From atom to space: A region-based readout function for spatial properties of materials
Authors: Jiawen Zou
Weimin Tan
Zhongyao Wang
Hao Qi
Bo Yan
Venue: ICLR 2026 Poster (2026)
Keywords: porous material
metal organic framework
readout function
graph neural network
material property prediction
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We convert the atom-decomposable inductive bias of MPNNs to a region-based one, broadening their applicability to spatial material properties.

Abstract: The message passing–readout framework has become the de facto standard of graph neural networks (GNNs) for material property prediction. However, most existing readout functions are built on an atom-decomposable inductive bias, i.e. the material-level property or feature can be reasonably assigned to contributions of individual atoms. This is a strong bias and may not hold for all properties, limiting the application scenarios (e.g. gas adsorption or separation of Metal Organic Frameworks, MOFs). In this work, we propose a region-based decomposition perspective, reformulating material properties as integrals over space and pooling contributions from spatial regions rather than atoms. Specifically, we propose a novel readout function named SpatialRead. SpatialRead introduces additional spatial nodes to represent a voxelized space, transforming the atomic isomorphic graph into a heterogeneous atom–space graph with unidirectional message flow from atoms to spatial nodes. To combine the two types of inductive bias, multimodal methods can be used to fuse the features of atoms the spatial nodes. Such a region-based readout function is especially suited for spatial properties such as gas adsorption capacity, separation ratio. Extensive experiments demonstrate that a simple PaiNN–Transformer-based SpatialRead trained from scratch outperforms state-of-the-art pre-trained foundation models on these special tasks. Our results highlight the importance of designing physically grounded readout functions tailored to the target property. The code and dataset can be found in github https://github.com/nankusa/SpatialRead.

Published as a conference paper at ICLR 2026
FROM ATOM TO SPACE: A REGION-BASED READOUT FUNCTION FOR SPATIAL PROPERTIES OF MATERIALS
Jiawen Zou', Zhongyao Wang Hao Qi', Wemin Tan Bo Yan
"College of Computer Science and Artificial Intelligence, Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai, China
SShnggaii Innovation Institution, Shanghai, China
SShanghai Academy of AI for Science, Shanghai, China i "Corresponding author
ABSTRACT
The message passing-readout framework has become the facto standard graph neural networks (GNNs) for material property prediction. However, most cxisting readout functions are built on an atom-ddcomposable inductive bias, i.c. the material-level property feature of individual atoms. This is a strong bias and may not hold for all properties, lim- iting the application scenarios (e.g. gas adsorption separation of Metal Organic Frameworks, MOFs). In this work, we propose a region-based decomposition perspective, reformulating material properties integrals over space and pool- ing contributions from spatial regions rather than atoms. Specifically, we propose a novel readout function named SpatialRead. SpatialRead introduces additional spatial nodes to represent voxelized space, transforming the atomic isomorphic graph into heterogeneous atom-space graph with unidirectional message flow from atoms to spatial nodes. To combine the two types of inductive bias, multi- modal methods can be used to fuse the features of atoms the spatial nodes. Such region- based readout function is especially suited for spatial properties such as gas adsorption capacity, separation ratio. Extensive experiments demonstrate based SpatialRead trained from scratch outper- forms state-of-the-an pre-trained foundation models on these special tasks. Our results highlight the importance of designing physically grounded readout fune- tions tailored to the target property. The code and dataset can be found in github attps://githus: com hwtp//nnnnuuaaaa http///annnaaaaaiilaeeeeeee Sttp/SSttaaaaaaeaeee
be reasonably assigned to contributions
that simple
INTRODUCTION
The field of material artificial intelligence is fundamentally shaped by the task of material property prediction. Accurate predictions can significantly accelerate the screening and design of novel mate- rials by bypassing costly and time-consuming experiments. In this domain, Message Passing Neural Networks (MPNNs) have emerged as the state-of-the-art paradigm for both property prediction and material generation. MPNNs represent a material graph, with atoms as nodes and edges con- necting neighboring atoms. A typical MPNN consists of two stages: (1) message passing, where node features are iteratively updated through local aggregation, and (2) readout, where the final node features are aggregated into graph-level property.
