	Match Rate ( $\uparrow$ )	RMSD ( $\downarrow$ )	OccMAE ( $\downarrow$ )	Validity ( $\uparrow$ )
Sub-20	Ada-FlowMM	0.49%	0.0335	0.0457	96.90%
	Ada-OMatG	11.78%	0.0331	0.0813	88.30%
	Ada-DMFlow	45.17%	0.0416	0.0096	92.10%
	EP-Flow	60.82%	0.0483	0.0075	97.80%
Sub-50	Ada-FlowMM	0.18%	0.0501	0.0391	98.10%
	Ada-OMatG	5.25%	0.0242	0.0594	93.50%

Table 2: Disordered CSP Performance on the MPDS dataset. When match rate is zero (i.e., all failed), RMSD and OccMAE are undefined (N/A). The best results are marked in bold. “Ada” denotes the adapted baselines.

Variant	Match Rate ( $\uparrow$ )	RMSD ( $\downarrow$ )	OccMAE ( $\downarrow$ )	Validity ( $\uparrow$ )
w/o Formula	20.01%	0.0529	0.0106	96.7%
w/o Energy-diff	35.23%	0.0527	0.0159	93.50%
w/o Double Centering	46.06%	0.0537	0.0115	91.0%
w/o Projection	15.78%	0.0434	0.1277	90.5%
w/o DC & Sinkhorn	0.00%	N/A	N/A	0.00%
Full Model (Reference)	60.82%	0.0483	0.0075	97.8%

Table 3: Ablation results on MPDS-Sub-20.

Figure 3: Reconstruction results for disordered functional materials. Each row compares predicted vs. ground-truth structures, with RMSD and OccMAE. Key-site occupancies are annotated; the rightmost column shows the site with largest occupancy error. Three disorder types are covered: oxygen-vacancy, interstitial/partial-occupancy, and mixed-cation/anion substitutional.

Figure 4: Site-wise occupancy-pattern analysis.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Database	Setting	Train	Val	Test	Total	Ratio (Train:Val:Test)
COD	Substitution 20	7,100	879	921	8,900	$7.98:0.99:1.03$
	Substitution 50	9,483	1,141	1,190	11,814	$8.03:0.97:1.01$
	Disorder 20	9,408	1,185	1,217	11,810	$7.97:1.00:1.03$
	Disorder 50	13,825	1,689	1,725	17,239	$8.02:0.98:1.00$
MPDS	Substitution 20	53,492	6,648	6,786	66,926	7.99 : 0.99 : 1.01
	Substitution 50	65,187	8,412	8,499	82,098	$7.94:1.02:1.03$

Appendix

Table 4: Dataset statistics for COD and MPDS databases.

Figure 5: Element index map and MPDS Disorder-50 distribution. Left: the element-to-position index map used in this work. Right: the element-occurrence heatmap for the MPDS Disorder-50 subset.

$\mathrm{occtol}$	Successful Matches	Match Rate	Normalized RMSD	OccMAE
0.30	4,127	60.82%	0.0483	0.0075
0.25	4,067	59.93%	0.0483	0.0058
0.20	4,012	59.12%	0.0480	0.0048
0.15	3,916	57.71%	0.0480	0.0033
0.10	3,804	56.06%	0.0479	0.0020
0.05	3,653	53.83%	0.0475	0.0010

Appendix

Table 5: Performance of EP-Flow on the MPDS Sub-20 dataset under varying occupancy tolerances.

Figure 6: Unified representation of diverse disorder types via the vacancy-extended Occupancy Distribution Matrix (ODM). In an ordered crystal (left), the site count $N$ is well-defined by the formula with discrete occupancies. When transitioning to disordered systems (right), this assumption breaks down. For substitutional disorder (top right), occupancies become continuously fractional while $N$ remains unchanged. For vacancy (middle right) and interstitial disorder (bottom right), the site count cannot be determined from the raw formula alone. To resolve this ambiguity, a vacancy pseudo-atom ( $\mathrm{Vac}$ , depicted as grey/white spheres) is introduced to ensure that the occupancy at each site sums to one. This yields an extended formula $\mathbf{c}$ that incorporates the total vacancy count, satisfying $\sum_{j=1}^{M}c_{j}=N$ . This formulation explicitly conditions the structural representation on both the formula and the site number, allowing all disorder types to be modeled within a unified ODM framework.

