MatRIS: Toward Reliable and Efficient Pretrained Machine Learning Interatomic Potentials
Authors: Yuanchang Zhou
Siyu Hu
Xiangyu Zhang
Hongyu Wang
Guangming Tan
Weile Jia
Venue: ICLR 2026 Poster (2026)
Keywords: Foundation Machine Learning Interatomic Potentials
Training Efficiency
Accuracy
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: MatRIS, foundation MLIP

Abstract: Foundation MLIPs demonstrate broad applicability across diverse material systems and have emerged as a powerful and transformative paradigm in chemical and computational materials science. Equivariant MLIPs achieve state-of-the-art accuracy in a wide range of benchmarks by incorporating equivariant inductive bias. However, the reliance on tensor products and high-degree representations makes them computationally costly. This raises a fundamental question: as quantum mechanical-based datasets continue to expand, can we develop a more compact model to thoroughly exploit high-dimensional atomic interactions? 
In this work, we present MatRIS (\textbf{Mat}erials \textbf{R}epresentation and \textbf{I}nteraction \textbf{S}imulation), an invariant MLIP that introduces attention-based modeling of three-body interactions. MatRIS leverages a novel separable attention mechanism with linear complexity $O(N)$, enabling both scalability and expressiveness. MatRIS delivers accuracy comparable to that of leading equivariant models on a wide range of popular benchmarks (Matbench-Discovery, MatPES, MDR phonon, Molecular dataset, etc). Taking Matbench-Discovery as an example, MatRIS achieves an F1 score of up to 0.847 and attains comparable accuracy at a lower training cost. The work indicates that our carefully designed invariant models can match or exceed the accuracy of equivariant models at a fraction of the cost, shedding light on the development of accurate and efficient MLIPs.

Published as a conference paper at ICLR 2026
MATRIS: TOWARD RELIABLE AND EFFICIENT PRE- TRAINED MACHINE LEARNING INTERATOMIC POTEN-
TIALS
Yuanchang Zhou Siyu Hu2.2
Xiangyu Zhang
Hongyu Wang
Guangming Tan 1.2 Weile Jia 1,2,-
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences
University of Chinese Academy of Sciences
Schooo of Advanced Interdisciplinary Sciences, UCAS
zhouyuanchang223s, jiaweile) @ict.ac.cn
ABSTRACT
Foundation MLIPs demonstrate broad applicability across diverse material sys- tems and have emerged as powerful and transformative paradigm in chemical and computational materials science. Equivariant MLIPs achieve state-of-the -frt accuracy in a wide range of benchmarks by incorporating equivariant inductive bias. However, the reliance tensor products and high-degree representations makes them computationally costly. This raises a fundamental question: as quan- mechanical-basee datasets continue expand, can develop a more com- pact model to thoroughly exploit high-dimensional atomic interactions? In this work, we present MatRIS (Materials Representation and Interaction Simulation), an invariant MLIP that introduces attention-basee modeling of three-body inter- actions. MatRIS leverages a novel separable attention mechanism with linear complexity O(N), enabling both scalability and expressiveness. MatRIS deliv- ers accuracy comparable to that of leading equivariant models on wide range Seatbench-Discoveee MatPES, MDR phonon, Molecular of popular benchmarks Diiieevvrr-icoorrryyy as an example, MatRIS achieves an Fl dataset, etc). Taking
score of up to 0.847 and attains comparable accuracy at lower training cost. The work indicates that our carefully designed invariant models can match or exceed the accuracy of equivariant models at a fraction of the cost, shedding light the development of accurate and efficient MI.IPs.
INTRODUCTION
Quantum Mechanics (QM)-based calculations are the cornerstone of modern drug and material re- search, providing highly accurate modeling of interatomic interactions. However, their prohibitive computational cost makes large-scale simulations intractable (De Vivo et al. 2016 Jain et al. 2013b) Merchant et al. 2023). Machine learning interatomic potentials (MLIPs) have emerged as a pow- erful alternative, enabling accelerated, tingttineccale molecular dynamics (MD) simulations while retaining caar--uantum-chheeee
accuracy. With the increase QM-based reference data and model innovations, MLIPs have demonstrated remarkable accuracy and generalization in property predic- tion and materials discovery (Merchant et al. 2023 Barroso-Luque et al. 2024 Zhang et al. 2024
Yang et 2024 Zhang et 2025 Fu
2025 Wood al. 2025).
Graph neural networks (GNNs) have been widely adopted for 3D molecular modeling, where atoms are represented as nodes and interatomic interactions as edges (Qu & Krishnapriyan, 2024: Liao & Smidi, [223) Liao et al.) 22024b] Fu et al.) 2025). Through Message Passing (MP), node features are iteratively updated to capture local and global structural interactions. To enhance model expressive- ness and generalization, many MLIPs incorporate domain-specifie inductive biases (e.g., translation,
*Corresponding author

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Published as a conference paper at ICLR 2026
rotation, permutation, reflection invariance, equivariance). Depending how these symme- tries are encoded, MLIPs are broadly divided into invariant, equivariant and unconstrained architec- tures (Duval et al. 2024 Jacobs et al. (025). In invariant models, the structural descriptor is encoded based on attributes such as interatomic distances, bond angles, and dihedral angles (Gasteiger et al.) 2020 2022b; Klicpera et al. 2021 Deng al. 2023 /Zhang al.) (025). In equivariant models, higher-order equivariance is typically enforced through computationally intensive tensor products of rotation order L (Batzner et al. 2022
Batatia et al. 2022 Liao & Smidt 2023 Liao al. 2024b).
Previous work demonstrates that equiv- ariant models often deliver superior accu- racy (Batzner et al.} (2022). In contrast,
0.85
M
6.4x
unconstrained models do not explicitly encode
Fl score (higher is better)
13.0x
symmetries; instead, the model learns them
0.81-
from data or approximates them through data augmentation or auxiliary losses, leading to more flexible architectural design (Duval
+8%
0.77
jet al.]
2024
Neumann et al., 2024] Rhodes
0.73
eSEN-30M
Hi HIENet
eqi2 5 DeN5
Eanorm
eet al.
Meanwhile, the results in popular benchmark (Matbench-Discovenn leaderboard (Riebesell et al., 20255) indicate that equivariant GNNs achieve higher accuracy. When the training data is MPTri (Deng et al. 2023), eSEN-30M-MP (Fu al. 2025) and eqV2 S DeNS (Barroso-Luque et al.) 2024) reach Fl scores of 0.831 and 0.815, with energy errors of 0.033 and 0.036 m/atom, respectively. Despite the accuracy gains, the heavy equivariant operations make equivariant methods significantly more computationally expensive and memory-intensive. vestigation into the training cost of several mainstream pretrained models is summarized in Figure I cSEN-30M-MP and eq S DeNS show superior FI score while requiring 335 and 228 GPU days, respectively. This high computational demand can be caused by three main factors: 1) the intensive equivariant operations such as tensor products, 2) the large number of model parameters, and 3) the prolonged training schedules (e.g.. 100, 150, and 600 training epochs are reported for eSEN-30M-MP, eq DeNS, and SevenNet-13i5 (Kim al.) 2025). respectively).
2025).
0.69
MACE-MP-O
Nequii
DPA3 MatRIS
SevenNet
0.65
120
180
240
300
360
Training time (A100 GPU-days)
Figure 1: Trade-off's between training time and FI score of foundation MLIPs. Training times for eSEN and eqV2 estimated Nvidia-A100 Nequix (Koker & Smidi (025) was trained on JAX; all others PyTorch. Larger marks in- dicate models with more parameters.
Our in-
Incorporating equivariance into the GNN MIIPs serves as an implicit form of data augmentation. AlphaFold has shown that, with sufficient data, non-equivariant models can accurately predict pro- tein secondary structures (Jumper et al. 2021 Abramson et al. 2024). In MLIPs, the rapid increase in QM-based reference datasets (Barroso- Luque al.) 2024 Levine al. 2025 Gharakhanyan
et al. 2025 Sriram et al.] (025) motivates us to ask: Is the equivariance indispensable as the QM-based dataset continues to increase? Can we develop
compact architecture to capture the high-dimensional atomic interactions encoded in QM-based data sufficiently?
We have the following findings: 1) Recent studies (Yang al. 2024 Zhang et al. 2024
2025
show that invariant models offer reliable property predictions and enable a wide range of scien- tific applications while maintaining computational efficiency. 2) For a more compact architecture (i.e., one that can fully exploit QM-based data). Element types and pairwise interactions have been shown be insufficient for distinguishing graphs with different chemical properties (Xu et al. 2019). Incorporating three-body interactions is needed to exploit the knowledge in QM-based data. Self-attention mechanisms (Mazitov et al., 2025; Qu Krishnapriyan 2024) have proven be promising method in improving model expressiveness, also benefiting in model scalability.
Building on these insights, we introduce an invariant MLIP: interatomic potential for Materials Representation and Interaction Simulation (MatRIS). To the best of our knowledge, our model the first to explicitly leverage an O(N) attention mechanism to model three-body interactions. MatRIS consists of graph generation, feature embedding, graph attention, refinement, and readout block. We provide the ablation study of these modules in this paper. Putting all these modules ofth-ot-thhe-rt
together, MatRIS achieves state-of-the the-art (SOTA) accuracy and efficiency across wide range of
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chemical applications. Additionally, the novel separable attention has lower complexity (O(N) compared to full attention (O(N2). Across diverse benchmarks, MatRIS achieves competitive results. MatRIS-L achieves SOTA results compliant Matbench-Diseovery with F1 of 0.847 and a root mean square displacement (RMSD) of 0.0717. Moreover, MarRIS-S and MatRIS-M deliver accuracy comparable to eqV2 S DeNS and eSEN-30M-MP, respectively, while improving training efficiency by 13,0x and 6,4x, respectively. These results demonstrate MatRIS's strong potential for applications in materials science and drug discovery.
