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

1 Introduction

• Nearly 47% of verified ICSD structures contain disorder, including substitutional mixing, vacancies, and interstitial species that can govern material properties.Most existing generators instead assume each crystallographic site contains a single deterministic element.Source: §1 · p. 1
• The Occupancy Distribution Matrix (ODM) represents site-wise species distributions subject to full-site occupancy, mass conservation, and non-negativity.A vacancy pseudo-species accounts for vacancies and allows every site to be included in the composition constraints.Source: §1 · p. 2
• Formula-dependent transportation polytopes couple site occupancies globally, so changing one site's allocation changes the feasible allocations at other sites.The feasible domain also varies with composition and site budget, complicating direct generation in a shared Euclidean space.Source: §1 · p. 2
• EP-Flow maps each formula-dependent polytope into a shared double-centered space, learns a marginal-preserving flow, and uses Sinkhorn recovery to enforce occupancy constraints.It jointly generates occupancies, fractional coordinates, and lattice parameters.Source: §1 · p. 2
• EP-Flow achieves state-of-the-art performance on COD and MPDS benchmarks, improving structure match rate, occupancy accuracy, formula consistency, and validity over adapted baselines.Case studies show sparse, chemically meaningful disorder motifs rather than only global composition matching.Source: §1 · p. 2

Ordered Crystal Structure Prediction

• Ordered-crystal models assign each site a single discrete species, whereas EP-Flow targets fractional occupancies coupled by global stoichiometric constraints.Earlier generative approaches progressed from latent-variable and diffusion models to flow-based and Bayesian flow methods.Source: §2 · pp. 2–3

Disordered Crystal Structure Prediction

• Dis-Gen and DMFlow address disordered structures, but rely on site-level disorder annotations that are unavailable when only the global formula is given.EP-Flow jointly models the occupancy distribution without explicit site-specific supervision.Source: §2 · p. 3

Constrained Generative Modeling on Manifolds

• Prior constrained-generation methods use manifold, simplex, or polytope models, but projection corrections can break trajectory smoothness and barrier or mirror-map methods can be numerically fragile.EP-Flow instead uses a marginal-preserving flow in shared double-centered space followed by Sinkhorn recovery.Source: §2 · p. 3

3.1 Transportation Polytopes and Sinkhorn Projection

• A transportation polytope consists of nonnegative matrices satisfying prescribed marginal constraints; Sinkhorn alternates row and column rescaling to project onto it.For strictly positive inputs, the differentiable projection coincides with the unique KL projection, and finite iterations can meet a prescribed tolerance.Source: §3.1 · p. 3

3.2 Double-Centering Operator and Zero-Marginal Space

• Double-centering maps matrices to a zero-marginal subspace that provides a shared representation across formulas.The subspace is Euclidean and its tangent space is isomorphic to itself, making it suitable for generative vector fields.Source: §3.2 · p. 3
• The EP-Flow overview shows occupancy canonicalization from a formula polytope to double-centered space, followed by Sinkhorn recovery and joint structure generation.The joint flow generates occupancy, coordinates, and lattice parameters conditioned on the formula.Source: Figure 2 · p. 4

4.1 Task Definition

• EP-Flow models a disordered crystal as an occupancy matrix, fractional coordinates, and lattice parameters, generating the full structure conditioned on stoichiometry rather than fixed site labels.The occupancy matrix must satisfy site-wise occupancy, mass conservation, and non-negativity constraints; vacancy is represented as a pseudo-species.Source: §4.1 · p. 4

4.2 Occupancy Distribution Matrix Generation

• Formula-dependent occupancy polytopes are canonicalized into a shared double-centered space by applying a mapping to the log occupancy matrix and removing row- and column-wise components.The centered space has zero-sum marginals regardless of the underlying formula.Source: §4.2 · pp. 4–5
• For fixed stoichiometry, each centered occupancy state maps uniquely to an interior ODM, recovered by Sinkhorn projection onto the target polytope.The projection minimizes KL divergence to the exponentiated centered matrix while restoring physical stoichiometry.Source: §4.2 · p. 5
• Occupancy flow uses a linear path between a prior sample and the data state in centered space, with projected velocities and a formula-derived mask restricting training.The projected trajectory remains in the zero-marginal space.Source: §4.2 · p. 5

