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PXtal: Learning to Align Powder X-Ray Diffraction and Crystal Structures under Information Asymmetry across Modalities

Zhuoran Yang, Christopher M. Collins, Bei Peng, Luke M. Daniels, Matthew J. Rosseinsky, Vladimir V. Gusev

arXiv:2610.10653v1cs.LGcond-mat.mtrl-sci

TL;DR

PXRD compresses three-dimensional crystal structure information into diffraction patterns, leaving paired modalities unequally informative. PXtal uses adaptive unbalanced transport with coupling-level GKL supervision, and improves retrieval across test sets, especially for close but crystallographically distinct neighbors.

  • Problem

    PXRD patterns retain aggregate reflection intensities rather than the full three-dimensional crystal description, and similar patterns can correspond to crystallographically distinct structures.

    Source: §1 · p. 2

  • Method

    PXtal learns shared-space PXRD and crystal encoders using an unbalanced transport coupling with adaptive marginals and coupling-level GKL supervision.

    Source: §1 · p. 2, Unbalanced Optimal Transport Coupling · p. 3, From Conditional CE to GKL at the… · p. 4

  • Results

    PXtal improves retrieval across six test sets, including four zero-shot transfer sets, with the largest gains for patterns with close but crystallographically distinct nonpaired neighbors.

    Source: §1 · p. 2, §6 · p. 7

  • Takeaways & Limitations

    The results suggest that scientific multimodal alignment objectives should reflect what each modality preserves.

    Source: §6 · p. 7

    The coupling-level GKL derivation assumes strictly positive row sums and P_ij > 0 wherever Q_ij > 0.

    Source: From Conditional CE to GKL at the… · p. 4

Abstract

from arXiv · show

Scientific multimodal learning commonly assumes that paired views are comparably informative. Powder X-ray diffraction (PXRD) makes this mismatch explicit: compressing a three-dimensional crystal structure into a one-dimensional diffraction pattern loses information and makes the pattern harder to connect to the crystal structure that produced it. We introduce PXtal, a framework for learning aligned PXRD and crystal representations under this physically imposed information asymmetry. PXtal uses Unbalanced Optimal Transport (UOT) to adapt the cross-modal coupling and coupling-level generalized Kullback-Leibler (GKL) divergence to supervise the full transport plan. Across six test sets, including four zero-shot transfer sets, PXtal consistently outperforms the baseline models in PXRD-to-crystal candidate retrieval, with the largest gains when PXRD patterns have close but crystallographically distinct nonpaired neighbors, meaning similar input patterns associated with different crystals. The resulting crystal and PXRD encoders transfer more effectively to downstream materials and crystallographic tasks. These results identify information asymmetry as a general design problem in scientific multimodal learning: alignment objectives should reflect what each modality preserves.

1 Introduction

PXRD patterns preserve less structural information than crystal descriptions, making paired views unequally informative. PXtal addresses this alignment problem with adaptive transport and evaluates retrieval and encoder transfer.

1 Introduction

  • PXRD patterns collapse three-dimensional crystal structure information into aggregate intensities, complicating links between observed patterns and their underlying structures.Conventional analysis narrows candidate phases through database search, then uses indexing and Rietveld refinement.Source: §1 · pp. 1–2
  • Earlier PXRD–crystal models use contrastive embeddings to guide structure generation, while PXRDnet generates candidates; the joint representation problem remains the focus here.InfoNCE assigns every pair the same target confidence and treats all nonpaired batch members as negatives, regardless of pattern similarity.Source: §1 · p. 2
  • PXtal learns shared-space PXRD and crystal encoders using a batch transport coupling whose marginals adapt to competing matches.PXtal-UOTGKL directly supervises the full coupling with generalized KL divergence; PXtal-UFGOT also models within-modality relations.Source: §1 · p. 2
  • The evaluation tests PXRD-to-crystal retrieval, crystal-encoder prediction of materials properties, and PXRD-encoder prediction of crystallographic labels and geometry.These tasks connect retrieval to database search and test encoder transfer to materials and crystallographic properties.Source: §1 · p. 2

Contributions

  • PXtal improves retrieval most for patterns with close but crystallographically distinct nonpaired neighbors and transfers both encoders to downstream tasks.The study uses up to 2.05 million pairs and evaluates retrieval on six test sets, including four zero-shot transfer sets.Source: §1 · p. 2

2 Related Work and Preliminaries

The section reviews contrastive multimodal learning and optimal transport as foundations for cross-modal alignment. It also situates information asymmetry among representation-level, behavioral, and semantic issues.