RELATED WORKS
Invariant MLIPs are models whose intermediate representations are invari- ant under rotations and translations (Duval et al. (2024) This invariance is achieved by using internal coordinates instead of Cartesian coordinates, with features such as interatomic distances, bond angles, and dihedral angles remaining unchanged under rotations and translations of the sys- tem (Schütt et al. 2017| Gasteiger et al.,
Invariant MLIPs.
2020 Novikov et al. 2220 Fan et al. 2022 Gasteiger ct al.)
20226 Chen & Ong 2022 Deng et al.
2023 [023/Zhang
al. (025). Early invariant MLIPs, such as
SchNet (Schüitt et al., 2017). CGCNN (Xe & Grossman 2018), and PhysNet (Unke & Meuwly)
2019). employ relative distances between node pairs and encode local geometric information via learnable radial basis functions. More recent MLIPs enhance representational expressiveness by incorporating higher-order many-body scalar features. For example, the DimeNet (Gasteiger al. 2020 (2022a) series introduced directional message passing, allowing angular information to be em- bedded in edge updates between atoms. The GemNet series (Gasteiger et al.) 2022b) Klicpera et al. 2021 further incorporates dihedral angles to improve performance. SphereNet (Liu et al. (022 and ComENet (Wang et al., 2022) proposed methods to efficiently extract four-body angles within local neighborhoods, avoiding the need to iterate over all three-hop neighbors. DPA3 (Zhang et al. 2025) builds upon the line graph series (LiGS), capturing higher-order interactions. Invariant MLIPs are progressively increasing their representational expressiveness while maintaining inherent computa- tional efficiency. Building upon these insights, we design MatRIS from an invariant perspective. We further provide detailed discussion in Appendix A highlighting how MatRIS differs from other MLIPs that incorporate three-body encodings or attention bbaed mechanisms.
Equivariant MLIPs are models where intermediate representations are in- variant (e.g., scalars) equivariant (e.g.. vectors or higher-order tensors) under rotations (Duval et al. 2024). Current equivariant MLIPs can be divided into scalarization- bassed models (Schütt
Equivariant MLIPs.
et al. 1021 Du et al. [023 Thölke & Fabritiis 2022. Aykent & Xia (025) and high-degree steer-
able models (Musaclia et al. [223) Batzner et 2002 Batatia et al. 2022) Liao & Smidi (023).
Scalarization- MIIPP model interatomic interactions in the Cartesian coordinate system while
restricting the set of operations on geometric features to preserve equivariance (Duval et al., 2024
Wang et al. [2044] Yin et al., 2025). On the other hand, high-degree steerable equivariant MLIPs use irreducible representations (irreps) to encode features, ensuring equivariance under 3D rotations.
Each irrep of degree L corresponds to a (2L + 1) -dimensional vector space (Batzner et al., 2022 Batatia et al., 2022: Liao & Smidi 2023). In equivariant GNN-based MLIPs, MP involves trans- forming and combining these type- I. vectors. To interact across degrees during MP, tensor products (by using Seebsch-Gordan coefficients to combine) are employed. To avoid excessive computational complexity, these models typically employ only low-degree equivariant representations (Park et al, 2024 Liao et al., [202bb; Fu et al. 2025). Equivariant MLIPs continue to deliver SOTA accuracy on various benchmarks (Tran et al. 2023: Riebesell et al. 2025). while remaining computationally demanding.
Unconstrained MLIPs. Unconstrained MLIPs do not impose strict constraints on their interme-
diate representations. Instead, these models typically learn symmetries directly from the data or incorporate additional loss terms to encourage symmetry learning (Duval et al.) 2024 Rhodes et al.) 2025). For example, Qu & Krishnapriyan (2024); Neumann et al. (2024): Rhodes et al. (2025) use data augmentation (applying random rotations to training samples) to learn rotational equiv- ariance and have demonstrated promising results. In addition. Neuumann et al. (2024) enhances stability in MD simulations by removing net force and torque, while Rhodes et al. (2025) intro- duces an 'Equigrad' loss to incentivize rotational invariance of energy. Unconstrained MLIPs have inference efficiency comparable to invariant MLIPs and more flexible architectures. These models
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demonstrate competitive accuracy in multiple benchmarks (Chanussot et al. 2021 Tran al. 2023
Riebesell et al. (025).. However, studies indicate that they may lead to errors in certain property
prediction tasks [uu et al. 2023 Póta al. 2025 Bigi et al. 2025
MATRIS
In this section, we introduce the detailed architecture of MatRIS, The interaction between the atom graph and the line graph is described in Section 3.1 The Graph Attention module is depicted in Section 3.2 In Section 3.3 we describe other key components of MatRIS. An overview of MatRIS shown in Figure and the model formalizations are detailed in Appendix
3.1 LINE-ATOM GRAPH INTERACTION
To model three-body interactions, explicitly construct a Line Graph. In the Atom Graph, nodes represent atom types and edges represent pairwise interactions (bonds), whereas in the Line Graph. nodes represent edges of the atom graph and edges represent three-body interactions (an- gles) (Harary& & Norman 1960 Whitney 1992).
Specifically, given an Atom Graph G (V", Ea), where Va is the set of atoms and E is the set of edges within a cutoff distance reue the corresponding Line Graph G¹ = (V', E*) is
Atom graph
Line graph
constructed as follows: ① Each node in V corresponds to an edge in E An edge E added between two nodes if their corresponding edges in E" share common atom, rep- resenting the angular information formed by the three atoms (i.e.. the three-body interaction). The conversion from the atom graph to the line graph is illustrated Figure 2
12
C23
Figure 2: Conversion from an Atom Graph to Line Graph.
For graph information fusion, update the Line Graph to obtain edge and angular features that encode three-body inter- actions. These updated edge features then propagated back to the Atom Graph, enabling the atomic features to incorporate higher-order information from the Line Graph.
3.2 GRAPH ATTENTION
In this section, we introduce the design motivation and implementation of the Dim-wise Softmax and Separable Attention mechanisms.
Dim-wise Softmax. Recent studies have shown that attention mechanisms play an important role
improving both the accuracy and scalability of MLIPs (Liao Smidt 2023 Liao al. 2024b
Qu & Krishnapriyan 2024). Existing approaches (Liao & Smidt 2023 Wang et al. 2024 Shao
al. 2025 Liao al. (2024b) typically compute attention weights agd to weight the value vectors
V E Degghoors where D denotes the hidden dimension. The weights depend on the fea-
tures of node and its neighbors (ii,, while the values V are obtained by applying nonlinear transformation to the fused edge and node features.
In these methods, the same attention weights are applied to all feature dimensions, implicitly as- suming equal importance across dimensions. However, this assumption limits the model's ability to distinguish the independent contributions of different feature dimensions. Our proposed Dim- wise Softmax computes attention scores independently for each feature dimension. Given input feature x E Reiighooss and neighbor list. N,, the Dim-wise Softmax is computed as follows:
exp(Fia)
Oid Dim- wise Softmax(z.a.NN ΣkEN(O) exp(xxa)
where a E x the attention weight matrix, Fid denotes the d-th feature of node i, and ((()) represents the set of neighboring nodes of node i. This approach preserves the independence
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of feature dimensions while emphasizing the relative importance of different neighbors in each di- mension, thereby enhancing the model's ability to capture local structural information.
Separable Attention. In molecular systems, interatomic interactions are directional, and each atom plays two roles: target node and source node. Most existing methods only aggregate informa- tion from the source node to the target node (Liao & Smidt 2023] Liao et al. 2024b, Wang et al, 2024 Shao al. 2025), which assumes symmetric information flow. However, this
always true in real physical systems. For example, in polar bonds, charged environments, or local defect structures, the effect of the source node on the target node can differ from the effect of the target node on the source node (Bengtsson 1999 Kühne & Khaliullin 2013). To address this, we introduce two independent sets attention weights: source attention weights and target attention weights. The first models how neighboring nodes affect the central node, while the second captures how the tral node influences its neighbors. In this way, the two roles of nodes are explicitly separated during aggregation. The overall workflow is illustrated in Figure B(b). Given the interaction Eij between target node v; and a source node Ujj we compute the attention weights as follows:
lij Linear(es) (ii)) and ldij Dim-wise Softmax (4g (4)
(2)
Sij = Linear(exs) and s04j Dim-wise Softmax(s:;,
(3)
Here, (i) and N(j) denote the indices of the target and source nodes, respectively. The final attention outputs are obtained as the weighted sum of taij, 80.j. and c'y- Here, d, is obtained by concatenating Gij. Uts and Vy. followed by target and source feature fusion through a gMLP (see Figure 3 dd). The two attention branches share the same computational flow and can therefore be executed in parallel. We also implement optimized kernels to improve training efficiency.
Generality Analysis. As mentioned earlier, many MLIPs are either unconstrained or equivari- ant. Unconstrained MLIPs are flexible, allowing Dim-wise Softmax and Separable Attention be applied directly. For equivariant MLIPs, symmetry must be preserved. To ensure this, Dim-wise Softmax computed invariant features (e.g., L 0), producing attention weights that are in- variant. These weights are then applied to equivariant features within each irrep channel, without mixing components of different orders, ensuring the features remain equivariant under geometric transformations. Separable Attention extends this approach with two branches, computing attention equivariantly and aggregating information separately over the indices of the target and source nodes.
3.3
OVERALL ARCHITECTURE
(a) MatRIS Architecture
(b) Graph Attention
(c) Refinement
(e) Feature Embedding
Cij
i
V
Fiy
Graph Generation
Concat
nn. Embi
Bessel
Fourier
Linear
Linear
Concat
8MLP
Feature Embedding
Envelope
Linear
Linear
Linear
Dim-wise Softmax
gMLP
Line Graph Attention
Agg
Atom Graph Attention
st.j
F
MLP
MLP
(f) Readout Block
Ao
Line Graph Refinement
Aae
Aaa
(d)
gMLP
Norm
Atom Graph Refinement
Concat
MLP
MLP
Pooling
M
XN
gMLP
SiLU
Sigmoid
Readout Block
Autograd
E.F.S,M
S
Figure 3: Overview of MatRIS. The model architecture (a) consists of feature embedding (c), graph attention (b), refinement (c), and readout block(f).

