Beyond Structure: Invariant Crystal Property Prediction with Pseudo-Particle Ray Diffraction
Authors: Bin Cao
Yang Liu
Longhan Zhang
Yifan Wu
Zhixun Li
Yuyu Luo
Hong Cheng
Yang Ren
Tongyi ZHANG
Venue: ICLR 2026 Poster (2026)
Keywords: Crystal representation
Invariance
Pseudo-Particle
Diffraction
Reciprocal space
Long-term interaction
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: Pseudo-Particle Ray Diffraction for Crystal Property Prediction

Abstract: Crystal property prediction, governed by quantum mechanical principles, is computationally prohibitive to solve exactly for large many-body systems using traditional density functional theory. While machine learning models have emerged as efficient approximations for large-scale applications, their performance is strongly influenced by the choice of atomic representation. Although modern graph-based approaches have progressively incorporated more structural information, they often fail to capture long-range atomic interactions due to finite receptive fields and local encoding schemes. This limitation leads to distinct crystals being mapped to identical representations, hindering accurate property prediction. To address this, we introduce PRDNet that leverages unique reciprocal-space diffraction besides graph representations. To enhance sensitivity to elemental and environmental variations, we employ a data-driven pseudo-particle to generate a synthetic diffraction pattern. PRDNet ensures full invariance to crystallographic symmetries. Extensive experiments are conducted on Materials Project, JARVIS-DFT, and MatBench, demonstrating that the proposed model achieves state-of-the-art performance. The code is openly available at \url{https://github.com/Bin-Cao/PRDNet}.

Published as a conference paper at ICLR 2026
BEYOND STRUCTURE: INVARIANT CRYSTAL PROP- ERTY PREDICTION WITH PSEUDO-PARTICLE RAY DIFFRACTION
Bin Cao-.a, Yang Liu"; Longhan Zhang', Yifan Wu', Zhixun Lie, Yuyu Luo", Hong Cheng?, Yang Ren 2 Tong-Yi Zhang
Guangzhou Municipal Key Laboratory of Materials Informatics, Advanced Materials Thrust,
The Hong Kong University of Science and Technology (Guangzhou)
The Chinese University of Hong Kong
3 Data Science and Analytics Thrust, The Hong Kong University of Science and Technology (Guangzhou) Department of physies, The City University of Hong Kong
Materials Genome Institute, Shanghai University
ABSTRACT
Crystal property prediction, governed by quantum mechanical principles, is com- putationally prohibitive to solve exactly for large many-body systems using tradi- tional density functional theory. While machine learning models have emerged as efficient approximations for large-scale applications, their performance is strongly influenced by the choice of atomic representation. Although modern graph- based approaches have progressively incorporated more structural information, they often fail to capture long-range atomic interactions due to finite receptive fields and local encoding schemes. This limitation leads to distinct crystals being mapped to identical representations, hindering accurate property predic- tion. To address this, introduce PRDNet that leverages unique reciprocal- space diffraction besides graph representations. To enhance sensitivity to ele- mental and environmental variations, employ data-driven pseudo-particle to generate a synthetic diffraction pattern. PRDNet ensures full invariance to crystallographic symmetries. Extensive experiments are conducted on Materi- als Project, JARVIS-DFT, and MatBench, demonstrating that the proposed model achieves state-of-the performance. The code is openly available at ht tpss / //github. com Bin- cot.c//m//i-CCC//RDDee...t CFDNe....
INTRODUCTION
Material properties are fundamentally determined by quantum mechanical equations such as the Schrödinger equation (Kohn et al., 1996), which are intractable for many-body systems and thus commonly approximated by Density Functional Theory (DFT). However, the high computational cost of DFT limits its application to large atomic systems. Recently, machine learning (ML) force fields have emerged as scalable surrogates (Chen et al., 2019; Chen & Ong. 2022), trained on DFT data to enable efficient structure relaxation and property prediction across diverse materials. These models achieve near-DFT accuracy while allowing exploration large-scale atomic correlations.
In partieular, the performance of ML models strongly depends on how atomic systems are represented (Xie & Grossman, 2018; Schütt et al., 2017: Chen al., 2019; Louis al., 2020; Choudhary & DeCost, 2021; Yan et al., 2022: 2024; Taniai al.: Ito et al.. 2025). Early efforts relied on hand-engineered descriptors to capture atomic interactions (Isayev et al., 2017;
Crystal
Diffraction
Probe
Su et al., 2024; Zhu al., 2024), which offered interpretabil- ity but suffered from human bias and poor transferability. With the emergence of data-driven models, Graph Neural Networks
Figure I: Relationship between
crystal and reciprocal space.
Corresponding authors (yangliu005@euill @euhk hk, mangren@cityu.cdee mezhangt hkust-gz.edu.cn)

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Figure 2: Representation limitations of projecting different atomic periodicity into the same graph representation using (a) multi-edge graphs, (b) atomic angular embedding, and (c) a periodic vector- based reference system. The gray region represents a lattice cell of the 2D crystal.
have become mainstream crystal encoding models. Specifically, they treat crystals graphs and perform message passing (Chen et al., 2019) to aggregate node information. Initial solutions fo- cus solely on pairwise local interactions, leveraging graph convolution networks (Xie & Grossman, 2018; Chen et al., 2019) and graph attention networks (Louis et al., 2020). To capture higher-order geometry, later works incorporated atomic angular embedding (e.g.. bond angles and multi-body descriptors) (Choudhary & DeCost, 2021; Chen & Ong, 2022), improving structural fidelity but at the expense of efficiency and symmetry invariance. Most recently, equivarian//invarinn GNNs (Yan et al., 2022; 2024; Taniai et al.; Ito et al., 2025) have emerged as the leading paradigm, systematically overcoming prior limitations by embedding symmetry constraints such as message passing, thereby achieving state-of-the-art performance in Crystal Property Prediction (CPP).
However, crystals are, in principle, infinite three-dimensionss atomic systems. Current real-space encoders often fail to capture long-range atomic interactions (Tantivasadakarn et al., 2024), which essential for determining physical properties. In particular, these methods typically struggle to represent them, as real-space embeddings with finite receptive fields in multi-edge graph construc- tions limit their ability to account for long-range effects. As result, they map distinct crystal struc- tures into the same representation. Figure 2 illustrates these examples. In DFT, these interactions are addressed through boundary conditions and supercell configurations (Makkar & Ghosh, 2021). A promising alternative is diffraction space encoding: since every atom contributes to diffraction along specific lattice planes, ideal representation is guaranteed to embed complete real-space information without loss (Bouwkamp, 2016), enabling more faithful modeling of long-range correlations. Fig- ure 1 illustrates the relationship between the crystal graph and its diffraction. Diffraction arises from the periodic arrangement of atoms and their interactions with the probing beam, described by form factors. Traditional methods, such as X-ray diffraction, rely on fixed, tabulated atomic form factors that depend only on the scattering vector and atomic species, deriving from historically collected, averaged quantum mechanical calculations and experimental measurements. This makes them in- capable of distinguishing between atoms of the same element situated in different local chemical environments, as they ignore crucial structural variations that affect material properties.
Reciprocal-space representation is an effective strategy for capturing long-range interactions while remaining compact. This is because the complete diffraction pattern can be analytically derived from single real-space unit cell, without requiring large supercell constructions, due to the trinsic periodicity of crystals. However, achieving a lossless and physically valid reciprocal-space representation requires addressing several key gaps that existing works (Kosmala al., 2023; Lin et al., 2023; Nie et al., 2025) have not systematically resolved. (1) The diffraction result must strictly satisfy the invariance constraints required by crystallographic symmetry. (2) Beyond real physical particles, which are inherently limited by physical laws and therefore not ideal as universal probes, more flexible and learnable probing mechanism needed. (3) The form factor must preserve its full physical dependencies, namely atomic species, local chemical environment, and difffaction-vector dependence, which are often simplified or omitted in prior methods. In addition, diffraction is fundamentally global property of the entire structure, and therefore must be integrated at the modality level rather than treated as atom-wise feature fusion.
To exploit this, introduce novel -pautccl representation, named PRDNet, that serves as a unique probe, learned through neural networks, and shown to be more effective for differentiat- ing atoms and structural environments than conventional physical probes such X-rays, electrons,
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neutrons. Building this idea, we develop multimodal framework that integrates both the graph and diffraction information into the learning process. In addition, we show that the proposed model fulfills the geometric constraints, invariant to rotations, reflections, and translations of crys- tals. Quantitative evaluations on large-scale datasets, including Materials Project, JARVIS-DFI, and MatBench, demonstrate that method consistently surpasses state-of-the-ar models in common crystal property prediction tasks. In summary, our main contributions are as follows:
We highlight the fundamental challenge of crystal representation, noting that existing methods can map distinct crystal structures to the same representation, and clarify the previously omitted premise complete reciprocal peiprocal-space information. To address this, we introduce diffraction-based representations, which are physically guaranteed to be unique.
We propose PRDNet, a novel architecture that integrates graph embeddings with a learned pseudo- particle diffraction module. It generates synthetic diffraction patterns that are invariant to crystal- lographic symmetries.
We extensively evaluate PRDNet multiple large-scale benchmarks, including Materials Project, MARVIS-DFT, and MatBench. Our model achieves state-of-the-arr performance across a wide range of crystal property prediction tasks, demonstrating its effectiveness.