Contrastive Multimodal Learning

  • Contrastive learning uses objectives such as InfoNCE for paired observations, while later work scales multimodal pretraining or changes its objective.CLIP combines cross-modal InfoNCE with its reverse direction for image–text learning.Source: Contrastive Multimodal Learning · pp. 2–3
  • Prior analyses describe modality gaps, information imbalance, concept association bias, and weak binding of objects to attributes, relations, and order.These address representation-level, behavioral, and semantic manifestations of information asymmetry.Source: Contrastive Multimodal Learning · p. 3

Optimal Transport and Inverse Alignment

  • Optimal transport compares weighted discrete measures through couplings with prescribed marginals; entropic OT adds an entropy term and uses Sinkhorn scaling.Inverse optimal transport learns costs from observed assignments, and related contrastive implementations supervise normalized couplings with conditional cross entropy.Source: Optimal Transport and Inverse Alignment · p. 3

3 Methods

PXtal aligns paired PXRD patterns and crystal structures through batch-level transport, with unbalanced coupling and coupling-level GKL as its primary objective. It also evaluates balanced and relational variants.

3 Methods

  • Across the method variants, PXtal moves from symmetric InfoNCE to balanced or unbalanced batch coupling, with the relational extension adding within-modality structure.PXtal-UOTCE supervises normalized conditionals, while PXtal-UOTGKL supervises the full measure.Source: §3 · p. 3

Common Setup

  • The common setup uses paired PXRD patterns and crystal structures encoded as embeddings, with a diagonal target coupling and uniform prior marginals unless stated otherwise.PXtal-CLIP applies symmetric InfoNCE to the embeddings.Source: Common Setup · p. 3

Balanced Optimal Transport: PXtal-OT

  • PXtal-OT replaces separate row and column normalizations with one balanced batch coupling under prescribed uniform marginals.It provides a balanced reference for testing whether sample-dependent marginal mass improves alignment.Source: Balanced Optimal Transport: PXtal-OT · p. 3

Unbalanced Optimal Transport Coupling

  • UOT treats the batch plan as a nonnegative measure with variable marginals and total mass, penalized against prior marginals.PXtal-UOTCE applies conditional cross-entropy after normalizing the plan, whereas PXtal-UOTGKL applies an outer GKL loss to the full measure.Source: Unbalanced Optimal Transport Coupling · p. 3

Relational Extension: PXtal-UFGOT

  • PXtal-UFGOT adds within-modality relational geometry to coupling-level GKL, rewarding couplings that preserve cross-modal relational neighborhoods.The method unrolls a fixed number of log-domain updates and is used when batch geometry is informative at scale; PXtal-UOTGKL remains the default direct coupling objective.Source: Relational Extension: PXtal-UFGOT · pp. 3–4

From Conditional CE to GKL at the Coupling Level

  • Coupling-level GKL retains marginal supervision that conditional cross-entropy loses when UOT allows marginals to vary.The objectives agree when marginals are fixed; under finite UOT penalties, conditional cross-entropy leaves the marginal GKL term unsupervised.Source: From Conditional CE to GKL at the… · p. 4

4 Experimental Setup

The experiments pretrain PXRD and crystal encoders on MP-100 and CrystaLLM-v1, then assess retrieval and downstream transfer across in-domain and external datasets. The setup compares PXtal objectives with external retrieval baselines where checkpoints are available.