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Graph Generation. Given crystalline system G(Z, X, L). where Z R denotes the atomic numbers, X E [ represents the atomic coordinates, and L E R refers to the lattice. We first perform periodic repetition of the structure and then employ a radius-cutoff graph construction to represent it. Inspired by CHGNet (Deng et al.) 2023), we construct the atom graph Ga (with atoms nodes and bonds edges) based Your and the line graph G (with bonds as nodes and angles as edges) based on rou for the crystal structure.
Feature Embedding. Atomic numbers are initialized using trainable embeddings, while pairwise distances are encoded via learnable radial Bessel basis with envelope functioon (Gasteiger et al. 2020). For bond angles, we employ a learnable Fourier basis expansion following Deng et al. (2023). These input features are then passed through linear layer to generate initial node, edge, and three- body representations. This block is illustrated in Figure B)e).
The output of Graph Attention vi and e'y is concatenated and processed by gMLP operator (shown in Figure 3))).. This representation is further refined by an Envelope func- tion (Gasteiger et al., 2020) (ensures smoothness). The refined signals are then aggregated over neighboring interactions, enabling each edge to integrate local information. Finally, the output is transformed by a MLP operator to produce the residual node features Au; and edge features Sesj.
Refinement.
Readout Block. The Readout Block (Figure f)) takes node features is from the last layer. Af- ter the normalization layer, the features are passed through MLPs to predict atomic energies and magnetic moments (M). Atomic energies are summed to obtain total energy (E). To ensure the reliability (Bigi et al. 2025), forces (Fi) and stress (σ) are computed by Equation 4 V is volume.
OE
10E
F,
0 = V de 0x,
(4)
4 EXPERIMENTS
We evaluated MatRIS on Gatbench-Discovern benchmark (Section 4.1 MatPES benchmark (Sec-
tion (42)). MDR phonon benchmark (Section 4.3), Molecular zero-shot benehmark (Section (44)) The best results are in bold, and the second best are underlined, Moreover, we conducted ablation studies on the network modules and training methods (Section 4.5). Finally, we analyzed the eff- ciency of MatRIS (Section 4.6 In addition, the results of experiments on the Matbench-Discoven benchmark (non-compliant), LAMBench benchmark (Peng et al.) (026). Zeolite benchmark (Yin) et al. 2025), DPA2 test sets (Zhang et al. (2024) and MD stability evaluation are provided in Ap- pendix D The training strategies are introduced in Appendix C and the training hyperparameter settings are detailed in Appendix
4.1 МАTBENCH DISCOVERY
Dataset and Setting, The Matbench-Discoverry benchmark (Riebesell et al.) 2025) is well- established benchmark for evaluating the ability of models new material discovery. In this bench- mark, all models are required to optimize the geometry and predict the formation energy of each of the 256k structures in the WBM test set (Wang et al.) 202I). These results are then used to as- sess thermodynamic stability the ground state (0 K). In the "compliant" setting, all models required to use MPTrj (Deng et al. 2023) as the training dataset, whereas in the "non-compliant" setting, this requirement is relaxed (the results reported in Appendix [D.I]. More details the hyperparameters can be found in Table 12 Moreover, inspired by Zaidi et al. (2023) and Liao al. (2024a), we apply denoising pretraining MatRIS-M and MatRIS-L. The specific details
described in Appendix EE] Notably, this denoising is also adopted in several other works (ORB v2 MPTrj (Neumann et (202)). V2 S DeNS, and cSEN-30M-MP) Structures are relaxed using MatRIS and the FIRE (Bitzek et al. 2006) optimizer, with convergence reached after 500 steps or when the maximum force falls below 0.05 eV/Á.
Results of the Compliant Benchmark We summarize the comparison with other models in Ta- [ MatRIs-S/M achieves performance comparable to eq S DeNS/eSEN-300 30M-MPPPPPPPP while using fewer parameters and lower computational costs (see Figure for comparison of training cost).

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Table I: MatRIS performance on the compliant Suubench--Dssoveer benchmark with results on unique structure prototypes. "f'/""" stands for higher/lower is better.
Model
Param.
Flf
DAFT
Precisionf
Recallf
Accuracy"
MAE]
R21
K Kyrmet
RMSDL
CHGNet
0.41M
0.613
3.361
0.514
0.758
0.851
0.063
0.689
1.717
0.0949
MACE-MP-0
4.69M
0.669
3.777
0.577
0.796
0.878
0.057
0.697
0.647
0.0915
GRACE-2L
15.3M
0.691
4.163
0.636
0.757
0.896
0.052
0.741
0.525
0.0897
Allegro-MP-L
18.7M
0.751
4,516
0.690
0.823
0.915
0.044
0.778
0.504
0.0816
Nequix MP
0.71M
0.751
4.455
0.681
0.836
0,914
0.044
0.782
0.446
0.0853
SevenNet--3i5
1.17M
0.760
4.629
0.708
0.821
0,920
0.044
0.776
0,550
0.0847
NequIP-MP-L
9.6M
0.761
4.704
0.719
0.809
0,921
0.043
0.791
0.452
0.0856
ORB v2 MPTrj
25.2M
0.765
4.702
0.719
0.817
0,922
0.045
0.756
1.725
0.1007
HIENet
7.51M
0.777
4.932
0.754
0.801
0,929
0.041
0.793
0.642
0.0795
Eqnorm MPTrj
1.31M
0.786
4.844
0.741
0.838
0,929
0.040
0.799
0.408
0.0837
DPA-3,I-MPTH
4.81M
0,803
5.024
0.768
0.841
0,936
0.037
0.812
0.650
0.0801
eq DeNS
31.2M
0.815
5.042
0.771
0.864
0,941
0.036
0.788
1.676
0.0757
eSEN-30M-MP
30.1M
0.831
5.260
0.804
0.861
0,946
0.033
0.822
0.340
0.0752
MaIRIS-S
4.3M
0.811
5.127
0.784
0.840
0,940
0.036
0.803
0.730
0.0766
MaRIS-M
6.3M
0.833
5.363
0,820
0.847
0,948
0.033
0.820
0.542
0.0742
MatRIS-L
10.4M
0.847
5.422
0.829
0.865
0,951
0.031
0.829
0.489
0.0717
These results demonstrate the effectiveness of MatRIS. Moreover, MatRIS-L. achieves SOTA per- formance across all metrics, with an FI score of 0.847. It also achieves RMSD 0.0717 when comparing relaxed structures to DFT reference values.
4.2 MATCALC BENCHMARK
Table 2:
Summary of model performance on the MatCalc benchmark. All results were obtained using the MatCalc package and associated dataset.
Equilibrium
Near-qqullibrium
Model
Param.
d]
Efj
K4
G
CV
f/SDert
MatPES-trained models
M3GNet
0.66M
0.42
0.11
25
27
0.97
CHGNet
2.7M
0.43
0.082
24
21
23
0.91
TensorNet
0.84M
0.37
0.081
18
15
13
0.93
MatRIS
1.4M
0.54
0.068
13
0.96 16
MPTrij-taiind models
CHGNet
2.7M
0.51
0.092
17.0
30.0
24.0
0,830
MACE-L
5.7M
0.43
25.3
22.5
11.6
0.829
SevenNet-I3i5
1.17M
0.55
0.057
13.2
170.2
8.03
0.922
egV2 DeNS
31.2M
0.25
0.033
27.2
29.1
0.964
cSEN-30M-MP
30.1M
0.34
0.039
18.8
77.3
4.66
0.986
MarRIS-M M
6.3M
0.32
0.041
12.4
16.4
7.39
0.983
OAM-trained models
SevenNet-MF-ompu
25.7M
0.502
0.028
13.3
32.2
4.60
0.976
egV2 M
86.6M
0.235
0.017
25.4
17.5
80.4
0.999
cSEN-30m 30M-0AM
30.1M
0.299
0.089
11.9
14.8
4.35
0.996
MatRSS 1OM-OAM
10.4M
0.316
0.025
10.6
13.3
3.97
0.985
Dataset and Settings. MatCalc benchmark (Kaplan et al. 2025) covers equilibrium properties (relaxed structure similarity d, formation energy Ef), near-equilibrium properties (bulk modulus K, shear modulus G, constant-volume heat capacity CV, force softening f/fDrr). and is con- structed from test data collected from the Materials Project (Jain et al., 2013a). Alexandria (Schmidt) et al. 2024), WBM high-energy states (Wang et al. 2021). We compare models trained on MatPES- PBE Kaplan et al. 2025), MPTrj and OAM, with the results reported in Table We maintained con- sistent evaluation settings for all MPTrj- and OAM --trriiieed models. Specifically, for elastic moduli calculations, we applicd normal (diagonal) strain magnitudes of +0.01 and +0.005 for é 622, E33+ and off-diagonal strain magnitudes of ++0066 and +-0.33 for 623, 133 612-

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Results and Analysis. Table summarizes the performance of models trained on the MatPES- PBE, MPTrj and OAM datasets. For models trained on MatPES- SAPPES-BEE, MatRIS achieves the best overall performance, clearly outperforming other models in predicting formation energy Ef-
For models trained on MPTrj. MatRIS attains SOTA or near-SOTH results on 83% of the evaluated metries. Notably, MatRIS remains robust "Near-equilibrium" tasks regardless of the training dataset. Although the higher fraction of near-equilibrium structures in MPTrj amplifies PES "Soft- ening" (F/ (Deng al., 2024 Kaplan al., 2025), MatRIS maintains stable performance. Comparisons with CHGNet further indicate that changing the training dataset does not result in significant softening, underscoring the effectiveness of the architectural design.
The same trend is observed for models trained on OAM, where MaURIS-LOM-OAM performs ro- bustly across all metries and achieves SOTA performance overall.
