BDIP-Net: Dual-Interaction Graph Learning for Property Prediction of Bilayer Materials

1 Introduction

• Stacking configuration, interlayer distance, lattice mismatch, and twist angle can change bilayer properties and produce electronic and optical properties absent from isolated monolayers [11].DFT with dispersion corrections such as DFT-D3 is commonly used to optimize bilayer structures and evaluate their properties, but multi-stage optimization is computationally expensive and limits large-scale screening and dataset construction.Source: §1 · p. 1
• MatterSim-D3 replaces costly DFT-D3 structural optimization, producing optimized bilayers at a fraction of its computational cost while maintaining comparable structural quality.The workflow uses MatterSim with DFT-D3 dispersion corrections as an MLIP-D3 surrogate.Source: §1 · p. 1
• BDIP-Net separates intra-layer and inter-layer edges, encodes them with Coulomb-based and London-dispersion-based potentials, and fuses their messages through interaction-message attention.This distinguishes strong intra-layer interactions from weak inter-layer van der Waals interactions while extending PotNet’s potential-based framework to stacked bilayers.Source: §1 · p. 1
• Across BiDB, HetDB, and SAMBA, BDIP-Net consistently outperforms BiMat-ML, conventional bilayer-based baselines, PotNet, and SE-PotNet.MatterSim-D3 also achieves nearly identical predictive performance to DFT-PBE-D3, supporting the proposed efficient optimization workflow.Source: §1 · pp. 1–2
• The framework covers BiDB homobilayers, HetDB van der Waals heterobilayers, and SAMBA twisted homo- and heterobilayers, which contains over 18,000 structures.The databases are built on the Computational 2D Materials Database (C2DB).Source: §1 · p. 1

2 Related Work

• BDIP-Net extends PotNet’s potential-based message passing with separate intra-layer and inter-layer interaction graphs, interaction-specific potential encodings, and interaction-message attention.Unlike PotNet [12], which uses a unified representation for local interactions, BDIP-Net distinguishes strong intra-layer bonding from weak inter-layer van der Waals coupling.Source: §2 · p. 2
• Crystal GNNs such as SchNet, CGCNN [21], MEGNet [2], and OrbNet [16] commonly use RBF-expanded interatomic distances and message passing for property prediction.PotNet [12] instead transforms distances into potential values for message passing, with experiments indicating that physically informed interaction modeling can enhance crystal property prediction.Source: §2 · p. 2
• SE-PAINN separates intra-layer and inter-layer edge sets but depends on computationally intensive optimized structures, whereas BiMat-ML [19] avoids structure construction but cannot learn stacked-geometry atomic interactions directly.The proposed work combines efficient MLIP-based structural optimization with interaction-aware graph learning on reconstructed bilayers.Source: §2 · p. 2

3.1 MatterSim-D3-based Bilayer Structure Construction

• MatterSim-D3 supplies energies and forces for z-scan, optional xy-scan, and relaxation across all datasets, including SAMBA.The initial stack applies dataset-specific 3 × 3 transformations to each layer and translates the top layer. Prior results [17] report best overall performance among evaluated MLIP-D3 models for reproducing DFT-PBE-D3 structures, with about 0.02 eV average difference in bandgap calculations.Source: §3.1 · pp. 2–3

3.2 BDIP-Net For Bilayer Property Prediction

• BDIP-Net prepares interaction-specific graph inputs and performs message passing with separate intra-layer and inter-layer aggregation followed by adaptive fusion.The model predicts bilayer properties from structurally optimized bilayers.Source: §3.2 · p. 3, Figure 2 · p. 4

3.2.1 Graph-Level Input Preparation for Node Message Passing.

• BDIP-Net builds separate adaptive-neighbor intra-layer and inter-layer edge sets, encoding them with local Coulomb and London potentials.The local potential features retain most of the performance improvement over raw distances, while infinite potential features add only modest improvement; the model therefore uses local features. The potentials are −εCoulomb/dij and −εLondon/(dij)^6.Source: §3.2.1 · pp. 3–4

3.2.2 Message Passing and Property Prediction.

• Across T layers, edge attention aggregates intra- and inter-layer messages separately, then interaction-message attention fuses them; mean pooling and an output MLP predict properties under MAE training.The two interaction-message weights are learned to determine each interaction type’s contribution to node updates.Source: §3.2.2 · p. 5

4.1 Experiment Settings.

• Experiments use bandgap prediction on BiDB, HetDB, and SAMBA, including aligned homo- and heterobilayers and twisted bilayers.Four-fold cross-validation splits at the monolayer-pair level, so bilayers from a given pair remain in one fold and test generalization to unseen pairs.Source: §4.1 · pp. 5–6