2 RELATED WORK
Current crystal representations can be broadly categorized into three types: (1) multi-edge graph models; (2) atomic angular embedding; (3) periodic vector-based reference system. De- spite their differences, they share a common limitation: dis- tinct crystal structures may be mapped onto identical represen- tations, as shown in Figure 2. Specifically, multi-edge crys- tal graphs, as in CGCNN (Xie & Grossman, 2018), SchNet (Schütt et al., 2017), MEGNet (Chen et al., 2019), and GAT- GNN (Louis al., 2020), define atomic connectivity through cutoff radii, continuous filters, or meessag--passing and attention mechanisms. They are unable to distinguish distinct crystal systems, since angular information not considered (Figure a). To address this, models such as ALIGNN (Choudhary & DeCost, 2021) and M3GNet (Chen & Ong, 2022) incorporate bond-angle features, although restricted cutoffs may still yield identical angular relationships (Figure b). Finally, mul- tiple methods extend beyond lattice vectors by introducing periodic vectors, as in MatFormer (Yan et al., 2022) and (Conforner/eConfrmeee (Yan et al., 2024), or by employing Transformer-based frameworks, such as Crystalformer (Taniai et al.) and Crystalframer (Ito et al., 2025), to capture broader correlations (Figure c). These approaches can satisfy basic crystallographic invariances, but they may still map distinct crystals to the same local neighborhood, particularly when the differences are governed by long-range periodicity or subtle symmetry variations.
Structure Real space
Reciprocal space
Figure 3: Analogy to light diffrac- tion demonstrating how reciprocal space encodes long-range interac- tions.
Recent studies have highlighted the importance of long-range correlations in crystal representations and have sought to capture them through diffraction- or reciprocal-space descriptions. The early attempts aimed to introduce long-range descriptors in reciprocal space capture the long-range interactions in Coulombic and van der Waals potentials, that are often omitted in MI force fields (Yu al., 2022). The framework is straightforward: they apply Ewald summation to construct the reciprocal-space potential, which is then concatenated with real-space features to produce more comprehensive predictions. Following this line of work, models such as EwaldMP (Kosmala al., 2023). PotNet (Lin et al., 2023), and ReGNet (renamed ReciNet in the latest version) (Nie et al., 2025) integrate Ewald summationnFourrer transform into message passing to model long-range in- teractions. However, existing methods often treat Ewald summation as a Fourier-like information fusion and overlook the unique and invariant nature of form factors, which should not be propa- gated through information aggregation across blocks or layers. Physically, form factors are func- tional quantities determined solely by the crystal structure and probe; they are unique, determined, and invariant.
In this work, we construct a complete invariant diffraction representation based on the fusion of structure and spectrum modalities, rather than atomic interaction information infusion, to explicitly encode long-range interactions. The diffraction- description reflects the physical process of probing periodic atomic structures with incident particles. As illustrated by the analogy in Figure 3,
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the diffraction intensity at reciprocal vector Q reflects the aggregated contribution of the entire real-space structure along different diffraction frequencies. That is, the response of an infinitely periodic atomic arrangement along that direction in crystal diffraction. By aggregating neighbor- ing reciprocal vectors in a close group, we obtain multiple directional slices of the structure, which together provide a compact and physically consistent description of long-range interactions. A com- plete reciprocal-space representation depends three key factors: (1) periodic atomic types f (2) diffraction basis QQ. and (3) atomic local environments Gg( (representing charge-density dis- tributions), as detailed in Section 4.3. Existing models such as EwaldMP and PotNet approximate infinite-range potentials but ignore Gg( in EwaldMP and both Gg( and |Q] dependencies in PotNet, while ReGNet similarly applies a Fourier-like transformation in each reciprocal block with- out considering dependencies on Gg(g) and Q). Such omissions can lead to inconsistent reciprocal representations by violating the structure dependence and invariant of form factors, as discussed in Appendix A.3.
3 PRELIMINARIES
Crystal Structures Crystal structures can be described using fundamental unit, such as prim- itive cell, conventional unit cell, Bravais lattice, asymmetric unit, together with its associated atoms. The unit is defined by a 3 x 3 lattice matrix that specifies its geometry in real space. Formally, a crystal structure is represented as M (A, L), where A denotes atomic types, P = Pp..... ,N]l E R3xN contains the fractional or Cartesian coordinates of the N atoms, and [ei, l2, Ez] R3x3 defines the periodic lattice in real space. A crystal is an infinite periodic structure, expressed as
P= (p: p. - - + kili + 12222 + kills, 61, hi EZ, 1 ISiSN,,
(I)
where P denotes the complete set of atomic positions generated by lattice translations. A crystal structure can also be represented as graph g (V.E), where vertices V = {ri, u2,...,, uN en- code the atomic attributes, and edges & C V x represent atomic interactions defined by interatomic distances derived from P and L, V (v: v; G(v.), 1 < i S N} where v contains the shared atomic features atoms occupying symmeery--equivaenn sites under the point group operation G.
Diffraction Diffraction refers to the interaction of probe particles with periodic structures (Cao et al.: Guo et al., 2025). In crystals, diffraction maps the real-space atomic arrangement into mo- mentum (reciprocal) space, where each diffraction feature encodes structural information along spe- cific periodic directions. Since diffraction is uniquely determined by the electron distribution for a given probe, and the electron distribution itself is governed by the potential field arising from the infinite atomic arrangement, diffraction provides a unique representation of the real-space crys- tal structure (Kohn et al., 1996; Cao et al., 2025; 2026). This process is governed by the structure factor (Greenfield et al., 1971), which describes how atoms scatter incident radiation (c.g.. X-rays in XRD). Formally, diffraction is the Fourier transform of the electron density p(r) in real space, giving
F(Q) ((Q==/
p(r) -Q dr,
unit cell
where r is the Cartesian position vector and Q the basis in the reciprocal-space. The electron density p(r) reflects the spatial distribution of electrons and governs the dispersion properties of the crystal.
For a crystal of discrete atoms at positions r with atomic form factors fj, the structure factor reduces to (Hubbell et al., 1975)
F(Q) = Σ ((4) 1T
where is the number of atoms in the unit cell. The atomic form factor (Q) specifies the scatter- ing amplitude of atom j and is subject to physical constraints (see Appendix A.3). Thus, each F(Q) encodes the real-space atomic arrangement along a crystallographic direction.
Unlike the Fourier transform recently applied for value extension, latent-space signal processing, or denoising, which decomposes an arbitrary signal into a frequency components to enrich the rep- resentation (Jiao et al., 2023; Choromanski al., 2024; Dong al., 2024; Gao et al., 2024), the
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Fourier transform in crystal diffraction is physical scattering process. It maps the periodic elec- tron density onto a discrete reciprocal lattice and provides a reciprocal view of the crystal struc- ture under a different basis Q, which physically reflects the crystal's 's directional arrangement rather than its real-space Cartesian coordinates. Although the mathematical foundation is similar, the diffraction transform represents the accumulated phase differences arising from the periodic distri- Oution/arrangemeet of electrons/atoms. The former yields a general frequency distribution, whereas the latter produces Bragg intensities that are strictly constrained by lattice periodicity and serve as physical representation of the underlying crystal structure.
Sseudo-particle Conventional diffraction probes, including X-ray photons, electrons, and neu- trons, have intrinsic limitations in reciprocal-space representations (Bouwkamp. 2016). X-rays (Epp, 2016; Cao et al.) and electrons (Glauber & Schomaker, 1953) scatter from the electron cloud, with intensities determined by the atomic form factor fj. making atoms with similar charge nearly indis- tinguishable. Neutron diffraction (Shull Wilkinson, 1953), in contrast, relies on nuclear scattering cross-sections, but heavy atoms of similar mass often show weak contrast. In a numerical frame- work, these limitations can be addressed by introducing a pseudo-particle with learned particle at- tributes, acting unique probe for reciprocal-spar crystal representation, enabling differentiation of atomic arrangements in distinct local environments.
Problem definition Many material properties can be derived from the fundamental equations of electronic behavior, where the dynamics of electrons are fully determined by the external po- tential imposed by the atomic arrangement in crystal (Martin, 2020). Crystal property predic- tion thus aims to estimate a physical property y from the crystal structure, represented as a tuple R) in regression tasks or categorical
(A, L). The target property y may be continuous (y E (1.2...., C.) in classification tasks with classes.
METHOD
4.1 OVERALL FRAMEWORK
Motivation for Learned Pseudo-Particles Unlike conventional X-ray diffraction, which relies on fixed tabulated form factors, our approach introduces learned pseudo-particle that incorporates environmental dependence, enabling greater sensitivity to chemical environments and elemental differences. In XRD, the interaction between X-ray photons and atoms (i.e., the form factor) is obtained from the International Tables for Crystallography (Prince, 2004). It is fixed tabulated value that depends on two factors:
X-ray
X-my (IQ), PT
(4)
where Q is the magnitude of the scattering vector Q (the reciprocal-space basis), and fr denotes the atomic species. Thus, real physical particles cannot distinguish between identical atoms located in different local environments for given scattering vector Q. Moreover, the intrinsic physical properties of X-ray photons limit their sensitivity to specific atoms.
This limitation motivates to design environment while retaining strong discriminability across different elements, making it well-suited for CPP tasks. Unlike real partieles, the learned pseudo-particle is not constrained by the physical limitations of X-rays. We define its form factor as:
particle that is explicitly sensitive to the local atomic
Preudo f-eeudo (IQ), J.)
(S)
where Gg( encodes the local chemical environment learned by the graph network.
Graph embedding A crystal structure is represented graph = (V,E), where vertices V (V1, ......, Enn correspond to atoms one-hot encoded attributes (Xie & Grossman, 2018) and edges& 8 C V x V encode atomic interactions within a cutoff radius. Each atom is characterized by node feature h = Embed(u:) us) E Rd and coordinates = between atoms i and are defined as:
(xi, Mi, zi) E R33 Edge features
Cjj- (B,, ① SBF(B)gk) @ dij, dy= = ||, ryll,
(6)
where RBF(.) denotes a radial basis function, SBF(-) a spherical basis function (Gasteiger et al.), Oigk is the angle between bond ij and bond ik, and @ indicates concatenation.