4.3 MDR PHONON BENCHMARK
(a) MDR phonon benchmark
(b) Phonon dispersion
MACE
GRACE
SevenNet
(mp-149)
BeSe (mp-1541)
ORBv3
eSEN
MatRIS
Nakus
60
61
traiiee models
50
60
40
30
26
Frequency (THZ)
32
10
CsCl (mp-22865)
GaAgS, (mpp-432)
1919
OAM-trained models
15
10
5-
Wmax
Cy
Wave vector
Figure 4: (a) Summary of model performance on the MDR phonon benchmark. The evaluation metries include Wmaz (K), S (J/K/mol), (kJ/mol), and Cv (J/K/mol), where the reported values represent the MAE between the model predictions and the DFT results. (b) Predicted phonon dis- persion obtained using MatRIS with 0.01A displacement. The DFT results are taken from the phononDB dataset.
Dataset and Settings. The MDR phonon benchmark (Loew al. 2025) used evaluate the ability of MLIPs to predict phonon properties. It requires MLIPs to compute the maximum phonon frequency (Wmez), entropy (S), free energy (F"), and heat capacity constant volume (Cv) for approximately 10,000 structures. To ensure a fair comparison, we follow the evaluation protocol adopted Loew al. (2025). Specifically, we first optimize the structures using the FIRE optimizer (max steps=500, f.ax=)..005) Displacements are generated with a magnitude of 0.01 Á, and the properties are computed 300
Results and Analysis. Results for both the MPTrj-trained and OAM-trained models are shown in Figure Ma). a). MatRIS achieves competitive results among the MPTri-trained models. As the training dataset grows, MatRIS- 1SSS-00--AM achieves SOTA accuracy on most metrics, with a particularly significant improvement in the maximum phonon frequency (Wmaz). ). is worth noting that MatRIS- 10M-OMat achieves higher accuracy, with values of Wmaas S, F. and Cy being 7.08, 7.12, 12, and 1.91, respectively.
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We also selected four representative structures from the phononDB dataset (Togo) for visualization, as shown in Figure 4(b). We find that MatRIS-MP-I not only reproduces the highest frequencies but also aligns well with the DFT results overall.
4.4 MOLECULAR ZERO-SHOT BENCHMARK
Table 3: MatRIS performance on the Molecular zero-shot benchmark. Energy (E) is reported in kcal/mol and Force (F) in kcal/mol/A.
MD22 Ao--Ala-NHMoo
TorsionNet-500
ANI-Ix
AIMD-Chig
Model
E MAE
RMSE
MAEB
NABH
E
F
AIMNe12
0.38
0.55
0.58
82
DPA2-Drug
0.24
0.35
0.36
18
MACF-OFF-I.
0.14
0.21
0.23
2.29
5.205
5.82
3,948
8.25
3.853
DPA3-L24
0.06
0.09
MaIRIS-M
0.04
0.07
0.07
2.23
5.140
4.15
4.414
1,55
3.604
The MAE of the torsional barrier height is defined as the energy difference between the minimum and maxi- mum along the torsional rotation. The number molecules with barrier height errors exceeding kcal/mol.
Dataset and Settings. 2025) and Yang et al. (2055) respectively, while the other models were pre-trained on SPICE-
AIMNet and DPA2-Drug were trained on the datasets from Anstine et al.
MACE-OFF23 (Kovács al. 2025). which contains 951,005 small-molecule configurations. The
energies and forces of all configurations were computed the (11777-D3(31)/d62--222PPP level. Following Peng al. ) we selected the Ac-Ala3-NHMe from MD22 (Chmiela et al. 2023).
ANI-Ix (Smith et al. 2018). and AIMD-Chig (Wang et al., 2023) datasets, which contain 2,212, 8,861, and 19,800 configurations, respectively. The training parameters can found at Table I2 in appendix. For the MD22, ANI-Ix, and AIMD-Chig benchmarks, we corrected inconsistencies caused by differences between DFT functionals.
Results and Analysis. We selected available molecular pre-trained models (MACE-OFF- (Kovács et (025) and DPA3-L24 for comparison, and the results are summarized in Table 3 MatRIS-M performs well across four downstream datasets. On the TorsionNet-50k test set, its en- ergy prediction error reduced by about 22.2%-33.33 compared the current SOTA model DPA3. In the remaining three datasets, MatRIS-M ranks first in five out of six metrics and second in the re- maining one, demonstrating that effectively leverages the knowledge from the pre- pre-training dataset to achieve more reliable performance on downstream tasks.
4.5 ABLATION STUDY
We conduct ablation studies various modules of MatRIS and their corresponding training meth- ods. We train the MatRIS-S on MPTrj and evaluate on 15,000 randomly sampled structures from the WBM dataset, reporting performance using the MAE of formation energies.
Table 4: Ablation studies. Formation energy (Ef) MAE is in me V/atom,m, lower is better.
Index
Dim-wise softmux
Separable attention
Leamable envelope
Ef (MAE)
Index
Denoising pretraining
Graph-level loss
Ef (MAE)
magmom
/
28.0
27.2
2
/
28.4
X
28.0
3
X
29.1
3
X
/
29.7
X
X
31.3
(a) Effect of modules: dim-wise softmax, separable attention and learnalle envelope.
(b) Effect of training methods: denoising pretraining, magnetic moment prediction, and graph-level loss.

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Module Ablation. We evaluated the impact of Dim-wise Softmax, Separable Attention, and Learnable Envelope on model accuracy, as shown in Table 4a). Specifically, replacing Dim-wise Softmax with standard softmax (i.e., sharing the creased MAE to 28.4 me (Index 2). Subsequently, restricting attention-based aggregation to the source-to-target direction further increased MAE (Index 3). Finally, replacing the learnable envelope function with fixed one also degraded performance (Index 4).
weight across all feature dimensions) in-
Training Method Ablation. We further analyze the contributions of different training strategies to model performance, as shown in Table ()).. Pretraining with denoising (see Section C.3 in ap- pendix) effectively improves performance, as predicting noise helps mitigate over-smoothing (God- win et al. 2022; Zaidi et al., 2023). Notably, predicting magmoms also enhances accuracy, since magmoms help distinguish features in different chemical environments and, being a node-Ievel task, likely reduce over-smoothing during training (Deng et al., 2023). Finally, performing loss reduc- tion the graph level prevents the force loss from being blased by differences in system size (see Section C.2 in appendix for more detail), further improving performance.
4.6 EFFICIENCY-ACCURAG ANALYSIS.
In this section, we compare the computational efficiency of MatRIS with several SOTA models (benchmarked without t torch. compi le optimized kernels).
(a) System scaling test
Relaxation test
10a
Energy MAE (mev/aton (atu/att))
55
better
Inference time (ms)
M
better
50-
45-
10._
MarRiS family
MACE
SevenNet Iai5
35-
eqV2
CSEN 30M
100 200
500
1000
0.8
1.2
System size (atoms)
Throughput (samples/s)
Figure Efficiency-accuracy comparison.
The inference time across system sizes (excluding graph construction) is summarized in Figure 5 a). MatRIS demonstrates higher efficiency than V2 and eSEN but lower than MACE-L, as MACE-L utilizes only 2 layers while MatRIS-S and MatRIS-M employ and layers, respectively.
We further evaluate the models in practical applications. We randomly select 500 structures from the WBM dataset (Wang al. 202T) and perform relaxations using ASE, recording the throughput and energy MAE for each model. As shown in Figure b]b), MatRIS achieves a favorable balance between speed and accuracy, maintaining high precision while enabling faster computation.
CONCLUSION AND FUTURE WORK
In this work, we review the characteristics of invariant, equivariant and unconstrained MLIPs in the era of rapidly expanding QM-ba datasets. Equivariant MLIPs deliver superior accuracy while their reliance on tensor products and high-degree representations leads to prohibitive computational costs. Motivated by the question of whether strict equivariance remains indispensable, we introduce MatRIS, invariant MLIP that leverages attention to model three-body interactions. Across mul- tiple benchmarks, MaIRIS attains SOTA results, opening a new path toward accurate and efficient MLIPs
In future work, we will scale MatRIS to even larger QM-based datasets to further validate its ex- pressiveness. We aim to develop reliable distillation strategy to develop student MLIPs. We also plan to expand MatRIS to incorporate long-range electrostatics, thereby enhancing its applicability to more complex downstream tasks.
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ACKNOWLEDGMENTS
This work is supported by the following funding: National Science Foundation of China (T2125013, 92270206, 62372435), the Innovation Funding of ICT, CAS. Part of the training is performed on the robotic Al-Scientist platform of Chinese Academy of Sciences. The authors thank the ICT operations team for their strong support.
We thank Duo Zhang for helpful discussions. We also thank Lijun Liu (Osaka University) for evaluating the Liao et al. (2024b); BBrrroo-LLuqqe et al. (2024) and eSEN Fu et al. (025)
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APPENDIX
Additional related works
Explicit three-body modeling in MLIPs
Attention-based MLIPs
Details of MatRIS
Training strategies
Load-balance strategy
Loss-balance strategy
Details of denoising pretraining
Additional evaluation
D.
Mathench-Discover benchmark (non-compliant)
LAMBench benchmark
D3
Zeolite benchmark
D
DPA2 test sets
D.5
MD stability evaluation
Training details and hyper-parameters
A ADDITIONAL RELATED WORKS
A.1 EXPLICIT THREE-BODY MODELING IN MLIPS
Three-body information is important for predicting various material properties (Choudhary & De-
Cost 2021 Choudhary et al. 2023 Zhang et al. 2025). Several MLIPs explicitly model three-
interactions, including the ALIGNN series (Choudhary & DeCost 2021 Choudhary et al.) 2023), )
DimeNet series (Gasteiger et al. 2020 2022a). GemNet series (Gasteiger et al., 2022b; Klicpera al. 2021), M3GNet series (Chen & Ongl 2022 Yang et al. 2024). and DPA3 (Zhang al. (025), where bhree-body features are updated within the interaction block and then used to refine edge and node features. MatRIS also explicitly models three-body interactions, but differs in its update and aggregation strategy:
1. In MatRIS, three-body features are updated using an attention mechanism with O(N) com- plexity, which improves model accuracy while maintaining computational efficiency (see Ta), Index and 3).