Datasets

  • Training uses MP-100 and CrystaLLM-v1, which contains 2,285,702 crystal structures paired with simulated PXRD patterns.The four transfer sets cover distinct classes of crystalline materials, and structures shared with the CrystaLLM-v1 pretraining split were removed before evaluation.Source: §4 · pp. 4–5

Models and optimization

  • MP-100 pretraining pairs a PXRD Transformer with CSPNet, while CrystaLLM-v1 pretraining uses the same PXRD encoder with GemNet-T at batch size 8192.The CrystaLLM-v1 pretraining comparison uses matched time conditions across objectives.Source: §4 · p. 5, Table 2 · p. 5

Tasks and evaluation protocols

  • Evaluation covers PXRD-to-crystal retrieval and downstream encoder transfer to five Matbench properties, symmetry, and unit-cell prediction.Retrieval ranks a fixed crystal library by cross-modal cosine similarity.Source: §4 · p. 5

Baselines and ablations

  • External retrieval baselines are evaluated on MP-100, while CrystaLLM-v1 comparisons match architecture, training data, and optimization across PXtal objectives.The CrystaLLM-v1 comparison includes matched InfoNCE PXtal-CLIP; Tables 2 and 3 describe in-domain and zero-shot retrieval evaluation, respectively.Source: §4 · p. 5, Table 2 · p. 5, Table 3 · p. 5

5 Results

PXtal-UOTGKL leads retrieval on both pretraining domains and all four zero-shot transfer sets, with larger gains in crowded PXRD neighborhoods. Its pretrained crystal and PXRD encoders also transfer to downstream tasks.

PXRD Neighborhood Structure and Local Ranking

  • From Q1 to Q4, closer nonpaired PXRD neighbors have lower space-group and crystal-system label entropy, yet Q4 neighborhoods average 4.36 space-group and 2.43 crystal-system labels.Figure 2 defines Q1–Q4 by increasing average cosine similarity to the ten nearest nonpaired patterns.Source: Figure 2 · p. 6, PXRD Neighborhood Structure and Local… · pp. 6–7
  • At zero margin in Q4, PXtal-UOTGKL ranks the paired crystal above every neighboring candidate for 70.74% of patterns, versus 65.03% for PXtal-OT and 61.34% for PXtal-CLIP.Positive-margin fractions decrease from Q1 to Q4 for all three models, linking greater neighborhood similarity to more local ranking reversals.Source: Figure 2 · p. 6, PXRD Neighborhood Structure and Local… · p. 7

Crystal and PXRD Encoder Transfer

  • Every PXtal-pretrained crystal encoder initialization improves all six Matbench metrics over training from scratch, although ALIGNN and coGN lead on most metrics.Table 4 reports five-fold means for five Matbench v0.1 tasks; PXtal rows fine-tune GemNet-T.Source: Table 4 · p. 6, Crystal and PXRD Encoder Transfer · p. 7

PXRD-to-Crystal Candidate Retrieval

  • 74.45% exact Acc@1 on MP-100 and 98.82% on CrystaLLM-v1: PXtal-UOTGKL leads every reported retrieval metric on both pretraining-domain test sets.On MP-100, it also raises space-group Acc@1 from 81.41% to 86.73% over PXtal-CLIP.Source: PXRD-to-Crystal Candidate Retrieval · p. 6
  • On all four zero-shot transfer sets, PXtal-UOTGKL leads PXRD-to-crystal Acc@1; on JARVIS-DFT-3D, it raises Acc@1 from 61.66 to 70.18 over PXtal-CLIP.It also raises JARVIS space-group Acc@1 from 72.30 to 82.18, and leads on QMOF and Perov-5 despite their distribution and material-class shifts.Source: PXRD-to-Crystal Candidate Retrieval · p. 6

Crystal and PXRD Encoder Transfer

  • Every pretrained PXRD Transformer outperforms training from scratch; PXtal-UFGOT and PXtal-UOTGKL lead probe results, while PXtal-UOTGKL improves most displayed fine-tuning metrics over PXtal-CLIP.Table 5 distinguishes frozen-encoder probes from fine tuning on JARVIS-DFT-3D.Source: Table 5 · p. 7, Crystal and PXRD Encoder Transfer · p. 7

PXRD Neighborhood Structure and Local Ranking

  • In Q4, PXtal-UOTGKL reduces local ranking reversals by 9.40 points and improves full-library Acc@1 by 10.32 points over PXtal-CLIP.Acc@1 and MRR decrease from Q1 to Q4 for every model, while PXtal-UOTGKL leads each quartile and its advantage over PXtal-CLIP grows toward Q4.Source: PXRD Neighborhood Structure and Local… · p. 7

6 Conclusion and Future Work

PXtal treats PXRD–crystal representation learning as alignment under physical information asymmetry, and adapts transport coupling to how strongly patterns constrain structures.