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(1+3)
Short-range
Long-range
Fommost
MLP
Updating
Re(F))Im(F)
Linear
Fusion
Fcoscctt
Structures Factors
Concat
hy
Scalled Dof-Product
Broadcast Operation
Altention
Multi-Head
Attention
(HC
(2()
(
(003)
MLP
MLP
Linear
Linear
Linear
Graph
Tex
(110)
hj
(3)
Embedding
Multi-Head Attention
Inputs
Pseudo-Partice Diffraction
Figure The configuration of PRDNet, which integrates crystal attention and pseudo-particle diffraction to capture both short- and long-range atomic interactions.
Multi-Modal Feature Fusion The integration of graph-based crystal real-space features h and physics-based diffraction representations F comcat is achieved through fusion mechanism:
(L)
GlobalPool (b( d MLP (FeaaiFFFFFoo concat))
Zfused MLPfusion d))
(7)
GlobalPool(-) aggregates node features across the entire crystal structure, MLPain() maps physics- based descriptors to the embedding space, and MLP fusion (-) produces the final fused representation.
Model Invariance Crystals are inherently symmetrical. Following existing work (Yan al., 2024; Liu et al., 2024; Zheng et al., 2024), we force the model prediction to be invariant to E(3) group (i.e., the rotation, reflection, and translation of input crystals). Specificallyy according final representation output Zused depends on the and d. Since depends on the relative geometric features dij and Bigk (Eq.6) that remain unchanged under E(3) operations, g is E(3)-invariant. Consequently, because d is also E(3) -iv)ainn,,,,,n,, the model output is E(3)-invarian) as shown in Section 4.3.
Eq. the
4.2 STRUCTURE MODELING
Multi-Head Attention Projections The PRDNet message-passing layer augments conventional graph attention mechanisms by integrating edge features and physics-informed constraints. Using multi-head attention framework, it captures information from local atomic environments. Given
node features h h E Rd and edge feature eiý at layer I, we first compute the query, key, value, and edge projections for each attention head /t E {1, (1, H):
hb) ( (c))_ = h ((h) w h Q
E
eijs
(8)
where W (h) K the attention scores are computed as:
w( E Rdd xd and (h(h) E Bddx are learnable projection matrices. Then
(b)
(h)
k.
K(M
+ K (h) +
(h) a
(A)
4
kij
3d
where denotes concatenation and represents element-wise multiplication.
Gated Message and Aggregation The message aggregation incorporates both value information and attention modulated defined as:
autention-modulated gating. For each attention head h, the value vector and gating factor are
(N) v(() V
E
(b) E
(b) gij = (L(LyyeNoom((a)))) (h)))
m (b)
W msp v (h)
G Gggg (10)
(h)
where (-) is the sigmoid activation, O denotes elementwise multiplication, and W masg 3dd, x3d, (h) learnable transformation matrix. The gating term giy acts adaptive filter, modulating the

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contribution of each message according to the attention scores. Messages from all neighbors and attention heads then aggregated and projected back to the hidden dimension:
mig
h (1+1)
B, h + (1 - B.) @ SiLU(BatchNormm (W
(Womea concan ())) (11)
(EN
where (i) denotes the neighbors of node i, + indicates concatenation, H is the number of atten- tion heads, and W concat projects the concatenated features into the hidden dimension.
4.3 PSEUDO-PARTICLE RAY DIFFRACTION MODELING
The meeudo-pattccle ray diffraction module incorporates particle diffraction theory into the neural network architecture. This module computes structure factors from the learned atomic representa- tions and fuses this physical information with graph-based features.
Learnable Atomic Form Factors Unlike traditional crystallographic calculations that rely on tab- ulated atomic form factors, the pseudo-particle form factor (i) is defined as
(Q) fi Go())
(12)
where [Q] depends the magnitude of the scattering vector Q, Gg(g) captures modifications due the local chemical environment, and fi encodes species-specific attributes. In PRDNet, these adaptive form factors are not fixed but learned through a dedicated form-factor layer that maps node embeddings to scattering strengths:
Jr (H) = MLP (h!!4) E R MiNi
(13)
Nu,, and (,(2) denotes the final node representation. Let f denotes dimension corresponding to a specific (h, 1) !) E H. with (h, k, () and H introduced below.
where MLPfrmm Rdd -
Miller Index Selection and Strueture Factors To ensure comprehensive coverage of reciprocal space, employ systematic selection of Miller indices. Each (h, k. 1) triplet specifies crystal- lographic direction, corresponding to a unique scattering vector Q.
Ho = (h, |4] S Cmax: ged(14), [4,111 = =1}
(14)
ged(-) denotes the greatest common divisor (GCD). The constraint ged( |Rs [T]) ensures that all fundamental reflections are considered, avoiding redundant higher-order reflections. The parameter Ciaax = 8 controls the resolution in reciprocal space. To guarantee that the index set closed under symmetry. we define
H = { permmhh, k) (h, A. E Ho).
(15)
where perm(h, k 1) denotes all permutations of (h, k. I) and the prefactor + accounts for inversion. By construction, H is closed under all crystallographic operations.
The complete diffraction signature constructed by computing structure factors for all selected Miller indices F(H) [Faad((,, The structure factors of pseudo-particle ray diffraction, denoted as F(H), are converted to real values by applying the real part operator. The real and imaginary components are obtained by summing over atoms with the corresponding form factor weights:
N Σ ft cos 1=1
Re(Fhit)
IT), - Im(Fae)) = Σ ft sin (2mh rf)
(16)
the reciprocal lattice vector, and r; = (xe, Mis 2;) represents the fractional coordinates atom i. The real and imaginary parts then flattened construct the diffraction feature tensor:
where ft" is the learned atomic form factor for atom i. h (h, 1)T is
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concal flatten(ee((H))) + Im (H)))
R
(17)
The forward propagation process is presented in Algorithm I, and case study on the crystal Cal2 provided in Appendix C.
Invariance in pseudo-particl diffraction Let G be the crystallographic operations acting on fractional coordinates by y r + g. Rg + Egz E G, where Rg (i.e, rotations and reflections) is an integer unimodular matrix (det Rg =1) that maps Miller indices to integer triplets. And t the translational part. Denote Miller index by h (h, k,)" E 2 and the set of indices by H. The set is closed under the action of i.e.
Vg G, Vh H
g g- R2 to RghEH H.
(18)
Under 9 the coordinates become g Hence
Rgr by and the Miller index maps g h R.g.
N Fanllg r:)) = Σ
fi exp 2 (Rgh) (Rgr, +tt)
==1
N
exp 2m (Ryh) tqg
Σ
fie Griber = ( Fa((r;))
(19)
The phase ((g, h) (R t, arises from the translation. Because always an integer, the factor 2rid))gh)) 1, which ensures that the representation remains invariant under crystallographic operations.
5 EXPERIMENTS
We benchmark the models on three standard evaluation suites, all independently re- trieved from the official websites. The definitions and background of the regression and classifi- cation targets are introduced in the Appendix
Datasets
Materials Project (MP) Database: We use stable structures retrieved from the Materials Project (Jain et al., 2020), comprising 122,959 entries with annotated formation energy, band gap, and metal/non- metall classification labels. Additionally, 9,473 of these entries include mechanical properties such as bulk modulus, shear modulus and Young's modulus. The dataset is managed using the Atomic Simulation Environment (ASE), and is available at https: //hugg fagfcce co dwwaaateett....... cwwbbiiioirro//ooo....c CCPpbenchaarrre
JARVIS-DFT (dft_dd): This dataset contains 75,993 entries, each annotated with formation energy, band gap (calculated using either the OptBS8vDW functional, denoted as OPT, or the TBMBJ functional, denoted as MBJ), bulk modulus, shear modulus, total energy (calculated us- ing the OptB88vDW functional), and energy above the hull (Enul) (Choudhary et al., 2020). We also evaluate the baselines on the SARVIS-DFT-3D-22 serves as an important supplementary benchmark.
dataset (55,723 entries), which
Matbench (MB) Suite: We evaluate the models on several tasks from the Matbench suite (Dunn et al., 2020): matbench_jddfe 2 (636 entries; exfoliation energy), matbench__pp_e e_rorm (132,752 entries; formation energy), matbench__ose lal__loog_vaaa gvrr (10,987 entries; shear modulus), and ma matbench. wtwttteenccccheecceeeeeeee dielectric (4,764 entries; refractive index).
Implementation details follow the original open- settings without additional tuning. Model performance is evaluated using the Mean Absolute Error (MAE) and accuracy. All implementations are based on the Py ToTch framework (Paszke et al., 2019) and trained on NVIDIA GeForce RTX 3090 GPUs.
The specific hyperparameters used are listed in Appendix D. All models
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Table 1: Mean Absolute Error and Accuracy comparison on the MP.
Param Num Form, Energy Bundgap Balk Modulus Shear Modulus Young's 's Modulus Metal/Nun-metal M million
Method
eViatom
log(GPn)
log(GPa)
lop(GPa)
Acc(%)
CGCNN (Xic & Grossmon, 2018)
200
0.041
0.262
0.079
0.162
0.155
87.5
SchNet (Schort at. 2017)
0.3M
0.038
0.344
0.092
82.8
MEGNET (Chen et al., 2019)
0.2MM
0.053
0.307
0.134
0.206
0.189
86.6
GATGINN (Louis al., 2020)
0.631
0.083
0.343
0.051
86.3
Matformer (Yan et al., 2022)
154M
0.035
0.193
0.089
0.306
0.291
92.7
Crystalformer (Taniai et al.)