2. During message aggregation, learnable envelope function is used to smooth the three- body contributions, instead of using a simple distance- based attenuation. This design yields better performance (see Table 4a), Index 3 and 4).
Beyond the above distinctions, MatRIS also achieves SOTA near- Sear-SOTA performance across mul- tiple benchmarks. Notably, on Matbench -issnchv-- (Riebesell et al. 2025), MatRIS M attains accu- racy comparable to eSEN-30M-MP but with significantly fewer parameters and substantially lower
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training cost. This demonstrates the effectiveness of combining three-body interaction modeling with attention mechanisms for materials modeling.
A.2 ATTENTION-BASE MLIPS
Full Attention MLIPs. In potential energy surface modeling, full-attention methods have been widely used, with representative models including DPA2 (Zhang et al. 2024). PET (Pozdnyakov & Ceriotti] 2023), and ESCAIP (Qu & Krishnapriyan, 2024). In these models, all features are concate- nated into a message tensor of shape [natoms, neighbors, DJ. where updated using full attention mechanism (asswai et al. 2023). This mechanism can fully leverage information from neighboring atoms and enable feature interactions. However, due to the O(N²) computational cost of full attention, the computation quickly becomes expensive the system size grows. To maintain efficiency, these MLIPs typically consider only two-body interactions.
the hidden dimension, and
In contrast, MatRIS employs attention mechanism with O(N) complexity capture both two- body and three-body interactions. This allows MatRIS to incorporate more geometric information and enhance the model's expressive power while maintaining computational efficiency.
Comple-Cymplyxity Attention MLIPs. Although there exist many MLIPs with attention mecha- nisms of O(N) complexity, such as Equiformer (Liao & Smidt 2023), ViSNet (Wang et al. 2024)) FreeCG (Shao et al. 2025). and MGT (Anselmi al.) 2024). we note that these architectures still differ from MatRIS in several technical aspects.
First, Equiformer is high-degree steerable equivariant MLIP that employs multi-layer perceptron attention (Liao & Smidt 2023) Brody et al. 2022) for feature encoding. with O(N) complexity, and can support vectors any degree However, unlike MatRIS, its attention mechanism not directly used to encode three-body interactions and only considers the source-to-targe direction.
ViSNet and FreeCG belong to equivariant MLIPs. Their attention mechanisms can capture up to four-body interactions and also have O(N) complexity. Nevertheless, their technical approaches differ fundamentally from that MatRIS: ViSNet and FreeCG implicitly extract and refine three- four-body features, without directly employing attention to encode and update such features. In contrast, MatRIS explicitly extracts three-body features and utilizes attention mechanisms to encode and update them.
is also worth noting that MGT employs attention and explicitly extracts three-body features. Nev- ertheless, its attention mechanism operates node and edge feature updates, and relies the ALIGNN module (Choudhary & DeCost 2021) to further refine three-body features. Similar to ViSNet and FreeCG, MGT leverages attention refine three-body features rather than directly en- code them.
In contrast to all the above attention- based models, MatRIS introduces separable attention mecha- nism that explicitly models three-body interactions. Its key innovations include:
1. Himension-wise: the attention weights vary across feature dimensions, distinguishing the relative importance of different dimensions.
2. Separable: it considers both source-to-target and target-to-source directions, generating separate attention weights for each direction.
3. Explicit and efficient modeling: due its O(N) complexity, MatRIS can efficiently and explicitly encode and update three-body features, thereby significantly enhancing the model's expressive power.
B DETAILS OF MATRIS
For completeness, in this section present the details of MaIRIS. Given material graph G (Z, X, L), the atomic numbers are denoted as Z E Rinatomxx the atomic positions as X m3 and the lattice as L 3333 In Graph Generation (see Figure 6). first perform periodic repetition, and then construct the Atom Graph G and the Line Graph G based on radial cutoffs ree and pf cut: respectively. We typically set < out for computational efficiency.
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G
Periodic repetition
4
GZZ,X,L)
0gjk
G
Figure 6: The detail of Graph Generation.
In Feature Embedding. the atomic types 2, are expanded using learnable embedding matrix A E IR [gmox_elementx D (see Equation 5) Following Gasteiger et al. (0220) the pairwise distances ri of G are expanded with Bessel basis functions, and smoothed with a envelope function(j((rg)) to ensure that pairwise interactions decay smoothly zero at the cutoff radius, as shown in Equation For the initialization of three-body interactions Bigk. follow Deng al. (2023) and employ the computationally simpler Fourier basis instead of spherical harmonics (we also tested spherical har-
monics but observed no change in accuracy), as shown in Equation 7 Finally, all initialized features undergo nonlinear transformations and are projected into a higher- ligees-dimensional space.
0 A(2 + +6
(5)
2 sin (
c = ((rrt)
V
(6)
ri
cos cos| (mn
m [0, Mmaa/ /2),
Loyk.
π
sin[(m - Mman/ / 2)0ss)] m E [Mman/ 2 + 1, Mmaz).
In the Graph Attention module, we consider the n-th layer, which contains the Atom Graph Go with features v. and eiy and the Line Graph G with features ei and k The two graphs share the edge features egy. To fuse information between the graphs, we first update G and then incorporate its output into Go to aid atomic feature updates (see Equation 8) Specifically, eij and 6ij are input
to the 'Line Graph Attention' module, producing Cij.l and 0 0.(m) The updated edge feature eig '()) encodes three-body interactions and is combined with of input the 'Atom Graph Attention" module, yielding u; (n)
'(n)
and egg (m))
(m) 0 (n)
< Line Graph Attention (eli) 0ixx)
ej Ejk
(8)
0 Cgg <
[n) ((n)
Atom Graph Attention (o]" Czj.1
((n)
Notably, in G", atoms are treated as nodes and pairwise distances as edges, whereas in G¹, pairwise distances are treated as nodes and boree-body interactions edges. Therefore, the computation of the Line Graph Attention and Atom Graph Attention follows the same computaion, differing only in their inputs. Here, we take the update of Go as an example and present the detailed operations of the Graph Attention module.
Given the input of the current attention layer, v; and Eij (for simplicity, we denote vf and e ((n) I as 0; and eij), we first generate the fusion feature:
gMLP(c.llu/lee))
Meanwhile, eij undergoes two nonlinear transformations. It is then passed through Dim-wist Soft- max to get the attention weights for the source and target nodes, sai and laij. These weights
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then element-wisg wise multiplied with dly and fused to produce the output of the attention layer, ty as shown in the equation:
tu,
Σ REN(1)
(tasy O dbg),
sv; Σ
(saks ckg))
kEN()
(10)
= DeoLLPtttollsss
After the Attention layer, we employ Refinement layer fürther enhance the geometric encod- ings. In this module, we also apply a learnable envelope function to smooth the potential energy surface. As before, we first update G and then G Taking the update of Go as an example, the inputs are of and Egj. and we fuse the features from the attention layer:
(n) mij
gMMM (MLP((/ll))le)))
(11)
(x) The fusion feature mij transformed nonlinearly to yield the edge update Acj-- For the node update Ae;, first apply the learnable envelope function (Equation [2], then aggregate, and finally apply a nonlinear transformation.
(n) Linear(e
(12)
A Au
MIP
Σ (Mi (n)
ml
kEN(()
(13)
Aei = MLP(m (m))
(N) used to directly predict the total energy (E) In the Readout block, the final atom features t, and magnetic moments (M);
F Σ MLP((.) ref(z))
(14)
M ML
Here, ref(2;) is obtained by performing least-squares fit to the dataset energies. The atomic forces F, and stress (or) are obtained via automatic differentiation of the energy with respect to the atomic Cartesian coordinates X; and the lattice strain tensor (E).
OE F = 0x,
=
(15)
V Fe
C TRAINING STRATEGIES
C.I LOAD-BALANCE STRATRGY
In most atomic datasets, the sizes of structures typically follow long-tailed distribution (Deng et al. 2023 Barroso-Luque et al. 2024). Figure a[)) visualizes the distributions of atom and edge counts in the MPTrj dataset (cutoff radius 6.0). In distributed training, allocating samples with fixed batch size may cause two issues: (1) GPU may receive only large samples, leading to out-of- memory (OOM) and limiting the maximum batch size; (2) assuming memory allows, some GPUs may have higher computational loads while others have lower loads, causing idle time and extra synchronization overhead. To address this, we adopt load balancing strategy:
1. Shuffle the entire dataset to ensure randomness, then split it into multiple chunks.
2. Within each chunk, sort samples in descending order of size.
3. Using greedy algorithm, assign samples sequentially to GPUs, prioritizing the GPU with the largest remaining capacity. If all GPUs are "full" or the chunk has no remaining sam- ples, batch is generated.
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(a)
(b)
10"
MetRI5-S
2.63
10
10
2.05 2.14
Sturctures
1.751.72
Speedup
100
200
300
400
Number of atoms
10
1
10
10a
5000
10000
15000
20000
Baseline
Load-balance
Batch
Number of edges
expansion
Figure 7: (a)Number of atoms and edges in MPTrj dataset. (b)Training speedup achieved by load balancing.
As shown in Figure 7b), with load balancing. the cumulative speedup reaches 1.35-1.75x, indi- cating that synchronization overhead due to load imbalance is significant during each training step- Building on this, the global batch size can be increased (denoted as "Batch expansion") resulting in a cumulative speedup of 2.05-2.63x.
C.2 LOSS-BALANCE STRATEGY
Besides load imbalance during training, the loss computation can also be imbalanced. This es- pecially clear for the force and magmom loss(e.g., node-level tasks). For example, the MSE loss (forces only) is defined as:
Nn Σ ppred
2
I
PFT
(16)
3NB
where Nn denotes the total number of atoms in this global batch.
This reduction method causes the losses of large samples to dominate during training, effectively making the model prioritize large samples and resulting in loss imbalance. To address this, we adopt graph-level loss reduction strategy (referred to as Graph-level loss). Specifically, when computing node-level losses such as force magmom, and then perform a second reduction across the global batch, thereby mitigating the loss imbalance. Ablation studies in Table b)) demonstrate the effectiveness of this method. The corresponding formula as follows:
first reduce the loss within each graph,
DFT
(17)
N
b=1
==1
where the number of graphs in this global batch, N₂ is the number of atoms in the b-th graph.