4.2.1 Prediction Performance.

• BDIP-Net achieves the best MAE, MSE, RMSE, and R^2 across BiDB, HetDB, and SAMBA, significantly outperforming the baselines.Against PotNet, BDIP-Net raises R^2 from 0.65 to 0.73 on BiDB, 0.89 to 0.93 on HetDB, and 0.73 to 0.83 on SAMBA; paired tests across 12 observations yield p<0.01 for all baseline comparisons.Source: Table 1 · p. 6, §4.2.1 · p. 6, Table 2 · p. 6

4.2.2 Structural Optimization and Its Effect on Prediction.

• Replacing DFT-PBE-D3-optimized inputs with MatterSim-D3-optimized structures causes only minor prediction variations, with comparable performance at substantially lower optimization cost.MatterSim-D3 preserves structural information required for bandgap prediction; DFT-PBE-D3 structural optimization typically takes hours.Source: §4.2.2 · p. 6

4.2.3 Ablation Study.

• BDIP-Net outperforms the ablated BDIP-Net− on all three datasets, with R^2 increasing from 0.69 to 0.73 on BiDB, 0.91 to 0.93 on HetDB, and 0.75 to 0.83 on SAMBA.The ablation replaces the proposed message passing with PotNet-style feature-wise sigmoid gates and removes scalar-softmax interaction-message attention in favor of direct summation; improvements are significant across metrics (p<0.01).Source: Table 3 · p. 7, §4.2.3 · p. 7

4.2.4 Runtime Analysis and Computational Efficiency.

• MatterSim-D3 structural optimization takes an average of 4.5 seconds per BiDB material, 3.5 seconds per HetDB material, and 13.5 seconds per SAMBA material.SAMBA takes longest because its twisted-bilayer workflow adds full z-scan, xy-scan, and relaxation; all methods have approximately 0.05 s per-sample inference time, while BDIP-Net training is modestly slower than several baselines.Source: §4.2.4 · p. 7

4.2.5 Cross-dataset Domain Generalization.

• On the 980-bilayer SAMBA test set, BDIP-Net outperforms PotNet with MAE 0.27, MSE 0.16, RMSE 0.40, and R^2 0.40 versus 0.34, 0.22, 0.47, and 0.15.After excluding 121 exact-pair-overlap bilayers, BDIP-Net achieves MAE 0.20, RMSE 0.29, and R^2 0.54 on the remaining 326 two-monolayer-overlap bilayers, versus 0.29, 0.40, and 0.12 for PotNet. In the HetDB-to-SAMBA setting, BDIP-Net also generalizes better to unseen combinations of known monolayers.Source: Table 4 · p. 7, §4.2.5 · pp. 7–8, Figure 4 · p. 8

• BDIP-Net consistently outperforms baseline models on BiDB, HetDB, and SAMBA under both structural optimization settings.Source: §5 · p. 8
• MatterSim-D3-optimized structures yield prediction performance nearly identical to corresponding DFT-optimized structures, while the workflow is significantly faster than full-DFT optimization.Source: §5 · p. 8
• The workflow constructs and efficiently optimizes bilayers with MatterSim-D3, then uses reconstructed bilayer graphs to model intra-layer and inter-layer interactions.BDIP-Net separates the interaction types and fuses their messages through an interaction-message attention module.Source: §5 · p. 8
• On unseen bilayer combinations formed from monolayers observed during training, BDIP-Net performs more strongly than PotNet; prediction is harder when either target monolayer is absent from training.Source: §5 · p. 8

• Public computational materials databases may carry biases, inaccuracies, and limited material diversity that affect model generalization.Source: Limitations and Ethical Considerations · pp. 8–9
• The framework’s predictions should complement, not replace, high-fidelity first-principles calculations and experimental validation.Source: Limitations and Ethical Considerations · p. 9
• The study involves no human participants or personal data, so privacy and informed-consent issues are not applicable.Source: Limitations and Ethical Considerations · p. 9

Generative AI Usage

• The authors used ChatGPT solely for language editing; they report that all scientific content and analyses were developed and verified by the authors.They state that generative AI was not used for research conception, experimental design, data analysis, interpretation, or scientific-content generation.Source: Generative AI Usage · p. 9

A Background on Stacked 2D Materials Representation

• Crystals are represented by atomic features, Cartesian positions, and lattice vectors; stacked bilayers combine layer structures after selecting an interlayer distance and relaxing atomic positions.For BiDB, the top monolayer may be rotated, translated, or flipped before stacking; the bilayer lattice includes layer thicknesses, interlayer spacing, and vacuum.Source: A Background on Stacked 2D Materials… · p. 10