0.8,1
0,049
0.251
0.064
0,140
0,127
86.5
Crystalfarmer (Ito et 2025)
0.9M
0.030
0.216
0.047
0.118
0,106
87,5
cComFormer (Yan al., 2024)
5.6.1f
0.033
0.153
0.088
0,269
0,311
82.5
EwaldMP (Kosmala al., 2023)
12.2M
0,052
0.098
84.2
PotNet (Lin et al., 2023)
18M
0.035
0.251
0.104
0.175
0.151
88.9
ReGNett (Nic et al., 2025)
3.3Mf
0,047
0.088
85.1
PRDNet
20.9M
0.028
0.151
0.035
0.108
0.104
93.3
Table 2: Mean Absolute Error comparison on the JARVIS-DFI (left) and MB (right).
JARVIS-DFT
MB
Method
Form. Energy <V/atom
Bandgip (OPT) eV
Balk Madulus log/CiPt)
Shear Modulus.
Total Energy
Bandgup (MBJ) eV
Fuat
jdft2d
ppp_e_oom
gvrh dielectric log(GPP)
log(Git)
eViatom
cV
VVitoon
<V/atom
unitless
CGCNN
0.038
0.144
0.125
0.163
0.041
0.402
0.049
0.050
0.027
0.066
0.541
SchNet
0.074
0.278
0.143
0:225
0.048
0.507
0.073
0.056
0.042
0.085
0.341
MEGNET
0.074
0.221
0.124
0.214
0.054
0.369
0.056
0.052
0.036
0.098
0,459
GATGNN
0,069
0.235
0.878
0.160
0,053
0.372
0.048
0.046
0.032
0.187
0,408
Matformer
0.033
0.207
0.101
0.157
0,041
0.341
0.034
0.072
0.023
0.071
0.764
Crystaforrmer
0.052
0.248
0.075
0.146
0.282
0.053
0.071
0.020
0.080
0.244
Crystalframer
0.048
0.208
0.065
0.144
0.047
0.270
0.115)
0.056
0.165
0.084
0.248
&ComFormer
0.119
0,260
0,118
0,243
0.035
0.335
0.034
0,031
0.020
0.085
0.543
EwalMP
0.052
0.256
0.191
0.057
0.451
0.047
0.052
0.053
0.117
0.485
PotNet
0.034
0.144
0.112
0.156
0.039
0,339
0.079
0.033
0.029
0.084
0.397
ReGiNe
0.054
0.201
0.092
0.178
0.041
0.355
0.051
0.045
0.033
0.079
0.512
PRDNet
0.032
0,140
0.064
0,122
0.032
0,267
0.011
0.038
0.019
0.058
0.242
5.1
RESULTS
Benchmarks Performance Across the MP, JARVIS-DFT, and Matbench benchmarks, PRDNet consistently delivers superior competitive performance compared with existing models. On the MP dataset (Table 1), PRDNet surpasses all baselines, achieving the lowest errors in formation en- ergy (0.028 eV/atom), band gap (0.15) eV), bulk modulus (0.035 log(GPa)), shear modulus 10.108 log(GPa)), and Young's 's modulus (0. (0. loP(G)))), while also reaching the highest metal/non-mett classification accuracy (93.3%). On JARVIS-DFT (Table 2). PRDNet again outperforms the base- lines, achieving the best results on most tasks, including formation energy (0.032 eV/atom), band gap (OPT, 0.140 eV), bulk modulus (0.064 log(GPa)), shear modulus (0.122 log(GPa)). total en- ergy (0.032 e eV/atom) and band gap (MBJ, 0.267 eV) while also achieving competitive performance in Emmil (0.041 eV). On the Matbench benchmark (Table 2), PRDNet achieves the lowest errors in formation energy (0.019 eV/atom), shear modulus (0.058 log(GPa)) and refractive index (0.242 (unitless)). while maintaining competitive performance in exfoliation energy prediction. PRDNet also achieves SOTA performance the JARVIS-DFT-30 (OPT) task) outperforming all baseline models (Table 8). These results show that PRDNet general- izes effectively across diverse datasets and provides balanced improvements in both energy-related and mechanical property predictions, underscoring its robustness and versatility for crystal property modeling.
dataset (except for the band gap
Ablation Study To validate the effectiveness of each component in the PRDNet architecture, we perform systematic ablation studies that isolate the contributions of individual modules and design choices. We evaluate PRDNet across diverse property prediction tasks using the Materials Project, JARVIS-DFT and Matbench databases, ensuring a comprehensive assessment. Table presents the ablation results on the MP dataset, while Appendix B, Table 5, and Table 6 report the results on JARVIS-DFT and Matbench, respectively. The results show that each component is essential for optimal performance. Removing the diffraction module causes the obversely performance drop, underscoring its critical role. Likewise, using single-head attention, removing residual connections, or excluding edge features consistently degrades performance, indicating that multi-head attention, residual updates, and edge features collectively strengthen representation learning. These findings confirm the effectiveness of our default architecture, in which all modules work synergistically to achieve the highest accuracy across material property predictions.
As of November 12, 2025, the ReGNet (ReciNet) code is unavailable; the results were obtained using our own implementation based on the paper's settings. We use a hidden dimension of 304 instead of 256, as this brings the total parameter count elose to the 3.3M reported in the paper, using 256 results in only 2.37M.

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Table 3: Ablation study on the Materials Project. Diff. indicates whether the diffraction module is used, MH denotes single-head or multi-head attention, Res. specifies whether residual connections are applied during node attribute updates, and EF refers to the inclusion of edge features (see Ap- pendix B).
Model
Diff. MH Res. EF
Form. Energy Bandgap Bulk Modulus Shear Modulus Young's Modulus Metal/Non-metal MAEleVattom MAB(eV) MAE(log(GPa)) MAE(log(GPaa) MAE(log(GPa))
Acc(%)
Default
<
V
0.028
0.151
0.035
0.108
0.104
93.3
NoDiff'
0,041
0.361
0.081
0.171
0.162
81.9
SingleHead
v 0.040
0.318
0.067
0.166
0.181
89.3
NoRes
0.038
0.297
0.077
0.204
0,222
82.7
NoEdge
0.043
0.355
0.071
0.198
0.179
80.1
Table 4: Comparison of Mean Absolute Error and Accuracy the MP dataset across baselines en-
hanced by diffraction modules. The models were trained on NVIDIA GeForce RTX 4090, GeForce RTX 4080, GeForce RTX 3090, NVIDIA A100 and A40 GPUs.
Form. Energy Bandgap Bulk Modulus Shear Modulus Young's Modulus Metal/Noon-ettal eVintom
Method
GPUs
log(GPa)
log(GPa)
lng(GPa)
Acc(S))
CGCNN (Xie & Grossman, 2018)
2x0400
0.0391
0.207
0.077
0.148
0.150
89.2 1
SchNee (Schilt et al., 2017)
2xA40
0.032
0.303
0.091 1
0.110. 1
0.145(-)
85.7
MEGNET (Chen et al.. 2019)
3x3090
0,040
0,291
0.087
0.202 4
0,177
87.1
GATGNN (Louis et al., 2020)
2x4080
0.062
0,309
0.044
0.169
0.113 (-)
87.1
Matformer (Yan al., 2022)
2xA100
0.034
0.187]
0,082
0.217
0,277
90.1
Crystalformer (Taniai et al.)
2x4090
0.047
0.252 (-)
0.059
0.167 1
0.126
89.6 1
Crystalfarmer (lto al., 2025)
3x40800
0.032
0,166 1
0,066
0.117
0,108
90.7
oComFormer (Yan et al., 2024)
2x3090
0.032
(-)
0.065 $
0.221 J
0.287 1
86.6 1
Pseudo-Particle vs. X-ray Photon For evideneing the enriched representation of Pseudo-Partich on crystal sensitivity, we compare two PRDNet variants: (1) PRDNet-Learned: employs learned pseudo-particle form factors fi (Eq. 13); (2) TabNae-Tabulate: substitutes S with tabulated X- ray photon form factors. As summarized in Table (Appendix B), the useudo-particle representation consistently outperforms the X-ray counterpart across all datasets and property prediction tasks. Specifically. it achieves lower MAEs for formation energy, bandgap, and mechanical moduli, while also delivering higher classification accuracy in metal/non-metal discrimination.
Diffraction Module Assembling Each real-space crystal representation baseline in Table 1 is integrated with our reciprocal-space diffraction module and the fusion mechanism defined in Eq.,7 and 14-11 This allows a systematic comparison of all baseline models after incorporating our diffraction module. Although several baselines are neither invariant equivariant, therefore cannot achieve the same level of physical consistency as PRDNet. We include them to provide a more comprehensive evaluation, as listed in Table 4. Overall, the results show that most models benefit from the unique reciprocal-space representation and the modality-level fusion mechanism. Exceptions include GATGNN and Crystalformer on shear modulus, Crystalformer on bulk modulus, and Matformer on metal/noon-metal classification, which show decreased performance; all other models show improved or comparable results.