Readout is a critical component of this architecture. Simple pooling functions, such as global sum or mean, have proven remarkably successful and even scaled to foundation models with hundreds of millions of training samples (Gasteiger et al. 2020 2021 Shoghi et al. (024). More complex readout functions, such GraphTrans (Wu et al. 2021) and GMT (Back al. 2021), introduce architectural sophistication but ultimately still treat nodes as the fundamental units of aggregation. inductive bias: graph-level property or feature can be decomposed into node contributions. While works well for many tasks, its broader applicability has not been carefully examined.
These designs reflect the implicit node

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A typical counterexample of the coom-dccomposabe inductive bias arises in porous materials such as metal orgaano frameworks (MOFs). Being promising to applications such as gas adsorption and clean energy storage (Snyder et al. 2023 Nugent al. 2013 Datta al. 2013; Zhao et al. 2018 Yang et al. 2012 Zhou et al., 2022), , these materials attract broad research interest. One of the key properties such material the gas adsorption capacity can naturally be expressed as the summa- tion of the adsorption capacities of all regions within the material. Other similar examples include accessible pore volume, adsorption heat, gas selectivity ratio, etc. In these cases, the target prop- erty. which we named as Spatial Properties should be decomposed into contributions of each spatial region instead of atom. The inherent mismatch between these properties and node-decomposable readouts highlights critical blind spot in current graph learning methods for materials.
To correctly consider such region-based inductive bias, we propose a novel perspective to reformu- late the node-decomposable property as the integral over space. We first prove the equivalence of the two expressions. Thus the reformulation simply a kind of inductive bias without harming the expressivity of the model. Based on this, we propose novel readout module named SpatialRead. The central idea is introducing spatial nodes that represent voxelized space. This transforms the conventional atomic graph into heterogeneous atom- trom-space graph. Messages are passed unidirec- tionally from atoms to spatial nodes. However, pooling features from spatial nodes may compromise the performance on those non-spatial properties. To make the network adaptively select the correct inductive bias of atomic and space-decomposition We use attention-based multimodal method to process both atom and spatial node features. This approach enables SpatialRead maintain com- parable performance in non-spatial attributes, thus providing a general flexible option.
ReatialRead is especially suited for spatial properties such as gas adsorption capacity without harm- ing the performance on other tasks such as the MatBench dataset (Dunn et al. (020). To evaluate the performance for spatial properties. we collect and release dataset, covering porous materials and 27 downstream tasks. Remarkably, on these special properties, a simple architecture equipped with SpatialRead, trained from scratch, outperforms a Gem founda- tion model pretrained on 120 million samples (Shoghi et al., (2024). In summary, our contributions are:
We reformulate the node-decomposable property as the integral over space and prove the equivalence. Such perspective provides a new inductive bias that the target property can be decomposed into contributions of each spatial region instead of atom.
We propose SpatialRead, which uses spatial nodes to construct heterogeneous atom space graph and then use multimodal method to achieve adaptive selection of the two kinds of inductive bias. SpatialRead achieves the state-of-the-art performance on spatial properties without pre-training and maintains comparable performance on non-spatial properties.
We release collected benchmark, covering 4 porous materials and 27 downstream tasks.
PRELIMINARY AND RELATED WORKS
Message Passing Neural Networks (MPNNs) are first formally defined by Gilmer et al. (2017) and have since achieved great success in fields like chemistry and material science. In these domains, molecule or a material is represented as a graph G = (V, E),, where V is the set of nodes (atoms) and E is the set of edges. Each node v E V is typically represented by its atomic number z; and coordinates pos;- Edges are usually constructed based on a distance cutoff Teat between adjacent atoms, often with an additional limit the maximum number of neighbors to ensure computational efficiency.
An MPNN consists of two main stages: message passing process and readout process. In the message passing process, each atom updates its own feature vector by aggregating "messages" from its neighbors. For typical T T-layer MPNN, this process be formally expressed
h++ U.b,, (hyy eo (myeN(a)))))))
where ht, is the feature vector of atom o; at layer t, (w:) denotes the set of its neighbors, cvy,, represents the edge feature between en and vy, and U, is the node feature update function of the 7th
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layer neural network. More advanced MPNNS may also incorporate edge updates, seen in models like DimeNet (Gasteiger et al., 2020) and GemNet (Gasteiger et al. (021), which use angular and dihedral information to enrich edge features. After T layers of message passing, each node possesses a rich representation vector h The final step is the readout process, which pools these node features generate graph-level property. The two most common readout strategies are:
Feature-level:
hgraph Pool( (he Joee),, 0; - (L.(
p MMP((ggraa
(2)
Numeric-level:
P - Pool((o (for)ne))
(3)
Since the numeric value can be regarded as one-onee one-dimension feature, we will not distinguish between two pooling methods hereafter. Although a variety of alternative readout functions exist, such as Set2Set (Vinyals al., 2015) and diffpool (Ying et al.) 20188) most state-of-the-an art methods like DimeNet (Gasteiger et al. 2020), GemNet (Gasteiger et al.) (021). ViSNet Wang et al. 2024), and
JMP (Shoghi al. 2024) still rely these simple pooling forms for most tasks.