C.3 DETAILS OF DENOISING PRETRAINING
In this section, present the denoising strategy used in work. Several studies, such Noisy Node (Godwin et al. 2022) and DeNS (Lao et al. 2024a), have demonstrated that denoising can mitigate the over-smoothing of GNNs and improve generalization. However, these methods have restricted applicability. For example, Noisy Node must be applied to equilibrium structures, while DeNS extends non-equilibrium states but is limited to equivariant neural networks. In practice, most mainstream atomic datasets are not fully in equilibrium (Deng et al.) 2023 Schhidtt et al.) 2024 Barroso-Luque et al. (2024), and MLIPs are not necessarily equivariant GNNs. The training of MatRIS faces this scenario. For coneevuivar GNNs non-equilibrium structures, key challenge in applying denoising is how to encode forces. Inspired by Liao et al. (2024a), we project forces onto edges to scalarize the vectors. The detailed denoising strategy is described step by step as follows:
1. Given corruption probability, randomly select atoms corrupt.
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2. For each corrupted atom, project its force onto the relative position edges by computing inner products, resulting in a scalarized force feature F(projected).
3. We sample a random timestep and obtain the noise standard deviation o, from a linear schedule. We then add noise as where is standard Gaussian noise.
4. The noisy structure, the projected force F(projected), and the timestep f are used as inputs to the MatRIS model, which directly predicts the added Gaussian noise.
Step1:
Step2: projection
Step3:
Step4: prediction
selection
ox sampling
F (projjecedd] embedding
Gaussian noise(c)
MatRIS
F
drE
Figure 8: The workflow of denoising pretraining on MatRIS.
The overall workflow is illustrated in Figure 8 further fine-tuned using the pretrained weights.
After the denoising pretraining stage, the model is
D ADDITIONAL EVALUATION
D.1 SATBENCH-DISSOOEER BENCHMARK (NON-COMPLIANT)
We report the performance of non-compliant MaIRIS-10M-OAM models, all of which were cessed before September 24, 2025. In this benchmark, the training data for each model is not restricted. We follow the setup used by most models, with pretraining on the OMat24 (Barroso- Luque et al. 2024) dataset and joint fine-tuning with sAlex (Schmidt et al. 2024, Baarroo-LLqqe 2022) and MPTrj (Deng et al.) (023). Results on the full, unique, and 10k most stable splits are shown in Table 5 Table and Table 7 respectively. Although MaURIS-10M-OA parameters than most models, still delivers competitive performance across all splits.
has fewer
Table 5: MatRIS performance on the non-compliant Matbench- Disnnchey benchmark with results on full test set.
Model
Param
FIf
DAP
Precisionf
Recall
Accuracy!
MAE]
R2f
RMSDL
GRACE-21-OAM-L
26.4M
0.862
5,093
0.874
0.851
0,953
0,022
0.856
0.064
DPA3-3M-FT
3.27M
0.864
4.912
0.843
0.887
0,952
0.022
0.862
0.069
Nequip-OAM-L
9.6M
0.870
5.060
0.868
0.872
0.955
0.022
0.858
0.065
Allegro-OAM-L
9.7M
0.873
4.876
0.837
0.912
0,954
0.021
0.861
0.065
SahaNee-vl--MM
4.6SM
0.883
5.000
0.858
0.910
0.959
0.023
0.827
0.079
GevenNet-MF-ompee
25.7M
0.884
5.082
0.872
0.895
0,960
0.021
0.861
0.064
ORB
25.5M
0.887
5.159
0.885
0.888
0,961
0.023
0.820
0.075
eqV2
86.6M
0.896
5.243
0.900
0.893
0,965
0.842
0.069
-SEN-30M-ON
30,2M
0.902
5,281
0,906
0.899
0.967
0.018
0.860
0.061
MaRRIS-I0M-OAM
10.4M
0.903
5.275
0.905
0.901
0.967
0.019
0.864
0.060
D.2 LAMBENCH BENCHMARK
We evaluated the performance of MatRIS- CILRRIS-OOM-AA 10M-OAM on LAMBench benchmark (Peng et al.) 2026). which mainly tests the generalizability of MLIPs. The benchmark covers test domains including molecules, inorganic materials, and catalysis, and includes tasks such as force field prediction and property calculation.
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Table 6: MatRIS performance on the non-compliant Maabench-Discoowe benchmark with results on unique structure prototypes.
Model
Param.
DAF
Precision"
Recall
Accuracy1
MAE]
R21
Karmek
RMSDL
GRACE-21 200--.-
5.840
0.893
0.964
0.022
0,064
DPA3-3M-FT
3.27M
0.884
5.667
0.866
0.903
0.963
0.023
0.869
0.469
0,069
Nequip-OAM-L
9.6M
0.893
5.832
0.890
0.895
0.967
0.022
0.865
0.166
0.065
L.legro-OAM-L OAM--
9.7M
0.895
5.674
0.867
0,923
0.966
0.022
0.868
0.319
0,065
AlphaNet-v1-OMA
4.65M
0,901
5.747
0.879
0,924
0.968
0.024
0,831
0.644
0.079
SevenNat-MF-oopa
25.7M
0.901
5.825
0.890
0.911
0.969
0.021
0.867
0.317
0.064
ORB v3
25.5M
0,905
5.912
0.904
0.907
0.971
0.024
0.821
0.210
0.075
cqV2 M
86.6M
0.917
6,047
0.924
0.910
0.975
0.020
0.848
1.771
0.069
eSEN-30M 30M-OAM
30.2M
0.925
6.069
0.928
0.923
0.977
0.018
0.866
0.170
0.061
MadRIS- 10M-OAM
10.4M
0.921
6.039
0.923
0.918
0.976
0.019
0.871
0.218
0.060
Table 7: MatRIS performance on the non-compliant Matbench-Discovery benchmark with results on 10k most stable.
Model
Param.
Flf
DAF
Precisiont
Recall
Accuracy!
MAFJ
R21
RMSD
GRACE-2L-OAM-L
26.4M
0.980
6,280
0.960
1.000
0.960
0.025
0.835
0.064
DPA3-3M-FT
3.27M
0.987
6.369
0,987
1.000
0.987
0.019
0.901
0.069
Ncquip-OAM-L
9.6M
0.985
6.344
0.985
1,000
0.985
0.021
0.854
0.065
Allegro-OAM-L
9.7M
0.987
6.371
0.974
1.000
0,974
0.018
0.908
0.065
AlphaNet-vI-OMe
4.65M
0.965
6.312
0,980
1,000
0,980
0.020
0.882
0.079
SevenNet-MF-ompa
25.7M
0.970
6.346
0.985
1.000
0.985
0.019
0.888
0.064
ORB
25,5M
0.964
6.307
0,964
1,000
0,964
0,021
0.860
eq
86.6M
0.988
6,382
0,976
1.000
0.976
0.015
0.904
0.069
eSEN-30M-OAM
30.2M
0.971
6.350
0.971
1.000
0.971
0.016
0.901
0.061
BaIRIS-I0M-OAN
10.4M
0.986
6.366
0.973
1.000
0.973
0.015
0.904
0.060
We first performed the force field prediction task, in which the model is evaluated on datasets from the molecules, inorganic materials, and catalysis domains, predicting energies, forces, and virials. For fair comparison, we selected MLIPs trained on OMat+sAlex+MPTij as reference. The results are reported in Table 8 Overall, MatRIS-1OM- 10MOAMM achieves the best accuracy, and for molecules and inorganic materials, it generally outperforms other models in force prediction.
Table 8: Summary of model performance on LAMBench (Force Field Prediction). Energy (E) RMSE is in me V/toom Force (F) RMSE in me V/Á, Virial (V) RMSE in me V/atom.
GRACE- 2AM
SevenNet MF-mop
ORBv3-mpu
MaRIS 10M
E
Molecules
AIMD-Chig Wang al 2023
3.1
239.8
3.4
247.9
2.4
200.2
2.6
200.0
ANI-Ix Smalh di 2018
32.1
365.5
26.1
337.7
20.0
247.9
22.1
246.1
235.9
4.7
2.8
Catalysis
Vandernause et aL.
2022
5.5
99.4
60.6
12.7
100.8
39.1
6.8
92.6
51.2
13.8
110.9
47.8
Zinin et 2019
251.6
723.0
392.2
937.5
464.9
1169.4
609.2
963.4
Fermante: Vilaantee
3.0
95.6
Inorganic materials
Lopanitsyna stal.
2025
55.4
294.7
200.1 1
55.4
268.8
226.9
56.4
324.2
198.2
61.0
257.2
212.9
A045
70.1
309.7
243.1
69.8
265.5
235.8
71.0
330.6
242.4
75.6
261.2
251.6
Batzher
0.7
107.9
Tollss 2r
201
3.2
161.2
2.8
167.9
2.8
141.9
2.6
148.5
17.8
135.7
164.1
17.7
104.8
161.1
20.4
167.4
141.4
14.9
83.9
150.1
Sours Kulkarm
2023
162.8
6.5
173.6
6.5
182.7
176.1
In addition, we also evaluated the performance on property calculation tasks, which include reac- tion barrier prediction (OC20-NEB task), elastic constant prediction (Elastic Properties task), and molecular conformer energy prediction (Wiggle150 task). The results are reported in Table 9 In the OC20-NEB task, MatRIS-10M-OAM achieves competitive results, but we observe that its pre-
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dictions for reaction barriers (Ea) and reaction energies (d) while its success rate remains relatively high. We speculate that this is because MatRIS uses MAE to compute energy loss during training, without placing extra weight outlier data points, which amplifies their effect in this task; even so, the success rate not significantly affected. In the elastic and Wiggle 150 tasks, MaIRIS-IOM-OA achieves the best performance.
than those of other models,
Table Summary model performance on LAMBench (Property Calculation). Metrics are lower- is-better unless otherwise noted.