Figure 7: Structural reconstruction trajectories of three representative disordered oxides. Each row shows the evolution from the noisy state at $T=1$ to the reconstructed structure at $T=0$ , compared with the ground truth.

Formula	RMSD ( $\downarrow$ )	OccMAE ( $\downarrow$ )	Max Occ Error ( $\downarrow$ )
$\mathrm{Sr_{0.2}Ca_{0.4}La_{1.4}Mn_{1.8}Cr_{0.2}O_{6}}$	0.0188	0.0006	0.0019
$\mathrm{Mg_{1}Zn_{0.6}Cr_{4}Co_{0.4}O_{8}}$	0.0191	0.0053	0.0371
$\mathrm{Mg_{1.04}Al_{0.76}Cr_{3.24}Fe_{0.96}O_{8}}$	0.0151	0.0052	0.0335

Appendix

Table 6: Reconstruction metrics for the representative disordered oxide cases in Fig. 7 .

Formula	Mixed site	Reference occupancy	Predicted occupancy	$\Delta$ Occ
$\mathrm{Sr_{0.2}Ca_{0.4}La_{1.4}Mn_{1.8}Cr_{0.2}O_{6}}$	B-site	$\mathrm{Mn_{0.900}Cr_{0.100}}$	$\mathrm{Mn_{0.8987}Cr_{0.1013}}$ , $\mathrm{Mn_{0.9013}Cr_{0.0987}}$	0.0013
	A-site	$\mathrm{Sr_{0.100}Ca_{0.200}La_{0.700}}$	$\mathrm{Sr_{0.1019}Ca_{0.1988}La_{0.6993}}$ , $\mathrm{Sr_{0.0981}Ca_{0.2012}La_{0.7007}}$	0.0019
$\mathrm{Mg_{1}Zn_{0.6}Cr_{4}Co_{0.4}O_{8}}$	Cation site	$\mathrm{Mg_{0.500}Zn_{0.300}Co_{0.200}}$	$\mathrm{Mg_{0.5371}Zn_{0.2726}Co_{0.1903}}$ , $\mathrm{Mg_{0.4629}Zn_{0.3274}Co_{0.2097}}$	0.0371
$\mathrm{Mg_{1.04}Al_{0.76}Cr_{3.24}Fe_{0.96}O_{8}}$	Mg/Fe site	$\mathrm{Mg_{0.520}Fe_{0.480}}$	$\mathrm{Mg_{0.5535}Fe_{0.4465}}$ , $\mathrm{Mg_{0.4865}Fe_{0.5135}}$	0.0335
	Al/Cr site	$\mathrm{Al_{0.190}Cr_{0.810}}$	$\mathrm{Al_{0.1884\text{--}0.1914}Cr_{0.8086\text{--}0.8116}}$	0.0016

Appendix

Table 7: Site-wise occupancy comparison for representative structural reconstruction cases. Only mixed-occupancy sites are reported; fully occupied sites are omitted for clarity.

Hyperparameter	Symbol / module	Value
Hidden dimension	$d_{h}$	512
Message-passing layers	$L$	6
Activation function	$\sigma$	SiLU
Learning rate	$lr$	$5\times 10^{-4}$
LR decay factor	$\gamma_{\mathrm{lr}}$	0.85
LR scheduler patience (epochs)	$P_{\mathrm{lr}}$	50

Appendix

Table 8: Architecture and training-related hyperparameters.

Database	Setting	Batch Size	Sampling $\bm{\mathrm{step\_lr}}$	$\bm{\lambda_{{\bm{A}}}}$	$\bm{\lambda_{{\bm{F}}}}$	$\bm{\lambda_{\mathrm{len}}}$	$\bm{\lambda_{\mathrm{ang}}}$
COD	Substitution	256	3	20.0	100.0	1.0	100.0
COD	Disorder	256	3	20.0	100.0	1.0	100.0
MPDS	Substitution	512	5	20.0	100.0	5.0	500.0
MPDS	Disorder	512	5	20.0	100.0	5.0	500.0

Appendix

Table 9: Dataset-specific training hyperparameters and generative loss weights.

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