CONCLUSION
In this work, we introduce a Pseudo-Particle Ray Diffraction Neural Network (PRDNet) for crystal structure property prediction. PRDNet outperforms previous approaches by capturing long-range atomic correlations through a physically invariant reciprocal-space representation embedded within the neural network. The learned pseudo-partick attributes act as novel probe, exhibiting higher sensitivity than any known physical particle to atomic type, local chemical environment, and diffrac- tion directions. Experiments demonstrate that PRDNet provides a principled framework for inte- grating crystallographic theory with deep learning, offering a novel conception of invariant crystal representation and achieving SOTA performance across multiple properties and databases. The lim- itation also arises from the fusion of complete reciprocal-space information, which implies that the inverse process for Crystal Structure Prediction (CSP), i.c.. generating valid structures real space by shaping distributions in diffraction space, can break invariance and thus cannot be directly applied to CSP. In future, we plan to: (1) Integrate PRDNet with the structure identification tool (XQueryer (Cao et al., 2025)) to enable a fully automated materials analysis workflow; (2) Extend PRDNet to simultaneously handle multiple length scales, from atomic to mesoscopic structures; (3) Extend relaxed invariance structures for CSP tasks.
2 As these are supplemental experiments conducted within limited time window, we were unable to train all models on the same GPUs Therefore, we explicitly list the GPUs used for training each baseline for this table.
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ACKNOWLEDGMENTS
This work was supported by the Guangzhou-HKUST((Z) Joint Funding Program (Nos. 2023A03J0003 and 2023A0310103). R. also acknowledges financial support from the City Uni- versity of Hong Kong (Project Nos. 9610533 and 9680351). Y. L. is supported in part by the Postdoctoral Fellowship Scheme of The Chinese University of Hong Kong.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our research. To this end, we provide a pub- licly accessible GitHub repository accompanied by comprehensive README file. The README provides detailed information on the PRDNet model, its hyperparameters, environment configu- ration, dependency installation, environment validation, datasets, training executor, and training procedures. Furthermore, Appendix D lists all benchmark model settings used for validation.
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A ADDITIONAL DETAILS
A.1 THE USE OF LLMs
In this work, we used LLMs as an auxiliary tool for polishing the writing and improving the clarity of the manuscript. Specifically, use review the entire text for grammatical accuracy, improve sentence structure, and ensure consistent phrasing and tone throughout the paper.
A.2 CRYSTAL INVARIANCE
The representation of crystal structure must satisfy several fundamental invariance properties to reflect the physical symmetries inherent in periodic materials (Li al., 2025). These include:
Permutation invariance: Let (9) denote a representation of a crystal graph g (V,E), where V is the set of atoms and & the of edges. Then, for any permutation of the node indices, require
S(G) = (())))),
(20)
meaning that relabeling atoms does not alter the representation. This property is naturally satisfied graph-based models.
Translation invariance: all atomic positions r are shifted by a vector t E R i.c., r - r +t, t, then
f((r)) f((r + +)),
(21)
ensuring the representation is invariant under global translations.
Rotation invariance: For any rotation matrix R SO(3), the representation must satisfy
f((r:)) = f((
(22)
i.e., the representation remains unchanged under rigid body rotations.
Periodic invariance: For any lattice translation vector e 23, the atomic positions r, and r, + should yield the same representation:
=(tr)))= f(fr: T)),
(23)
capturing the periodicity of the crystal lattice.
These four invariance properties are essential any representation matter based on a periodic unit cell. Beyond these, symmetry invariance plays crucial role in crystallographic systems.
Symmetry invariance requires that atoms related by a space group symmetry operation, such as those occupying the same Wyckoff position, must be treated equivalently in the representation. If symmetry operation maps atomic coordinates r to Grr), then the representation must satisfy
f(r) = f(G(t))
(24)
Such symmetry operations include:
Identity (E): G(r) = This operation is always present by group theory.
Inversion (I): G(r)
Crystals exhibiting this symmetry called centrosymmetric.
Rotation (C,): G(r) R, where Rn is rotation matrix corresponding to 3609 rotation
about a symmetry axis.
Mirror reflection (a): G(r) = r - 2(r n, reflecting the position across a mirror plane with normal vector n.
In crystallography, certain symmetry operations combine rotation or reflection with fractional lat- tice translations, giving rise to glide planes and serew axes (Fjellvåg, 1994). These compound operations are essential for defining space groups but are not directly relevant to the basis invariance required for representation learning.
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A.3 PHYSICAL CONSTRAINTS DIFFRACTION
Recent works endeavor to introduce Fourier transforms to capture long-range interactions (Kos- mala et al., 2023; Lin et al., 2023; Nie et al., 2025), incorporating reciprocal-space information to model long-range atomic correlations to some extent. However, a complete diffraction representa- tion uniquely invariant and physically determined; slight omissions or approximations may lead to incomplete information, often treated empirically as a form of supplemental atomic information fusion. our work, we systematically address these dependencies and construct an framework that preserves invariance. Taking ReGNet (renamed ReciNet in the latest version) (Nie et al., 2025) as an example, the model introduces a dedicated reciprocal block that approximates the structure factor, functioning similarly to Eq. 3, and achieves improved performance. We further decompose their key Fourier- transforr settings to discuss the underlying dependencies.
Dependence:
The atomic form factor (Q).. defined in Eq. 3, is inherently depends three fundamental factors: the atomic species, the local atomic environment, and the seattering vector Q. Preserving these dependencies is crucial for maintaining physical fidelity in diffraction-based modeling.
Consistency: According the Schrodinger equation & Shubin, 2012), the charge den- sity (or electron distribution) is uniquely determined by the atomic configuration. Therefore, the reciprocal-space representation must remain consistent with this determinism and cannot evolve independently.
In ReGNet, the reciprocal embedding is computed as:
F(Q) = e
(25)
where hn is learnable embedding that evolves through network layers. The following dependencies and invariances may be omitted in such formulations:
Form factor dependence: The complete diffraction factor (Q) should be expressed as
fy(Q)) = ( ()(),,
(26)
where [Q captures the dependence on the magnitude of the scattering vector, Go(9) reflects
the local atomic environment, and 1 encodes the atomic species. The R in Eq. 25 loses the dependence on Q and the local state (G), depending only on the atom and block orders.
Reciprocal-space uniqueness: The Fourier transform F(Q) of the charge density p(r) is uniquely defined by given atomic configuration. Eq. 25 evolves the diffraction expression across blocks way that breaks this invariance, thereby losing the physical uniqueness.
Modal fusion:
Diffraction captures long-range interactions by introducing complete reciprocal- space information, which reflects the periodic arrangement and physical interactions between the structure (i.e., electron attributes) and probing process. This reciprocal-space information is holistic representation the structure and thus should be fused at the modality level rather than at the atomic level.
A.4 TRAINING SETTINGS OF PRDNET
A.4.1 HUBER LOss FOR OUTLIER ROBUSTNESS
To accommodate the diverse nature of crystal property prediction tasks and improve training stabil- ity, PRDNet employs Huber loss (Meyer, 2021) for training.
The Huber loss combines the advantages of L2 loss for small errors and L1 loss for large errors:
5 y)²
if yl < 8
Ctuterr (1. yr 8)
(27)
و yl 882 otherwise
where 5 is hyperparameter that controls the transition point between quadratic and linear regimes. This formulation provides robustness against outliers while maintaining smooth gradients for small errors.
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A.4.2 ADAMW WITH DECOUPLED WEIGHT DECAY
PRDNet employs the AdamW optimizer, which decouples weight decay from gradient-based up- dates:
m, = f + (1- VoC.
(28)
B2 + (1 (1- (e
(29)
m1
V₁
m.
V: 1 $2
(30)
1 $
m
0t11 0, 0t
+ 20,
(31)
Vv. +
where 81 0.9, 82 0.999, 100 and À is the weight decay coefficient.
A.4.3 COSINE ANNEALING WITH WARM RESTARTS
The learning rate schedule follows cosine annealing pattern with periodic restarts:
Teur IT
7/min + (1max 7mmin)
1 cos
(32)
T Timax
Teur mod Tmax
(33)
where Teur is the current epoch within the restart cycle, Tmax is the restart period, Tamax is the maximum learning rate, and Mmmin is the minimum learning rate.
A.4.4 LAYER NORMALIZATION WITH LEARNABLE PARAMETERS
Layer normalization applied to stabilize training:
(34)
1=1
1
0²
11) 1=1
(x)
(35)
LayerNorm(x)
X 1
(36)
V0² +c
where y and are learnable scale and shift parameters.
B ABLATION STUDY
B.1 DIFFRACTION ABLATION
PRDNET-NoDiff removes diffraction integration entirely:
Zuodilf MLP (g))
(37)
PRDNET-Full incorporates complete diffraction physics integration:
Ziused MLP Fusionn (gg @ d))
(38)
where d MLP real + F Fimagl)
(39)
B.2 ATTENTION MECHANISM ABLATION
PRDNet-SingleHeedd
uses single-head attention:
single Attention Q Kj, V,, Egy) m
(40)
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PRDNet-Full employs multi-head attention:
H
mulii m Attention (A) (Q) K(A) V )
(41)
B.3 RESIDUAL CONNECTION ABLATION
PRDNet-NoRes removes residual connections:
h ((+1) SiL.U(BatchNor((h) ()
(42)
PRDNet-Full uses residuals:
h (++1 B 1 1hh +11 B:) SiLU(BathNorm(h) (((+1)))
(43)
B.4 EDGE FEATURE ABLATION
PRoNee-NoEdge excludes edge features from attention:
Q K
(44)
Vd
PRDNet-Full incorporates edge features:
QQ,
+
C (K, $ K2 + Eij)
O3
(45)
Table Ablation study on the JARVIS-DFT. Diff. indicates whether the diffraction module is used, MH denotes single-head or multi-head attention, Res. specifies whether residual connections are applied during node attribute updates, and EF refers the inclusion of edge features
Model
Diff. MH Res. EF Form. Energy Bandgap Bulk Modulus Shear Modulus
Default
0.032
0.140
0.064
0.122
NoDiff
0.055
0.333
0.287
0.291
SingleHead
X
0.071
0.291
0.236
0.311
NoRes
0.048
0.397
0.201
0.237
NoEigee
0.083
0.412
0.289
0.341
Table 6: Ablation study on the MB. Diff. indicates whether the diffraction module is used, MH
denotes single-head or multi-head attention, Res. specifies whether residual connections are applied during node attribute updates, and EF refers to the inclusion of edge features
Model
Diff. MH Res. EF Exfo. Energy Form. Energy Shear Modulus Refractive Index
Default
0.038
0.019
0.058
0.242
NoDiff
X
4
/ r
0.072
0.042
0.121
0.297
SingleHead
/
v
0.121
0.038
0.094
0.298
NoRes
V
x
v
0.064
0.038
0.101
0.311
NoEdge
V
v
0.099
0.051
0.099
0.304
Number of Miller Indices Each Miller index specifies crystallographic direction. The choice of a sufficiently large set of Miller indices, Ho, is crucial for adequately representing reciprocal space. We evaluate the performance of PRDNet using 100, 200, 300, 500, and 1000 Miller indices the MP dataset for both formation energy and band gap prediction tasks. As shown Figure 5, using 300 indices provides good balance between computational cost and predictive accuracy.