Model
OC20-NEB
Wander et 2025
Elastic Properties
Wiggle150 (Brew al. 2025
Ma (eV)
MAE.d (eV)
Thansfer (96)
Desarption
Dissociation
Shear Modullas (GPPa
Balk Modulus
MAE (kcal/mol)
RMSE (kcal/mol)
(96)
(96)
GRACE-2L-OAM
1.6
0.7
65.1
90.5
72.2
9.1
7.5
12.1
14.0
SevenNet-MF-ompa
2.1
1.3
66.9
92.9
68.3
9.5
7.5
11.0
12.8
ORBv3-mpa
2.3
61.7
87.4
72.2
7.6
12.9
MaiRIS-IUM- 10M-OAM
4.0
3.6
61.6
90.6
69.0
6.4
10.6
Overall, the cross-domain evaluation on LAMBench shows that 10M-SAMMM-AMMM performs con- sistently well across both force field prediction and property calculation tasks, achieving either the best or highly competitive results, demonstrating strong generalizability and application potential.
D.3 ZEOLITE BENCHMARK
The Zeolite Dataset Yin et al. 2025) comprises zeolite structures relevant to catalysis, ad- sorption, and separation applications. For each type, atomic trajectories were generated via AIMD simulations at 2000 K using VASP. We adopt the pre-par pre-partttioned training, validation, and test sets, containing 48,000, 16,000, and 16,000 structures per zeolite, respectively. The model's prediction targets are the total energies and atomic forces of the systems. The results are shown in Table 10 MatRIS achieves SOTA performance overall.
Table MatRIS performance on Zeolite dataset. Energy(E) MAE is in meV, force(F) MAE is in V/Å.
Deep Pot
AlphaNet
DPA3-L24
MatRIS-M
Type
E
F
F
F
E
F
AB Wopt
90
90
19
13.7
19.1
3.4
6.8
BCTopt
110
50
6.8
12
6.8
12.0
3.0
4.9
BPHopt
210
60
29
16
28.6
16.5
39.2
8.3
CANopt
150
90
18
15
17.3
15.1
10.2
6.5
EDlopt
40
50
8.7
13.5
10.8
15.1
3.6
6.2
FERopt
290
130
43
28
41.6
25.4
30.6
11.9
GlSopt
60
50
11
11
12.1
11.8
6.2
5.8
Wopt
150
70
13
9.7
12.4
4.0
5.9
LOSopt
110
70
21
12
21.0
13.1
17.9
6.5
LTAopt
150
64
12
21.3
13.5
20.5
7.0
LTTopt
70
110
15
11
13.4
11.2
14.4
6.7
NATopt
210
110
15
17.2
15.6
8.0
7.4
PARopt
90
70
18
24
21.4
24.2
9.5
6.8
PHIopt
60
120
20
14
19.6
14.4
15.9
7.5
SODopt
200
110
9.5
12
11.0
13.4
4.4
5.7
THOopt
160
60
17
15
20.2
16.9
7.5
8.0
D.4 DPA2 TEST SET
We evaluate the performance of the MatRIS model on the DPA2 dataset (Zhang et al. 2024 assess its ability to handle small-scale datasets. This composite dataset integrates 18 -opean- sub- sets (e.g., Alloy, Drug, HzO, OC2M) and is generated using various DFT software packages (e.g..
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Table II: MatRIS performance on DPA2 test from Zhang et 2024 Energy (E) RMSE
in metammm, force (F) RMSE is in meV/. 'OOM' indicates Out-Of-Memory error. 'NC' indicates that the model employs a non-conservative force prediction.
GemNet-OOCN
NequlP
MACE
cgV2NC
DPA3-L24
MatRIS-M
Dataset
Size
F
E
Alloy
71K
14.3
85.1
44,0
175.6
16.2
190.2
8.5
62.7
2_1
99.2
5.8
96.2
Cathode-P
59K
17.9
14.3
69.8
2.6
37.8
14.9
0.6
18.6
11.9
Cluster-P
139K
47.7
69.6
75.1
216.6
41.3
189.7
34.6
104.4
29.3
118.3
12.8
113.9
Drug
1380K
40.5
21.6
187.2
29.8
807.4
5.5
54.2
1.4
FerroBle-P
7K
1.5
17.9
1.1
23.0
2.3
31.7
1.1
13.0
0.3
13.4
0.2
10.9
2000K
129.1
97.4
226.1
45.2
9.0
128.0
77.9
SSE-PBE-P
15K
2.7
8.2
1.6
41.1
1.8
29.9
OOM
OOM
0.5
19.8
0.3
11.4
SemiCond
137K
20.5
180.7
182.8
40.8
108.3
H20-PD
46K
OOM
OOM
0.9
27.1
79.9
29.7
OOM
OOM
0.4
13.7
0.3
10.7
AgUAu-PBE
369.1
23.4
10.9
AlLMggCCu
24K
5.9
9.4
38.0
48.3
7.7
42,9
1.9
5.7
2.0
13.0
1.2
15.9
Sn
6K
8.4
33.7
18.2
62.2
5.2
19.6
2.9
49.8
2.3
45.9
V
15K
17.9
79.3
8.8
91.6
14,2
140.4
5.6
47,4
2.8
71.8
1,9
62.9
160.4
51.3
2.5
C12H26
34K
125.8
518.7
121.4
715.6
81.9
802.3
123.1
907.4
42.4
541.6
19.5
287.6
HIO2
28K
14.7
9.1
28.9
VASP, Gaussian, ABACUS), with the training data for each subset ranging from 6K to 2,000K. We evaluate MatRIS-A .M using the same training configurations and maintain consistency with the train- ing and test sets in /Zang al. (0244) to comparability. More detailed hyper-parameters are reported in Table
All results summarized in Table 11 with benchmark data for other models taken from Zhang iet al. (2024) 2025). Overall, MaRIS-M achieves better accuracy in both energy and force predic- lions, demonstrating its robustness across diverse domains. Notably, however, eq and GemNet- OC exhibit lower force errors than MatRIS in certain systems. This advantage likely stems from their con-conserv force prediction, where energy and forces are fitted separately the expense of physical consistency.
D.5 MD STABILITY EVALUATION
We assessed the stability of MatRIS by examining energy conservation in MD simulations under the NVE ensemble (microcanonical ensemble), using energy drift and thermostatted stability as evaluation metrics. The test data were taken from LAMBench (Peng et al. 22026), including organic molecules and inorganic material systems, which are out-of-distribution for MatRIS-M trained on MPTrj dataset. For each system, atomic velocities were first randomly initialized at 300 K using the Boxxeel-Boltzmang distribution. Subsequently, MD simulations were performed in the NVE ensemble for ps with a fs time step.
Figure 9 shows the results of the MD simulations. MatRIS-M is able to conserve energy over long simulations, with both the total energy and the kinetic temperature fluctuating around their initial values, and no large drifts are observed.
TRAINING DETAILS AND SYPEE-PARAMETERR
We summarize the hyperparameters of the models across different datasets and versions Table 12 For the MPTr dataset, we experimented with models of different sizes by varying the number of layers (S, M, and correspond to 4, 6, and 10 layers, respectively). The learning rate follows a cosine annealing schedule, decaying the minimum value at the last epoch. In distributed training.
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(a) C3N2H5
(b) H7C4NO
(c) NdPd3
2.0
Energy imeeitrom
Enengy onn (melaato))
1.5-
MatRIs-M
Energy drtt tmmtinatom
0.4
M
0.0
0.0
-0.5
-1.0
-0.1
10
30
40
50
90
70
80
93
(ps.
Time (p5)
600
400
Temperture (8)
500
300
400-
200
000-
200-
200
100
300
100
-100
80
(ps)
Time (95)
Figure 9: NVE MD simulations of three systems using MatRIS-M. The top shows the energy drift, and the bottom row shows the time series for the kinetic temperature.
we implemented load balancing scheme reduce synchronization overhead (see Section C. p. so the number of samples per GPU may vary; the table reports the average global batch size. Notably, the loss reduction distributed training performed using graph-level scheme to avoid bias benchmark, pre-trained the MatRIS-M and L models with a denoising process to enhance generalization, with details in Section CK
toward larger systems (see Section C.2). Additionally, for the Matbench-Disoon
Table 12: flyper-paraneee for MatRIS variants reported this paper. In MatRIS-OAE values outside parentheses denote the OMat configuration, those in parentheses the sAlex+MPTri fine- tuning configuration, and values without parentheses are shared.
Hy apper-parameters
MPIi-S
MPTi
MPIi L
MaiRIS OAM
SPICE M
MaPES- 1.4M
Others
Number of MaRIS layers
10
6
Dimenssio
128
Dimeasion
128
128
128
Dimension of three body features
128
128
128
128
128
64
128
Kadial basis function
Bessel
Bessel
Bessel
Bessel
Bessel
Bessel
Hessel
Pairwise curoff
600
Three body cutof!