B Initial Bilayer Structure Construction

• The appendix describes how initial stacked bilayer structures are constructed for BiDB, HetDB, and SAMBA before structural optimization.Source: B Initial Bilayer Structure Construction · p. 10

B.1 BiDB

• BiDB construction copies a monolayer into bottom and top layers, leaves the bottom unchanged, applies the specified stacking transformation and translation to the top, and combines their coordinates and features.The initial bilayer thickness is set from twice the monolayer thickness plus the chosen interlayer distance.Source: §B.1 · pp. 10–11

B.2 HetDB

• HetDB construction first searches for compatible supercells, then builds a heterogeneous bilayer; candidates are filtered by geometric and strain constraints and ranked by atom count, mismatch, and twist angle.The workflow fixes in-plane translations at (0, 0) and considers either both layers unflipped or both flipped.Source: §B.2 · pp. 11–12
• For HetDB, top and bottom supercells share a common in-plane lattice, and translating the top layer produces the combined initial bilayer.The bilayer’s out-of-plane length includes vacuum, interlayer distance, and both monolayer thicknesses.Source: §B.2 · pp. 13–14

B.3 SAMBA

• SAMBA retains supercell pairs meeting area, angle, and lattice-vector compatibility constraints, then removes duplicate bilayers by RMSD-based filtering.The filtering yields a non-duplicated set of initial SAMBA bilayers.Source: §B.3 · p. 14
• SAMBA keeps the out-of-plane direction unchanged and uses the specified interlayer distance with zero in-plane translation when constructing each candidate bilayer.The layers are placed in the bilayer lattice and combined into the initial structure.Source: §B.3 · pp. 14–15

C Structural Optimization

• MatterSim-D3 scans interlayer distance, optionally selects lateral registry, then relaxes the chosen bilayer until its dataset-specific force threshold is met.For SAMBA, the lateral scan follows the distance scan; other general workflow descriptions specify fixed in-plane translations during the z-scan.Source: C Structural Optimization · p. 15
• BiDB and HetDB use coarse-to-fine z-scans with 0.25 Å and 0.05 Å steps, but differ in scan start distance and final force threshold.BiDB uses a 5.0 Å start, total MatterSim-D3 energy without electron normalization, and a 0.01 eV/Å force threshold; HetDB starts at 10.0 Å and uses 0.05 eV/Å.Source: C Structural Optimization · pp. 15–16
• SAMBA uses a 0.5 Å coarse and 0.05 Å fine z-scan, then selects the lowest-energy translation from a 9 × 9 in-plane grid before relaxation.The final relaxation stops when the maximum atomic force falls below 0.01 eV/Å.Source: C Structural Optimization · p. 16
• Layer translation separates atoms at the midpoint z threshold, keeps the bottom layer fixed, and applies the required translation to the top layer.The same procedure supports z-scans and lateral scans, with in-plane translations fixed during the former and varied during the latter.Source: C Structural Optimization · p. 16

D.1 BDIP-Net Algorithm

• BDIP-Net constructs separate intra- and inter-layer edge sets and potential-based embeddings, then aggregates interaction-specific messages and adaptively fuses them into residual node representations.The algorithm uses Coulomb-based and London-based potentials for the respective edge embeddings.Source: §D.1 · pp. 17–18

D.2 Interaction-Specific Radial Basis Function Expansions

• BDIP-Net separately expands Coulomb and London potential values with RBFs over distances from 1 to 8, then maps the features to edge embeddings with independently parameterized networks.Source: §D.2 · p. 18

D.3 Adaptive Layer Neighbor Search

• Adaptive neighbor search filters neighbors by same-layer or cross-layer membership, keeps the complete shell at the N-th valid-neighbor distance, and doubles the cutoff until every atom has enough neighbors.The resulting interaction-specific neighbor lists define the edge set.Source: §D.3 · p. 18