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Table 7: Pseudo-Particle X-ray Photon
MP
JARVIS-DFT
MB
Method
Pseulo-Particle Particle
X-ray
Pseado-Particle
X-ray
Pheudo-Partics
X-my
Form. Energy (MAE eV/atom)
0.028
0.019
0.063
Bandgap (MAE eV)
0.151
0.271
0.140
0.331
Bulk Modulus (MAE log(GPPa)
0.035
Shear Modulus (MAE log(GPu)
0.108
0.322
0.122
0,256
0.058
0.111
Young's Modulus (MAR (og(GPa)
0.104
Exfo. Energy (MAE eVatoot)
0.038
0.089
Refractive (MAF unitless)
Mctal/Non-metal (Ace %)
93.3
78,9
0.55
MAE (Form. Energy)
0.22
0.50
MAE (Bandgap)
0.45
0.20
0.40
0.18
0.35
0.16
0.30
0.25
0.14
200
400
600
800
1000
Number Miller Indices
Figure 5: Performance of PRDNet on formation energy and band gap prediction across different numbers of Miller indices.
C CASE STUDY OF FORWARD PROPAGATION
To illustrate the complete PRDNet forward propagation process, we present a detailed case using concrete example crystal structure. This section demonstrates how atomic-level information flows through the network to produce property predictions.
Input Crystal Structure
Consider simple cubic crystal structure X'
Composition: CaF₂
Unit cell: atoms with fractional coordinates
Lattice parameters: b 5.46 Å,
Space group: Fm3m (face-centered cubic)
Atom 1 (A): Zi 20,
r1 (0.0, 0.0)
(46)
Atom 2 (B): Z2 = 9,
r2 (0.25, 0.25, 0.25)
(47)
Atom 3 (C): Z3 9,
r3 (0.75, 0.75, 0.75)
(48)
where Z₁ represents the atomic number and r; are fractional coordinates.
C.1
STEP I: GRAPH CONSTRUCTION AND FEATURE INITIALIZATION
Neighbor Detection
Using cutoff radius Teur 5.0 Å(for illustration), identify atomic neigh-
bors:
N(1) (2,3)
(A atom has 1 1 neighbor)
(49)
N(() = (1,33 (B atom has + neighbor)
(50)
N(3) (1,2) (C atom has A neighbor)
(51)
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Node Feature Initialization
Atomic embeddings initialized based atomic numbers:
h1 Embed Zi 20) IR 256
(0)
(Ca embedding)
(52)
h2 Embed( Zz = 9) E 256
(0)
(F embedding)
(53)
h3
(0) Embed Z3 9) E R 256
(F embedding)
(54)
Edge Feature Construction For each atomic pair (i,j), edge features encode geometric relation-
ships:
d12 ||r r2ll 2.36 Å
(55)
e12 RBF(d12) @ SBF(0123) @ d12 E RR" 12188
(56)
where RBF(-) is the radial basis function and SBF(.) is the spherical basis function (Gasteiger et al.).
C.2 STEP 2: MULTI-LAYER GRAPH CONVOLUTION
For each layer [ 0, 1, 1.....44 L
(with L 6 layers):
Hulti-Head Attention
For attention head h and atomic pair (1,
Q w@'n
RR
(Query for atom 1)
(57)
K w h R
(Key for atom 1)
(58)
KS (1) = W h2 E R232
(Key for atom 2)
(59)
V(I) = w( h
(1)
E R
(Value for atom 1)
(60)
( w h
(1) 32
(Value for atom 2)
(61)
E (1) w e12 E 32
(Edge projection)
(62)
Attention Score Calculation
41 Q
(1) (1)
(1) E
96
(63)
(1)
k(T)
+ K
4 E(L)
E R99
(64)
12
42 k
(1)
12
E 9
(65)
96
Message Aggregation
m (1)
W msg (V v
+ E @ a(a (H)
(66)
m m 12 + maa @
age
(1)
@ 2
m 13
(I)
(67)
Node Updating
h
(+1) = 3 +
+(1 (1- B.) @ SiLU(Bath W concar (EE)
(68)
C.3 STEP 3: PHYSICS INTEGRATION DIFFRACTION MODULE
Learnable Form Factor After I. graph convolution layers, compute atomic form factors:
(1(H) = MLP form (h (L) = (2.34,.) ...) (Ca form factor)
(69)
(2(H) MLP Farm (hg (b)) = (1.87,.) (F form factor)
(70)
(a(H) ML.Pom (b) (1.91....) (F form factor)
(71)
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Structure Factor
For selected Miller indices, e.g., (h, k. () (1,0,0) N:
Fio0-
Σ J exp(2 2y + yy + +02))) j=1
(72)
exp(22 0) + f2 0.25) fs exp(2 0.75)
(73)
2.34 + 1.876 /// 13/2
(74)
2.34 + 1.87i 1.91i 2.34 (-0.04);
(75)
where i is the imaginary unit.
Diffraction Feature Vector For NIRI 300 Miller indices:
F Freal [Re( [ Fio). Re(Foro), ...] = [2.34, 1.23,
(76)
F imag IIm( 100), Im(Fooo).... = 0.87,
(77)
F concat
F real + F imagg]
(78)
C.4 STEP 4: MULTI-MODAL FUSION
Graph Feature Pooling
g - (h) (4) + (4) + h E 250
(Mean pooling)
(79)
Diffraction Processing
MLPan Fomnca) E 256
(Diffraction features)
(80)
Feature Fusion and Prediction
Zfased MLP MiLPisioon/i (Ig + d]) E IR 128
(81)
D MLP out (rused) -2.47 eV/atom
(82)
The final prediction g 22477 e//atom represents the formation energy of the CaF, crystal struc- ture.
C.5 EXPERIMENTS
In Table 8. we present the performance of PRDNet and the baseline models on the SRRWVSSDFT-3D-200 dataset. All baseline models were independently retrained for a fair comparison. PRDNet consistently outperforms the baselines, achieving the lowest MAE forma- tion energy (0.025 eV/atom), bulk modulus (0.075 log(GPa)), shear modulus (0.132 log(GPa)), total energy (0.024 eV/atom), band gap (MBJ, 0.242 eV), and Enul (0.040 eV). It also demonstrates competitive performance on the band gap (OPT) prediction task, with MAE of 0.138 eV.
Furthermore, Table summarizes the reused baseline MAE results reported in the respective original papers. PRDNet achieves the best performance formation energy. total energy, and Emull, while ranking second on band gap (MBJ) and delivering competitive results on band gap (OPT).
C.6 EXPERIMENTS ON MATERIALS PROJECT (MEGNET) DATASET
In Table 10, summarize the performance of PRDNet and the baseline models the Materials Project (MEGNet) dataset. PRDNet achieves the lowest MAE for bulk modulus (0.031 log(GPa)) and shear modulus (0.062 log(GPa)). and competitive result for band gap (0. 187 eV).