Conclusion

  • PXtal addresses physically imposed information asymmetry by adapting transport coupling to how strongly each PXRD pattern constrains its structure.The conclusion reports improved PXRD-to-crystal retrieval, especially for patterns with close but crystallographically distinct neighbors, and transfer to downstream tasks.Source: §6 · p. 7

Future Work

  • The authors identify measured, multiphase, and open-set PXRD as future extensions of the approach.The conclusion also suggests that scientific multimodal alignment objectives should reflect what each modality preserves.Source: §6 · p. 7

Impact Statement

The paper aims to improve representation learning between PXRD patterns and crystal structures for materials characterization and discovery.

Impact Statement

  • The stated aim is better representation learning between powder X-ray diffraction patterns and crystal structures for materials characterization and discovery.The impact statement identifies the main positive impact as scientific.Source: Impact Statement · p. 8
  • Better PXRD–crystal alignment may make database search, structure retrieval, and downstream materials-property modeling more data-efficient, especially when diffraction evidence is ambiguous.These are stated as potential impacts, not measured outcomes in the impact statement.Source: Impact Statement · p. 8
  • In the long term, such tools may help researchers navigate large chemical and structural spaces, reduce trial-and-error in materials screening, and accelerate identification of candidate functional materials.The impact statement frames these as prospective benefits.Source: Impact Statement · p. 8

Supplementary Material

The supplementary material documents the GSAS-II diffraction forward model and simulation protocol, then proves how coupling-level GKL relates to conditional cross entropy and changing row mass.

Supplementary Material

  • The appendices expand the PXRD formulation, prove the main-text GKL decomposition, add retrieval and encoder results, and document data, models, training, solvers, and algorithms.Appendix A covers the formulation and proof; Appendices B–E cover results and reproducibility details.Source: Supplementary Material · p. 12

GSAS-II Crystallographic Forward Model

  • GSAS-II builds the simulated powder profile from CIF-derived crystal information, allowed reflections, structure factors, and instrument-dependent peak profiles.The compact main-text expression absorbs scale, volume, multiplicity, and correction factors into the reflection weight.Source: GSAS-II Crystallographic Forward Model · pp. 12–13

GSAS-II Simulation Protocol

  • The simulations use fixed GSAS-II settings and deterministic calculated profiles, with dataset-specific angular grids and MP-100 resampling before normalization.The protocol specifies the instrument parameters, crystallite size and microstrain, and defaults for missing CIF occupancy or displacement values.Source: GSAS-II Simulation Protocol · pp. 13–14

Formal Proof of Proposition 3.1

  • Conditional cross entropy is invariant to positive row rescaling, whereas coupling-level GKL changes with row mass and has a unique mass-matching minimizer at η_i=r_i/u_i.When target and predicted marginals match, coupling-level GKL and weighted soft cross entropy differ only by a target-dependent constant and have the same gradients with respect to P.Source: Formal Proof of Proposition 3.1 · pp. 15–16

B Additional Retrieval Results

The supplementary MP-100 retrieval table adds results for Matformer and GemNet-T crystal encoders alongside CSPNet results reported in the main text.

B Additional Retrieval Results

  • Table 7 adds MP-100 retrieval results for Matformer and GemNet-T crystal encoders, complementing the main text’s CSPNet results.The table identifies the best and second-best models within each encoder.Source: B Additional Retrieval Results · p. 16, Table 7 · p. 16
  • The main text gives the corresponding CSPNet MP-100 retrieval results, while Table 7 supplies the additional crystal-encoder comparisons.The table’s rankings are determined within each encoder.Source: B Additional Retrieval Results · p. 16, Table 7 · p. 16
  • Table 7 reports Acc@1/3/5 and MRR retrieval metrics, plus crystal-system Acc@1 and space-group Acc@1/3, all in percent.The table marks best and second-best models within each encoder.Source: Table 7 · p. 16

C Transfer, Neighborhood, and Encoder Results

The supplementary tables document zero-shot retrieval, neighborhood-quartile analysis, PXRD encoder probes, and crystal-encoder transfer evaluation.