4.3 Fractional Coordinates Generation

• Fractional-coordinate flow uses the shortest periodic displacement between noise and data samples, following a geodesic on the coordinate torus.The model predicts coordinate velocity from the interpolated coordinates so periodicity is respected throughout the trajectory.Source: §4.3 · p. 5

4.4 Lattice Parameters Generation

• Lattice lengths are represented in log space and angles as unconstrained logits mapped to the Niggli-reduced range [60°, 120°].This canonical representation is used for stable learning while preserving coupled statistics between cell geometry and atomic arrangement.Source: §4.4 · p. 5

4.5 Unified Training Objective

• The unified objective jointly regresses occupancy, coordinate, and lattice velocities, using weighted mean-squared errors to balance their learning.The lattice head is split into length and angle components to account for their different scales.Source: §4.5 · pp. 5–6

4.6 Stoichiometry-Conditioned Joint Velocity Network

• A stoichiometry-conditioned message-passing network predicts coordinate, occupancy, and lattice velocities from crystal states and formula, while satisfying SO(3) and translation invariance.Its inputs include centered occupancy, fractional coordinates, lattice state, and stoichiometry.Source: §4.6 · p. 6

5.1 Experimental Setup

• The benchmark uses COD and MPDS, with substitutional-only and vacancy-inclusive subsets capped at 20 or 50 sites, adapting ordered generators and evaluating matching, geometry, occupancy, and validity.The datasets contain 17,239 COD structures and 110,966 MPDS structures; FlowMM and OMatG are adapted alongside DMFlow.Source: §5.1 · pp. 6–7

5.2 Main Results on Disordered Crystal Generation

• Across eight subsets, EP-Flow attains 17.10%–60.82% Match Rate versus DMFlow’s 9.74%–45.17%, exceeding it by 7.36–23.78 percentage points.The reported tables cover COD and MPDS results.Source: Table 1 · p. 7, §5.2 · p. 7, Table 2 · p. 8
• Validity alone can mislead: FlowMM reaches 95.90% validity but 0.24% Match Rate on MPDS Disorder-20, while larger and vacancy-inclusive cases degrade.The degradation is reported for EP-Flow and Ada-DMFlow on COD and MPDS.Source: §5.2 · pp. 7–8, Table 2 · p. 8

5.3 Ablation Studies

• Removing formula conditioning drops Match Rate from 60.82% to 20.01%; removing Energy-diff lowers it to 35.23% and raises OccMAE to 0.0159.The ablations are reported on MPDS-Sub-20.Source: §5.3 · p. 8, Table 3 · p. 8
• Without double centering, Match Rate falls to 46.06%; without projection, OccMAE rises from 0.0075 to 0.1277; removing both normalization and projection yields 0.00% Match Rate.The reported ablations are on MPDS-Sub-20.Source: §5.3 · pp. 8–9, Table 3 · p. 8

5.4.1 Case Studies on Functional Disordered Materials

• Case studies recover vacancies outside cuprate CuO2 planes, coupled Ti/N disorder in a ThMn12 magnet, and mixed-cation/anion disorder in an oxyfluoride perovskite.Together the examples span oxygen-vacancy, interstitial, and multi-sublattice substitutional disorder.Source: §5.4.1 · p. 9

5.4.2 Disorder-Pattern Analysis

• Site-wise analyses show sparse, structured assignments: diagonal blocks indicate stable preferences, while localized off-diagonal responses capture mixed occupancy; MPDS verifies some COD-unseen patterns.Source: §5.4.2 · p. 10

• EP-Flow unifies ordered, substitutional, vacancy, and interstitial disorder through an Occupancy Distribution Matrix over formula-dependent transportation polytopes.Source: §6 · p. 10
• The framework canonicalizes occupancy polytopes into a double-centered space, learns a marginal-preserving flow, and maps samples back via Sinkhorn projection.It jointly generates occupancies, fractional coordinates, and lattice parameters from formula and site count.Source: §6 · p. 10
• Experiments report state-of-the-art performance across disorder types and site-count scales, with occupancy, geometry, and lattice modeled under exact compositional constraints.Source: §6 · p. 10

A.1 Proof of the Bijectivity of Canonicalization

• The canonicalization map is bijective: every centered matrix has a preimage, and a centered matrix uniquely determines its occupancy matrix through Sinkhorn recovery.Surjectivity follows from positive Sinkhorn scaling; injectivity follows from uniqueness of Sinkhorn scaling for strictly positive matrices.Source: §A.1 · p. 13