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Algorithm PRDNet Forward Propagation
Require: Crystal structure graph G = (V,&), atomic positions [r: lattice parameters
Ensure: Property prediction E l
Initialize: Node features h (0) Embed Zz) for ,....,N
Compute: Edge features Gtj = RBF(dg)) @ SBF(O:jk) dig
for = 0 L- do
for each attention head h to H do
Compute projections:
K (R)
(k) E (Eq. 8)
Calculate attention: (h) (Eq. 9)
(h)
Compute messages: m (Eq. 10)
end for
Aggregate multi-hean messages: m (Eq. 11)
Apply linear projection: h, (1+1) W comcar III a58
Normalize and activate: h (1+1)_ B, h (1) SiLU(BatchNorm((1) ()(
b((+1)
end for
Physics Integration:
Compute form factors: (H) MLP form (h) (1) for I,...,N
Calculate structure factors: Fakt = وド= fj exp(2rih r2) for (h, k. H. H. f,(H)
Construct diffraction map: F Fooncat Re(FF) @ Im())
Feature Fusion:
Pooling: MeanPool( (
Ji=1
Process diffraction: d = MLPPatt Pair concat)
Fuse modalities: Zfused = MLP (Ig + d))
Final Prediction: ò MLP our (Zfuse))
return Ú
Table 8: Mcan Absolute Error comparison on the JARVIS-DFT-3DD spta-split strategy CrystalFormer (Taniai al.), applying random seed 123 and dividing the dataset into training, validation, and test sets with an 8:1:1 ratio.
dataset. We adopt the same
Form. Energy
Bandgap (OPT) eV
Bulk Modulus Shear Modulus
Total Energy eV/atom
Bandgap (MBJ) eV
E eVV
Method
eVVatom
log(GPa)
log(GPa)
CGCNN
0.039
0.164
0.107
0.153
0.044
0.374
0.099
SchNet
0.048
0,219
0.109
0.162
0.067
0.459
0.198
MEGNET
0,045
0.175
0.120
0.201
0.048
0,409
0,089
GATGNN
0.042
0.201
0.107
0.155
0.051
0.450
0.079
Matformer
0.041
0.166
0.334
0,386
0.041
0.311
0.057
Crystalfommer
0.032
0.125
0.077
0.141
0.034
0.305
0.060
Crystalframer
0.038
0.132
0.081
0.134
0.042
0.330
0.070
eComFormer
0.037
0.165
0.200
0.275
0.035
0.301
0.055
EwaldMP
0.059
0.209
0.120
0.169
0.084
0.451
0.120
PotNet
0.034
0.141
0.106
0.155
0.062
0.301
0.079
ReGNet
0.038
0.195
0.211
0.162
0.047
0,336
0.071
PRDNet
0.025
0.138
0.075
0.132
0.024
0.242
0.040
D HYPERPARAMETERS
We provide detailed hyperparameter settings used in the benchmark models for fair comparison.
We use the released implementation from the official repo, with 300 training epochs, batch size of 64, an initial learning rate of 0.001 scheduled by OneCycleLR, and weight decay set to x 10- The graph encoder uses 256 atom features with convolutional layers, followed by a fully connected network with hidden dimension 128. Mean squared error (MSE) loss is used for regression and negative log-likelihood (NLL. loss with log-softmax for classification.
CGCNN:
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Table 9: Mean Absolute Error comparison the JARVIS-DFI:3D-2021 dataset. We adopt the baseline results reported by ReGNet (Nic et al., 2025) and Crystalframer (Ito et al., 2025).
Form. Energy
Bandgap (OPT) cV
Total Energy
Bandgap (MBJ) cV
Bhull
Method
cV/ttom
cV/atom
cV
CGCNN
0.063
0.200
0.078
0.410
0.170
SchNct
0.045
0.190
0.047
0.430
0.140
MEGNET
0.047
0,145
0.058
0,340
0,084
GATGNN
0.047
0.170
0.056
0.510
0.120
Matformer
0.033
0.137
0.035
0.300
0.064
Crystalformer
0.031
0.128
0.032
0.274
0.046
Crystalframer
0.026
0.117
0.028
0,242
0.047
eComFormer
0.028
0.124
0.032
0,280
0.044
PotNet
0.029
0.127
0.032
0,270
0.055
ReGNet
0.027
0.126
0.027
0.240
0.043
PRDNet
0.025
0.138
0.024
0.242
0.040
Table 10: Mean Absolute Error comparison on the Materials Project (MEGNet) dataset. We adopt
the baseline results reported by ReGNet (Nie al., 2025).
Form. Energy Bandgap
Bulk Modulus Shear Modulus
Method
eVaatom
eV
log(GPa)
log(GPa)
CGCNN
0.031
0.292
0.047
0.077
SchNet
0,033
0.345
0.066
0.099
MEGNET
0.030
0.307
0.060
0.099
GATGNN
0.033
0.280
0.045
0.075
Matformer
0,021
0.211
0.043
0.073
Crystalformer
0.019
0.198
0.038
0,069
Crystalframer
0.017
0.185
0.034
0.068
eComFormer
0.018
0.202
0.042
0.073
PotNet
0.019
0,204
0.040
0,065
ReGNet
0.017
0.189
0.033
0.063
PRDNet
0.020
0.187
0.031
0.062
We implement SchNet in SchNetPack and the default training script, with 500 training epochs, batch size of 64. The graph was constructed with a neighbor cutoff of 8 Aand Gaussian radial basis functions with 80 bases. The graph encoder used hidden size of 128 and 3 interaction blocks. Optimization is performed with AdamW using a learning rate of 1 x 10 10-3 and weight decay of 10-5 Training is with checkpointing validation loss, without learning rate scheduler. MSE loss used for training regression tasks and the cross entropy loss for classifica- tion task.
SchNet:
MEGNet: We adopted the MatGL implementation and the default training script. Crystal struc- tures converted into graphs with a neighbor cutoff of 4 Aand Gaussian radial basis expansion of 100 centers (width = 0.5). The embedding dimensions are 16 for nodes, 100 for edges, and 2 for global states. The network contains MEGNet blocks with hidden sizes of (64, 32) in the input MLP, (64, 64, 32) in the convolutional MLP, and (32. 16) in the output MLP. The softplus2 is used as activation function. The model trained with batch size for up 1000 epochs using Adam optimizer, mean squared eтror loss, and early stopping with patience of 500.
GATGNN: We used the released implementation with graph attention layers. The network con- tains 4 GAT layers, each with 128 neurons and attention heads. The default unpooling strategy is fixed-cluster. We train with batch size of 64, learning rate of x 10-3 and Adam optimizer with weight decay 10-4, A MultiStepLR scheduler is applied with milestones at 150 and 250 epochs and decay factor 0.3. Training runs for up 1000 epochs with 1 loss for regression tasks (CrossEntropyLoss for classification) and early stopping with patience of 150.
Matformer: We use the released implementation from the official repo, with 500 training epochs, batch size of 512, initial learning rate of 0.001 scheduled OneCycleI.R with 2000 warmup
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steps, and weight decay set to 1 x 10-5, The Matformer encoder uses 128 node features with 5 convolutional layers and attention heads per layer, followed by a fully connected network with hidden dimension 128. The model employs 02-dimensional atom imput features, 128- 28-dimensional edge features, and uses k-nearest neighbor strategy with cutoff distance of 8.0 Å and maximum 12 neighbors. MSE loss is used for regression and NLL loss with log-softmax for classification.
We use the released implementation from the official repo, with 500 training epochs, batch size of 256, an initial learning rate of 0.0005 scheduled by inverse square root without warmup, and weight decay set to 1 x 10 The CrystalFormer encoder uses 128 model dimensions with transformer layers. 8 attention heads, and a feed-forward dimension of 512. The model employs adaptive cutoff with sigma value of -3.5, lattice range of 2, and uses both real and reciprocal space representations with Gaussian basis functions. Position encoding is applied with sestance-based features (64 dimensions) and scale factors of 1.4 for real space and 2.2 for reciprocal space. LI loss is used for regression tasks and cross-entropy loss for classification tasks, with average pooling for graph-level predictions.
Crystalformer:
We use the released implementation from the official repo, with 500 training epochs, batch size of 256, an initial learning rate of 0.0005 scheduled by inverse square root without warmup, and weight decay set to x 10-5. The CrystalFramer encoder uses 128 model dimensions with 4 transformer layers, 8 attention heads, and a feed-forward dimension of 512. The model employs adaptive cutoff with sigma value of -3.5, lattice range of 2, and uses real space representations with Gaussian basis functions. Position encoding is applied with distance- based features (64 dimensions) and angle-based features (64 dimensions), with scale factors of 1.4 for real space and 2.2 for reciprocal space. LI loss is used for regression tasks and cross-entropy loss for classification tasks, with average pooling for graph-level predictions.
Crystalframer:
e eCoomormer:: We use the released implementation from the official repo, with 500 training epochs, batch size of 512, an initial learning rate of 0.001 scheduled by OneCycleLR with 2000 warmup steps, and weight decay set to 0. The eComformer encoder uses 256 node features with 3 convolutional layers and attention head per layer, followed by a fully connected network with hidden dimension 256. The model employs 92-dimensional atom input features, 256-dimensional edge features, and uses k-nearest neighbor strategy with cutoff distance of 8.0 Á and maximum 12 neighbors. The architecture incorporates equivariant updates with second-order representations and uses radial basis functions with range from -4.0 to 0.0 for edge encoding. MSE loss is used for regression and cross-entropy loss for classification.
We use the released implementation from the official repo, with 100-300 training epochs, batch size of 64, an initial learning rate of x 10-3 optimized by AdamW, and no weight decay. A OneCycleLR scheduler with 2000 warmup steps is used to anneal the learning rate. Crystal struc- tures are converted into graphs using 92-dimensional CGCNN atom features, a neighbor cutoff of 8 Á and at most 12 neighbors per atom. Local bond distances are expanded with a radial basis function (RBF) expansion into 256-dimensional edge features, while long-range periodic interac- tions are modeled by an and exponential potentials, which are further embedded into a 256-dimensional space. The PotNet encoder stacks gated message- paaggggg layers with 256 hidden channels, fol- lowed by global mean pooling and fully connected network with hidden dimension 256 and ShiftedSoftplus activation. MSE loss is used for all regression targets.
PotNet:
branch over R shells using combination of zeta
pressage--aasingg
EwaldMP: We adapted the released implementation from the official repo for property prediction, with 100-300 training epochs, batch size of 64, an initial learning rate or 0.0005 (task dependent: 0. 0.0001-0.001) scheduled by PlanaaeL with 0.8 fraction, 10 epochs patience, and 0.2 warmup steps. The AdamW optimizer was used with weight decay of 1e-6. Architecturally, we use the SchNet message passing network as backbone with 512 hidden channels, 256 filters, 4 interaction blocks, and 200 Gaussian radial basis functions, using a cutoff radius of 6.0 À and at most 50 neighbors constructed under periodic boundary conditions. Long-range electrostatics are modeled by Ewald message passing module with reciprocal space on a 3 x 3 x 3 voxel grid in (k}-space, applies Gaussian radial basis Fourier filters with a 16-dimensional linear bottleneck. The model uses additive pooling as readout, and MSE loss for regression and cross-entropy loss for classification tasks.