C Transfer, Neighborhood, and Encoder Results

  • Tables 8 and 9 report zero-shot PXRD-to-crystal retrieval on JARVIS-DFT-3D, Alex-MP-20, QMOF, and Perov-5 without target-set training or fine tuning.Metrics are reported in percent.Source: Table 8 · p. 17, Table 9 · p. 17
  • Table 10 divides JARVIS-DFT-3D PXRD neighborhoods into similarity quartiles, with Q1 lowest and Q4 highest, for retrieval-metric analysis.The main text uses the listed Acc@1/5/10 and MRR values in its quartile analysis.Source: Retrieval across PXRD Neighborhoods · p. 16, Table 10 · p. 17
  • Table 11 reports MP-100 PXRD encoder probes for symmetry classification and unit-cell regression, grouped by the crystal encoder used in joint pretraining.The main text defines the downstream tasks and probe and fine-tuning protocols.Source: Retrieval across PXRD Neighborhoods · p. 16, Table 11 · p. 18
  • Table 12 reports five-fold Matbench crystal-encoder transfer metrics, including MAE, AUROC, F1 for metal classification, and RMSE for four regression tasks.Values are means and standard deviations over the official folds.Source: Retrieval across PXRD Neighborhoods · p. 16, Table 12 · p. 18

D Reproducibility Details

The reproducibility details describe dataset selection and pattern simulation, followed by encoder configurations, training settings, and software.

D Reproducibility Details

  • MP-100 selects inorganic structures with at most 100 atoms per unit cell, an ICSD reference, and energy above the convex hull below 0.25 eV/atom.The split uses random seed 42; GSAS-II simulates the patterns using the described single-wavelength protocol.Source: D Reproducibility Details · pp. 18–19
  • CrystaLLM-v1 uses supplied CIF split manifests, simulates each raw CIF with GSAS-II, and obtains crystal metadata from source records or standardized structures.The source record provides space-group and crystal-system metadata when available; otherwise pymatgen infers them.Source: D Reproducibility Details · p. 19

Model and Training Details

  • The PXRD encoder is a dense-patch Transformer, paired on MP-100 with CSPNet, Matformer, or GemNet-T crystal encoders and mapped to a shared normalized 512-dimensional space.The configurations vary by dataset and crystal encoder, including layers, patch sizes, graph layers, and pooling choices.Source: Model and Training Details · p. 19
  • The supplementary details specify the software environment and pretraining settings, including optimizer, schedule, warmup, precision, gradient clipping, and validation cadence.Training and evaluation used Linux nodes with NVIDIA GH200 120GB Grace Hopper Superchips.Source: Model and Training Details · p. 19

E Optimal Transport Algorithms and Implementation

PXtal implements balanced and unbalanced transport variants with a unified proximal solver, using cross-modal costs and, for fused variants, a score that preserves within-modality similarities. The variants differ in whether they relax or enforce coupling marginals and in how they supervise the transport plan.

E Optimal Transport Algorithms and Implementation

  • The unified solver takes normalized PXRD and crystal embeddings and supports UOT, UFGOT, and FGOT, with fused variants combining cross-modal cosine distance and within-modality similarity preservation.The bilinear score rewards couplings that preserve within-modality cosine similarities; the outer-step cost also includes cross-modal cosine distance.Source: E Optimal Transport Algorithms and… · p. 20

Balanced Sinkhorn Scaling and PXtal-OT

PXtal-UOT and PXtal-UFGOT

  • PXtal-UOTCE applies conditional cross-entropy after row normalization, whereas PXtal-UOTGKL applies GKL to the full coupling and penalizes its marginal mass.Both use the same log-domain generalized Sinkhorn solver and relaxed row and column marginals.Source: PXtal-UOT and PXtal-UFGOT · p. 21

PXtal-FGOT

  • PXtal-FGOT uses the fused score with hard prescribed marginals, while PXtal-UFGOT uses unbalanced marginal penalties; one FGOT outer step with ρ = τprox = 0 recovers PXtal-OT.FGOT approximates each proximal subproblem with a finite K-step Sinkhorn solve; the variants differ in their inner marginal constraints.Source: PXtal-FGOT · pp. 21–22
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