A.2 Trajectory Feasibility in the Centered Space

• Projecting occupancy velocity into the zero-marginal space keeps the centered trajectory there, so inverse Sinkhorn recovery remains inside the target transportation polytope.The trajectory's row and column marginals remain zero in centered space, and the recovered matrix stays in P(1, c).Source: §A.2 · p. 13

A.3 SO(3) Invariance of the Joint Velocity Field

• The joint velocity field is rotation-invariant because lattice parameters and fractional coordinates do not change under a global rotation of the crystal.The lattice parameters encode lengths and inter-axis angles, while fractional coordinates are intrinsic to the lattice basis.Source: §A.3 · p. 14

A.4 Translation Invariance of the Joint Velocity Field

• The model's input representation is translation-invariant because it uses relative fractional displacements, which are unchanged by a common periodic shift.The other inputs are unaffected by shifting the crystal origin.Source: §A.4 · p. 14

A.5 Flow Matching

• Conditional flow matching trains a vector field to approximate a tractable conditional path between noise and data, then generates samples by integrating its ODE.The described linear path interpolates between a noise sample and a data sample; manifold-constrained settings can instead use Riemannian flow matching.Source: §A.5 · p. 14

B.1 Logarithmic Mapping as Energy Space Evolution

• The paper interprets log occupancy as an energy-like representation, with generation described as evolving from high-entropy noise toward a low-energy configuration rather than maintaining equilibrium at every step.This is presented as a Boltzmann-type interpretation of the trajectory.Source: §B.1 · p. 15

B.2 Double-Centering as Relative Potential Extraction

• Double-centering removes additive site and species background fields, leaving a relative-affinity representation that retains the information needed to recover the ODM under fixed marginals.The paper relates the site-specific and species-specific potentials to background fields and the remaining core term to intrinsic relative energy.Source: §B.2 · p. 15

B.3 Sinkhorn Recovery as Thermodynamic Annealing

• Sinkhorn recovery is analogized to annealing: scaling adjusts atomic distributions to satisfy global stoichiometric constraints imposed by the target formula.The scaling variables are described as dynamic chemical potentials in the analogy.Source: §B.3 · p. 15

B.4 Physical Interpretation of Energy-Difference Edge Features

• Log-occupancy differences encode relative site preferences and help distinguish locally compatible site–species configurations.The paper relates the difference to a log probability ratio and, under its Boltzmann-type interpretation, to a relative energy difference.Source: §B.4 · p. 16
• The ablation results identify the log-domain edge feature as critical for recovering accurate local disorder patterns.The feature is computed from the double-centered log-domain representation.Source: §B.4 · p. 16
• The thermodynamic connection is explicitly a heuristic framework, not a formal derivation.The authors describe it as an exploratory conceptual bridge.Source: §B.4 · p. 16

C.1 Datasets

• The dataset protocol deduplicates structures before splitting and filters structures for physical validity, computational tractability, and statistical support.The stated filters include interatomic distances below 0.5 Å, cell volumes below 0.1 Å3, or lattice parameters above 40 Å; rare elements are also removed.Source: §C.1 · pp. 16–17
• The benchmark uses four subsets: Sub-20 and Sub-50 for substitutional disorder, and Disorder-20 and Disorder-50 for settings that also include vacancies.The subsets are split approximately 8:1:1 for training, validation, and testing, and the formula includes an explicit vacancy pseudo-atom when needed.Source: §C.1 · pp. 16–17

C.2 Baseline Adaptations

• Adapted FlowMM and OMatG replace discrete species with continuous occupancy distributions and add an occupancy head alongside coordinate and lattice generation.At inference, Sinkhorn projection enforces composition and site-wise normalization; OMatG also receives EP-Flow's disordered formula embedding.Source: §C.2 · p. 17
• Adapted-DMFlow predicts site occupancies from the formula rather than receiving ground-truth site annotations, using spherical reparameterization for normalized site distributions.A Sinkhorn projection at inference enforces consistency with the target formula.Source: §C.2 · p. 17
• The baselines form a constraint progression: post-hoc enforcement only, row constraints during generation, then EP-Flow's joint row- and column-constraint preservation.This comparison is designed to assess incorporating the full transportation-polytope constraints into generative dynamics.Source: §C.2 · p. 18