4.0
4,5
4.5
4,5
4,0
4.0
4.0
Global batch size (arg)
512
512
320
512/320)
128
64
1/256
Optimizer
Adam/W
AdamW
AdooW
AdlamW
AdaonW
Adamn
Weight decay
le-2
1e-2
1o-3
Maximum Leuming
Se-4
3e-4
3e-4
3e-4 1e-4)
3e-4
Se-4
3e-1
Minimum I Learning rate
Se-6
36-6
3c-6
So-6i 3c-6i1o-6) Io-6)
3o-6
Se46
3o-6
Learning rate scheduling
Cod.R
CosLR
CoiLR
CosLR
CosLR
CodLR
CosLR
Number epochs
100
Loss function
Huber(5=000)
Hubert/-0.011
MAE/ MAE/LIMAE
MAE/ L2MEE
Hubent5-0.01)
MAE
Huber(ia0.01)
Gradient clirping nomn
0.5
0.5
0.5
0,5
0.5
0.5
0.5
Energy loss prefactor
5
5
prefactor
Stress prefactoe
0.1
0.1
Magmom loss prefacter
0.1
0.1
0.1
0.1
0.1
Denoising Settings
Comuption probubilay
Maximum number of steps
1000
1000
Maximum sigma
1.0
1.0
Minimum sigma
0.0
0.0
Number of spochs
28


Reviews:
[{"review_id": "yBclafd9YC", "reviewer": "Reviewer_KaLv", "summary": "The paper introduces MatRIS, a new invariant universal machine learning interatomic potential that achieves high accuracy at a lower computational cost than existing equivariant models. Instead of relying on tensor operations to enforce symmetry, MatRIS uses an attention-based architecture that explicitly models three-body interactions through a line graph and a separable linear attention mechanism with O(N) complexity. MatRIS matches or outperforms state-of-the-art models on benchmarks like Matbench-Discovery and MatPES. Ablation studies confirm that components like dimwise softmax, separable attention, and learnable envelopes significantly improve performance.", "questions": "1. What unique signals does the source/target split capture that a single asymmetric aggregator cannot? A controlled ablation that matches parameters and FLOPs would be nice (or does your ablation matched parameters and FLOPs?)\n2. In low-data regimes, how does MatRIS compare to strong equivariant baselines?\n3. Since forces/stress are energy-conservative, how does MatRIS behave in long NVE MD runs (energy drift, thermostatted stability)?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The architecture is new. The authors introduce a new architectural approach by explicitly modeling three-body interactions using a line graph attention mechanism. Each edge in the atom graph becomes a node in the line graph, enabling the model to attend over bond angles. This is claimed to be the first use of attention for three-body interactions in ML potentials, extending beyond traditional message-passing that usually considers only pairwise interactions.\n2. They employ a separable attention mechanism that splits the message-passing into two parallel branches: one capturing how each neighbor affects a central atom, and another for how the central atom influences its neighbors. Also, this \"attention\" is implemented with linear complexity in the number of atoms.\n3. The proposed model demonstrates competitive or superior results on a few benchmarks.", "weaknesses": "1. The evaluation on large-scale datasets is limited. A notable gap in the experiments is the absence of results on the latest massive and diverse datasets, specifically Open Materials 2024 (OMat 24) and Open Molecules 2025 (OMol 25). If the model is claimed to be scalable, then it should perform well on these datasets. At least a benchmark on OMol 4M would be nice to have.\n2. Despite the terminology, MatRIS’s mechanism differs from standard scaled dot-product self-attention with Q/K/V projections and global token mixing. Weights are computed per-dimension over local neighbors (a “dim-wise softmax”) and derived by applying a linear map to edge features, and followed by separate target/source softmax normalizations. In other words, the “attention” acts more like edge-conditioned, per-channel neighbor weighting than full QKV attention, and it is local (achieving linear O(N) complexity by avoiding all-pairs interactions). This design brings efficiency but may limit the capacity for global, content-based mixing compared to traditional Transformer attention.\n3. MatRIS is built on local, cutoff-based neighborhoods (pairwise and three-body/angle graphs), which means long-range electrostatics and other delocalized effects are not explicitly modeled or evaluated. To demonstrate robustness in regimes where long-range interactions matter (e.g., biomolecules, electrolytes), it would be great to test on OMol 25 and report results on biomolecule / electrolyte validation splits.\n4. The paper frames MatRIS as a universal MLIP, but the core pretraining and evaluations are materials-centric: Matbench trained on MPTrj, MatCalc using MatPES/MPTrj, and results on DPA2 materials test sets. While the abstract highlights “universal” scope and mentions a “Molecular dataset”, the molecular side is limited to small zero-shot style checks rather than large-scale molecular pretraining/evaluation. And there's no mention on other types of chemical systems, such as surface, organic crystals, polymers, MOFs, etc.", "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": "2atEoh7U95", "reviewer": "Reviewer_MUqg", "summary": "The paper proposes MatRIS, an invariant methodology to construct machine learning interatomic potentials that use three-body attention between edges with a common node. \n\nMatRIS is benchmarked on a broad range of benchmark datasets, including the Matbench Discovery benchmark (for \"compliant\" tracking fitting only on the MPTrj datasets), MatCalc, and others. \n\nAdditionally, the paper reports the computational efficiency of the model.", "questions": "Could you clarify the discussion in light of arranging models into invariant, covariant, and unconstrained ones?\n\nCould you provide more insights into the reasons why MatRIS is so computationally efficient?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The model, MatRIS, achieves state-of-the-art efficiency of the compliant track of Matbench Discovery with the F1 score of 0.844, and performs equally well on other benchmarks - MatCalc, Molecular Zero Shot Benchmark, and DPA2. \n\nThe paper provides several ablations on the model design and fitting procedure. \n\nThe computational efficiency of the MatRIS model is competitive in both training and inference.", "weaknesses": "The position of the available MLIPs architectures into invariant and equivariant groups is not correct. The paper groups all the models into only two groups, invariant and equivariant, missing the third big group of models - rotationally unconstrained architectures, and assigns many of the rotationally unconstrained architectures into the invariant groups. Because of this, some claims are not entirely true, for instance, \"Recent studies (Neumann et al., 2024; Qu & Krishnapriyan, 2024; Rhodes et al., 2025) show that invariant models allow more flexibility in exploring the potential energy surface while being computationally more efficient.\" The cited papers are unconstrained, not invariant, and thus the claim that ' invariant models allow more flexibility' is hardly correct. \n\nInvariant models are those where all the intermediate representations are invariant with respect to rotations - they are not changed at all if a system is rotated. Example - SchNet. \n\nEquivariant models are those where intermediate representations are invariant or covariant, which means that they transform in a specific manner with respect to the rotation of the input system, such as vectors, Cartesian tensors of higher rank, or spherical tensors. \n\nUnconstrained models are those where there is no restriction on intermediate representations, and they can be learned to transform arbitrarily with respect to rotations. \n\nIn general, invariant models are the least expressive and fastest; equivariant models are more expressive and more computationally demanding because of the associated tensor products, and unconstrained models are most expressive and simultaneously fast, roughly as invariant ones, but produce not rotationally invariant predictions, which can sometimes be a problem in molecular dynamics simulations. \n\nThe Edge-based design, while claimed to be new, has already been proposed in the community. Examples are Atomistic Line Graph Neural Network, Point Edge Transformer, and many others. \n\nSimilarly, there are multiple attention-based architectures with O(N) scaling that apply attention to local neighborhoods.", "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": "nBc8NjiO50", "reviewer": "Reviewer_npsV", "summary": "This work proposes MatRIS, a novel E(3)-invariant model architecture for machine learning interatomic potentials. They adopt dim-wise attention and separable attention for line graphs to encode three-body features. The model outperforms previous SOTAs on Matbench-Discovery, DPA2, and several molecular datasets in the zero-shot setting, performs on par or slightly better than others on MatCalc.", "questions": "1. How can framework be applied to equivariant models? The equivariance, and even the whole invariance will be broken if we manipulate the channel of a geometric tensor individually whose weight higher than 0. Does this framework have the potential to be applied in such scenario?\n\n2. From Eq. 1, the denominator does not contain $x_{ij}$ itself, as it is not in the neighboring list of itself. Is it a typo?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "1. Using self-attention in the line graph to encode three-body information is novel?, sound, and simple.\n2. The separable attention is also with reasonable motivation.\n3. The method performs well in several benchmarks.\n4. The experiments are sufficient.", "weaknesses": "1. Although I assume the methodology is novel, some evidences need to be given with respect to the claim in L94-95: ``$\\cdots$, our model is the first to leverage attention mechanisms for modeling three-body interaction''. Specifically, the authors should justify its differences to VisNet [1], FreeCG [2], and MGT [3], which all adopted self-attention mechanisms and captured up to four-body interactions. Is that the case that your model uses self-attention to encode the many-body interactions while they encode many-body interactions without using self-attention in this specific step, but using self-attention to refine the features that have already been encoded? A detailed methodological comparison can be added to the appendix.\n\n2. It appears to be a very rushed paper. Many typographical and presentational errors are spotted in it. I show here some of them:\n\na) Missing space before almost half of the left parenthesis;\n\nb) Misusing \\citep and \\cite in many sentences, e.g., ``GemNet Gasteiger et al. (2024) incorporated dihedral angles, and SphereNet Liu et al. (2022) further integrated torsional information. '' in L118-119.\n\nc) Table X or Tab. X. No Table.X, e.g., ``More details on hyper-parameters are given in Table.7 in appendix. Moreover, inspired by (Zaidi et al., 2022; Liao et al., 2024a),'' in L268-269.\n\nd) Omitting math mode, e.g., ``In equivarant models, the predominant strategy for enforcing higher-order equivariance relies on computationally intensive tensor products of rotation order L(Batzner et al., 2022; Batatia et al., 2023; Liao & Smidt, 2023; Liao et al., 2024b).'' in L62-64;\n\ne) The value vector $v$ should be explained in the main text;\n\nf) In Eq. 1, the $j$ is for feature dimension and $i$ for atom index. But in the adjacent context, $i$ and $j$ both represent atom index.\n\nThe typographical and presentational errors are too massive to list them all out here. I sincerely recommend the authors go through a complete writing check. I am not that kind of person who rejects a paper solely based on presentations, but the authors could pay more attention to such detail, or it would make a good work fail for such things instead of the methodology itself. It is not worth it for you.\n\n[1] Wang Y, Wang T, Li S, et al. Enhancing geometric representations for molecules with equivariant vector-scalar interactive message passing[J]. Nature Communications, 2024, 15(1): 313.\n\n[2] Shao S, Geng H, Wang Z, et al. FreeCG: Free the Design Space of Clebsch-Gordan Transform for Machine Learning Force Fields[J]. arXiv preprint arXiv:2407.02263, 2024.\n\n[3] Anselmi M, Slabaugh G, Crespo-Otero R, et al. Molecular graph transformer: stepping beyond ALIGNN into long-range interactions[J]. Digital Discovery, 2024, 3(5): 1048-1057.", "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)