E.1 Datasets

• The three datasets span aligned homobilayers, heterobilayers, and twisted bilayers, supporting evaluation of structural optimization and BDIP-Net across complementary bilayer systems.BiDB, HetDB, and SAMBA cover these respective system types.Source: §E.1 · p. 19
• 6,683 BiDB homobilayers from 940 unique monolayers have valid bandgap targets after CIF and bandgap filtering.The processed dataset initially contained 10,899 valid bilayer structures after removing 250 bilayers tied to 10 monolayers lacking CIF files.Source: §E.1 · p. 19
• 336 HetDB heterobilayers formed from 38 distinct monolayers have bandgap targets, with no repeated monolayer pair under different stacking configurations.The dataset follows Bimat-ML [19].Source: §E.1 · p. 19
• SAMBA contains more than 18,000 generated twisted bilayers, but only 980 have DFT structural optimization and property calculations, all with bandgap targets.The passage reports the DFT-calculated subset as 144 homogeneous and 836 heterogeneous bilayers; the dataset statistics passage gives 146 and 834, respectively.Source: §E.1 · p. 19
• Mean bandgaps are 0.93 eV for BiDB, 0.76 eV for HetDB, and 0.66 and 0.28 eV for SAMBA homo- and heterobilayers, respectively.SAMBA heterobilayers are concentrated at lower bandgaps, while its homobilayers cover a wider bandgap range.Source: §E.1 · p. 19
• With a 0.2 eV closeness threshold, most HetDB and SAMBA heterobilayers close to a monolayer bandgap are close to the smaller monolayer gap.HetDB has 149 samples close to the smaller gap and none close to the larger; among 605 SAMBA samples close to a monolayer gap, 603 are close to the smaller and 2 to the larger.Source: §E.1 · p. 19
• Bandgap distributions and bilayer–monolayer relationships vary across datasets: 76 HetDB and 435 SAMBA heterobilayers lie within the interval between their monolayer gaps.The remaining 260 HetDB and 399 SAMBA heterobilayers lie outside that interval; 75 SAMBA homobilayers are close to their monolayer gap and 71 are not.Source: §E.1 · p. 19

E.2 Running Time

• 575 s for BDIP-Net training on BiDB, versus 550 s for PotNet and 475 s for SE-MEGNET.BDIP-Net was slower than SE-CGCNN (400 s), SE-PAINN (300 s), and BiMat-ML (285 s).Source: Figure 6 · p. 19, §E.2 · p. 19
• 40 s for BDIP-Net training on HetDB, compared with 35 s for PotNet and 25 s for SE-MEGNET.SE-CGCNN required 20 s, while SE-PAINN and BiMat-ML each required 12 s.Source: Figure 6 · p. 19, §E.2 · p. 19
• 80 s for BDIP-Net training on SAMBA, compared with 70 s for PotNet and 65 s for SE-MEGNET.SE-CGCNN required 60 s, SE-PAINN 50 s, and BiMat-ML 48 s.Source: Figure 6 · p. 19, §E.2 · p. 19
• Approximately 0.05 s is the average per-sample inference time across all datasets and models.Training and inference runtimes are nearly identical for MatterSim-D3- and DFT-PBE-D3-optimized structures.Source: §E.2 · p. 19
• A few seconds for MatterSim-D3 structure generation contrasts with the hours typically required for DFT-PBE-D3 optimization from the same initial bilayer.This difference in structure-generation time makes the MatterSim-D3-based BDIP-Net pipeline computationally practical for bilayer property prediction.Source: §E.2 · pp. 19–20

E.3 Cross-Domain Generalization

• 63 unique monolayers appear in SAMBA, with (B) marking overlap with BiDB and (B, H) marking overlap with both BiDB and HetDB.These annotations identify monolayer overlap between training and test datasets in the cross-domain generalization analysis.Source: §E.3.1 · p. 20, §E.2 · p. 20

E.3.1 Monolayer Overlap.

• The 980-bilayer SAMBA test set is divided into 626 no-overlap, 284 one-overlap, and 70 two-overlap cases, including 36 exact-pair overlaps.Both models are trained on 336 HetDB heterobilayers with a learning rate of 2 × 10−4 and batch size 128; 13 HetDB monolayers overlap with SAMBA.Source: §E.3.2 · p. 20

Prediction Performance

• On 980 SAMBA bilayers, PotNet beats BDIP-Net: MAE 0.50, MSE 0.39, RMSE 0.63, and 𝑅2 −0.50 versus 0.56, 0.50, 0.71, and −0.93.This is the full SAMBA test set.Source: Table 8 · p. 20, §E.3.2 · p. 20
• Both models improve as monolayer overlap increases; PotNet has lower MAE on 626 no-overlap bilayers (0.57 vs. 0.65) and 284 one-overlap bilayers (0.43 vs. 0.46).Figures 7 and 8 show the overlap breakdown and absolute-error distributions.Source: §E.3.2 · p. 20, Figure 7 · p. 20, Figure 8 · p. 20
• Among 70 two-overlap bilayers, BDIP-Net has lower MAE than PotNet (0.12 vs. 0.15), while PotNet edges it on 36 exact-pair overlaps (0.07 vs. 0.10).Excluding exact-pair overlaps, BDIP-Net scores MAE 0.14, RMSE 0.18, and 𝑅2 0.98 on the remaining 34 bilayers, versus 0.24, 0.43, and 0.86 for PotNet.Source: §E.3.2 · p. 21
• Despite PotNet’s overall lead, BDIP-Net generalizes better to unseen bilayer combinations of known monolayers; overall performance is limited because most SAMBA bilayers have no or one monolayer overlap.Source: §E.3.2 · p. 21