ReGNet/ReciNet:WW We use the implementation with 100-300 training epochs, batch size of 64, an initial learning rate of 0.0008 (task-dependent: 0.0006-0.001) scheduled by OneCycleI.R with
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30% warmup steps. and weight decay set to le-5. The ReGNet architecture uses 304 hidden dimensions with blocks, where each block integrates a short-range geometric GNN and a re- ciprocal space block for long-range interactions, followed by fully connected decoder network with hidden dimensions [304, 304, 152). The model employs 92-dimensional CGCNN atom input features projected to 304 dimensions, 304-dimensional RBF-expanded edge features with scaling factor -0. -.755/dstannce, and uses k-nearest neighbor strategy with maximum 16 neighbors. The architecture incorporates dual-space modeling with Fourier transforms in reciprocal space using fractional coordinates and reciprocal lattice vectors, and uses radial basis functions with 256 ker- nels ranging from -4.0 to 4.0 for edge encoding. MSE is used for regression and cross-entropy loss for classification.
E BACKGROUND OF PREDICTION TARGETS
Formation Energy. The formation energy (Gillan, 1989) quantifies the thermodynamic stabil- ity of a material and is defined as the energy required to form a compound from its constituent elements in their reference states.
Eterm
Exot
Enapes niplis
(83)
where Exor is the total energy of the compound, n; is the number of atoms of element in the for- mula unit, and μ4 is the chemical potential (reference energy) of element i. A negative formation energy indicates that the compound is thermodynamicall favorable with respect to decomposi- tion into its elements. The unit we used is eVattom obtained by dividing the total energy by the number of atoms.
Band Gap and Metal/Non-metal Classification The band gap Eg (Zaanen al.. 1985) is funda- mental electronic property defined as the energy difference between the conduction band minimum (CBM) and the valence band maximum (VBM):
Fgg FCBM FVRM-
(84)
The unit used is eV. Materials with Eg > classified as semiconductors insulators depending on the magnitude of the gap, whereas those with Eg = 0 are considered metallic. Hence, the band gap also serves basis for the metal/non-metal classification of materials.
Bulk Modulus. The bulk modulus B (Makishima Mackenzie, 1975) represents resistance to uniform compression and is defined as:
OP 0v
B B
(85)
where V is the volume and P is the pressure.
Shear Modulus. The shear modulus G' (Makishima & Mackenzie, 1975) characterizes the mate- rial's response to shear stress and is calculated from the elastic stiffness tensor or averaged using the Voist-Reuss-Hill approximation.
G
(86)
wherer is the shear stress and y the shear strain. The unit used is GPa, and applied logarithmic transformation to its values for comparison with the leaderboard results.
Young's Modulus. The Young's modulus E (Kalra et al., 2016) measures the stiffness under uniaxial stress. For isotropic materials, it is calculated as:
9BG
(87)
3B + G
The unit we used is GPa, and we applied logarithmic transformation to its values for comparison with the leaderboard results.
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Exfoliation Energy. The exfoliation energy (Jung al., 2018) quantifies the energy cost to isolate a single atomic layer from bulk layered material, which is critical in evaluating the feasibility of two-dimensional material synthesis. It is defined as:
E Emono
Ebal/ /N
Bext
(88)
where Emons is the energy of the isolated monolayer, Ebul is the total energy of the bulk material containing N layers, and A the surface area of the monolayer. lower exfoliation energy indicates easier mechanical or chemical exfoliation. The unit we used is meV/atom, obtained by dividing the exfoliation energy by the number atoms.
26


Reviews:
[{"review_id": "di30ZntNZ7", "reviewer": "Reviewer_NZ8C", "summary": "This paper proposes PRDNet, a novel approach for crystal material property prediction. PRDNet integrates graph representation of materials with a learned diffraction module, which is invariant to crystallographic symmetries. Benchmark experiments show that PRDNet achieves better property prediction accuracy than baselines.", "questions": "No additional questions.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Propose a novel idea of introducing learnable diffraction module into the crystal graph neural network model. The good property of achieving crystallographic symmetries and discriminating distinct crystal structures is very useful.\n- Achieve state-of-the-art performance on benchmark datasets.\n- Generally good and well-organized paper writing.", "weaknesses": "- The idea of using learnable diffraction module is good, but authors could clarify more clearly about its advantages over existing approaches. Satisfying crystallographic symmetries is also focus of many existing approaches, but they may fail to distinct crystal structures (as mentioned in line 87). So authors are encouraged to discuss how existing approaches fail to do it and why the proposed learnable diffraction module succeed.\n- While the proposed PRDNet adopts a graph transformer architecture as backbone, could the learnable diffraction module be integrated into other backbone models, or even existing crystal material property prediction model (e.g., CGCNN and Matformer)? Authors are encouraged to discuss this generalizability and conduct experiments about integrating diffraction module into more different model architectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "etcv9w6HDG", "reviewer": "Reviewer_dNJc", "summary": "PRDNet augments a standard crystal graph encoder with a reciprocal-space “pseudo-particle diffraction” pathway. Instead of fixed X-ray form factors that can’t tell identical elements in different local environments apart, the model learns per-atom form factors that depend on |Q|, element type, and a graph-encoded local environment, making the diffraction signature itself discriminative for crystal property prediction. Across Materials Project, JARVIS-DFT, and Matbench benchmarks, PRDNet delivers superior or competitive performance.", "questions": "See weaknesses.", "limitations": "", "rating": 2, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. Reciprocal-space encoding captures long-range periodic interactions that finite-receptive-field graph encoders miss, which can otherwise collapse distinct crystals to the same embedding.\n2. It’s SOTA/competitive across MP, JARVIS-DFT, and multiple Matbench tasks, with best-in-table errors/accuracy on MP and strong wins on JARVIS/MB.", "weaknesses": "Majors:\n1. The paper discusses ReGNet/ReciNet and critiques its reciprocal block, but it does not cite or position against other highly related long-range approaches that also reason in reciprocal/Ewald space (e.g., EwaldMP [1] and PotNet [2]). Please add these citations and discuss and compare to them in the main text.\n2. You evaluate on your own MP/JARVIS selections and Matbench tasks, but we cannot directly compare against the MP/JARVIS setup used by all your other baselines, e.g., Crystalformer [3]. Please add a table on that benchmark or clearly justify why not.\n3. The message-passing and “reciprocal head + fusion” pipeline reads very close to ReGNet/ReciNet [4] as cited. The paper argues ReciNet’s reciprocal block omits key physical dependencies, but eventual formulation and implementation of this work is incremental to theirs. Please show the ReciNet's result over you benchmark to justify your novelty and contribution.\n\nMinors:\n1. Use \"long-range\" & \"short-range\" instead of \"long-term\" & \"short-term\".\n2. Use \"Matformer\", \"Crystalformer\" and \"Crystalframer\" in Appendix D.\n3. Line 161: \"on\" -> \"in the\"\n4. Line 205: \"task\" -> \"tasks\"\n5. Line 208: Remove \")\"\n\n\n[1] Ewald-based Long-Range Message Passing for Molecular Graphs. Arthur Kosmala, Johannes Gasteiger, Nicholas Gao, Stephan Günnemann.\n\n[2] Efficient Approximations of Complete Interatomic Potentials for Crystal Property Prediction. Yuchao Lin, Keqiang Yan, Youzhi Luo, Yi Liu, Xiaoning Qian, Shuiwang Ji.\n\n[3] Crystalformer: Infinitely Connected Attention for Periodic Structure Encoding. Tatsunori Taniai, Ryo Igarashi, Yuta Suzuki, Naoya Chiba, Kotaro Saito, Yoshitaka Ushiku, Kanta Ono.\n\n[4] ReciNet: Reciprocal Space-Aware Long-Range Modeling for Crystalline Property Prediction. Jianan Nie, Peiyao Xiao, Kaiyi Ji, Peng Gao.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vjyXQY4Dwk", "reviewer": "Reviewer_QDmo", "summary": "The paper presents a new architecture for processing crystalline graphs. The architecture makes use of Fourier/diffraction space features to enable capturing long-range dependencies. Empirical results validate this.", "questions": "- Is it possible to add on your diffraction modules to the existing GNNs to provide a fairer ablation? I'm interested to see if incorporating this information to the GNNs would improve performance. As a baseline, you could just add some more generic parameters (e.g., equivariant layers) to the GNNs and compare to that.\n- Can you match the number of parameters in evaluations? Basically, if you can show that for a given number of parameters, it is better to use diffraction modules (or some mix of diffraction modules and attention layers), the results would be convincing.\n- How is the diffraction module different from the Fourier transform employed in [1]? I think yours is probably more elaborate, but you should take some paragraphs to explain this, since Fourier features have been used for a while now.\n- You could consider generative tasks, although this could be tight given the rebuttal timeline. I don't expect such experiments to be done within the rebuttal period, but if they were to be done, I think the audience would be more interested.\n\n[1] Jiao, Rui, et al. \"Crystal structure prediction by joint equivariant diffusion.\" Advances in Neural Information Processing Systems 36 (2023): 17464-17497.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper provides a principled approach to modeling crystalline materials in ML. The use of diffraction-space features is well-motivated from the hundred years of crystallography which have relied on such features.\n- Empirical results show the improvement over many strong baselines.", "weaknesses": "see questions", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)