C.3 Evaluation Metrics

• The extended matching suite evaluates disordered structures using Match Rate, RMSD, OccMAE, and structural validity, incorporating continuous site-wise occupancies into matching.The reported RMSD is averaged over successful matches.Source: §C.3 · pp. 18–19
• Match Rate falls from 60.82% to 50.09% as occtol tightens from 0.30 to 0.01, while OccMAE decreases to 0.0004.The result is reported for EP-Flow on MPDS Sub-20 under increasingly stringent occupancy tolerances.Source: Table 5 · p. 18, §C.3 · p. 18

D.1 Disordered Crystal Structure

• The ODM treats each entry as a site's probability or average fraction for a candidate species, including vacancies, and represents diverse disorder types in one framework.It is a statistical summary of possible local arrangements, not a single static snapshot.Source: §D.1 · p. 20, Figure 6 · p. 20

D.2 Formula and Number of Sites

• A conventional formula may not reveal the crystallographic site count when occupancies are fractional, so the representation adds vacancies to complete each site's occupancy.The extended formula then sums to the number of sites, and the physical composition is recovered by marginalizing out vacancies.Source: §D.2 · pp. 20–21

D.3 Log-domain Double Centering

• Because real ODMs can contain zeros, the method smooths them before log-domain double-centering to obtain a shared zero-marginal representation for flow matching.Double-centering removes site-wise and species-wise additive effects.Source: §D.3 · p. 22

D.4 Sinkhorn Recovery

• Sinkhorn recovery can leave trace occupancies, so entries below 0.005 are truncated and a second projection restores marginals, yielding a sparse, formula-consistent ODM.The stated threshold removes occupancies below 0.5% of a site; the authors say this avoids treating negligible residuals as meaningful mixed-site disorder in CIF reconstruction.Source: §D.4 · p. 22

D.5 Structural Reconstruction

• Across three representative oxides, trajectories move from noisy T = 1 states toward T = 0 predictions that recover positions and occupancies and match ground-truth structures.The figure compares each trajectory with its ground truth; the text reports final predictions as valid and formula-matched.Source: §D.5 · p. 23, Figure 7 · p. 23
• All three reconstructions achieve RMSD below 0.02, while occupancy errors are small and largest mainly on mixed cation sites.The reported errors chiefly reflect competition among chemically similar species on mixed sites.Source: §D.5 · p. 23, Table 6 · p. 23
• Sr0.2Ca0.4La1.4Mn1.8Cr0.2O6 recovers coupled A-site Sr/Ca/La and B-site Mn/Cr disorder with maximum occupancy errors below 0.002.The authors describe the reconstruction as accurate for both sublattices.Source: §D.5 · pp. 23–24, Table 7 · p. 23
• For Mg1Zn0.6Cr4Co0.4O8, three competing species yield local Mg/Zn/Co redistribution between mixed sites despite preservation of the global formula.One site is slightly Mg-rich and the other relatively Zn/Co-rich.Source: §D.5 · p. 24
• For Mg1.04Al0.76Cr3.24Fe0.96O8, Al/Cr mixed sites have maximum occupancy error around 0.0016, while Mg/Fe sites redistribute locally but retain total Mg1.04Fe0.96.The text characterizes the local differentiation as chemically plausible, with the crystallographic average structure remaining the quantitative reference.Source: §D.5 · p. 24

D.6 Hyperparameters

• Training used 16-bit mixed precision; COD models ran on one NVIDIA GeForce RTX 4090 GPU, while MPDS models used two-GPU distributed data parallelism.Tables 8 and 9 summarize architecture/training hyperparameters and dataset-specific settings and generative loss weights.Source: §D.6 · p. 24, Table 8 · p. 24, Table 9 · p. 24

E Limitations

• The site count is specified or sampled from the benchmark distribution; determining it from formula alone remains open, especially when non-vacancy stoichiometry differs from the number of sites.This boundary arises because generation implicitly depends on the number of crystallographic sites.Source: E Limitations · p. 25
• EP-Flow does not support property-conditioned generation or directly optimize targets such as ionic conductivity, magnetism, superconductivity, or thermodynamic stability.Property-guided and inverse-design settings are identified as future work.Source: E Limitations · p. 25
• EP-Flow does not condition on experimental measurements such as PXRD, PDF, or spectroscopy data.The authors note that incorporating such information could further constrain generation and support experimental structure determination and refinement.Source: E Limitations · p. 25