The main requirement of the readout function is permutation invariance, i.e., the output should be independent of the input order of atomic features. Formally, p readout( (hh ),,1;, V. Existing
readout methods can be broadly classified into three categories: (1) Flat Pooling Methods directly pool all atom features, such GMT (Baek et al. (02T) and GraphTraas (Wu et al.
(202T) which use
attention mechanisms similar to the [CLS] token. (2) Node Clustering Pooling Methods divide nodes
into sets and pool hierarchically, including DifiPool (Ying et al., 2018), MinCutPool (Bianchi al.,
2020). and SEP (Wu et al. 2022). Recent advances include Cluster-wise Graph Transformer (Huang al. 2024) for cluster-node feature interaction, ORC-Pool (Feng Weber (0024) based on Ricci downsampling, and GPN (Song) ct al. 2024) for automatic pooling structure design. (3) Node Drop Pooling Methods select nodes to construct subgraphs, such as TopKPool Gao & Ji (112200)):((aeeee) et al. (2018); Knyazev et al. (2019) which selects top-K nodes by attention scores. Quan et al. (2024) combines TopK selection with clustering for protein representation. SSMA (Keren Taraday et al., (2224) treats neighbor features as 2D discrete signals for enhanced feature mixing, while Xu et al. (2018a) employs jumping-
Flow. K-MIS-Pool (Bacciu et al. (223)) for topology-presere
knowledge connections (Xu al. 2018b) to strengthen graph-level features.
3
SPATIAL READ
3.1 EMPIRICAL MOTIVATION: IMPLICIT REGION-BASE BEHAVIOR IN STANDARD GNNS
A
B
15
10
Contribution
4
IFEPUG07
Distance to pore core (A)
Figure Contribution of atoms for the adsorption capacity predicted by GNN. (A) Scatter plot between contribution and distance to the nearest pore. 100 materials from the test set from the CoREMOF dataset. Each dot represents an atom. The horizontal axis shows the distance of the atom from the nearest pore, while the vertical axis indicates the contribution of the atom to the target property. (B) Visualization of the IFEPUG07 in the CoREMOF dataset. Top 5% high- contribution atoms marked in red star.
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Before introducing spatial nodes, examine how conventional atom base- GNNs behave when predicting spatial properties such as gas adsorption in MOFs. To investigate this, we trained a standard PaiNN model on MOF adsorption tasks and computed per-atom contributions using the scalar outputs of its readout (see equation 3). We then compared these contributions with each atom's distance to the nearest pore. Fig. [ demonstrate the contribution versus distance to the pore (experiment details can be found in Appendix (. We observed strong alignment between high contributions and pore-adjacent atoms. Among the top 1% of atoms with the highest contri- bution rate, 86% of them are located within 0.05 angstroms of the pores. Visualization in Fig. [ demonstrates a case study, showing that the model attributes most predictive weight to atoms lining the pore channels. These results indicate that when predicting properties that are clearly spatially decomposable, GNNs implicitly learn the key regions and the surrounding atoms. Introducing di- rect representations of regions may reduce the learning burden of the network and thereby lead to performance improvement.
3.2 FROM NODE DECOMPOSITION SPATIAL INTEGRATION
Given that atom types and positions fully determine material structure, the property of material G = (V, E) can be regarded as a function of its vertices V. A typical MPNN applies a local description function e with limited receptive field: the feature of atom is determined solely by atoms within a finite radius. To obtain a material-level representation, these atom-level features are aggregated. Formally,
hgraph = f(hn H),
(5) where the readout function and H (hr, U. E V} the set of node features. Here, we reformulate the graph-level feature hggraph as an integral over the continuous spatial domain. To this end, we introduce a contribution function g(r S), where S denotes the material structure. Similar
hv. = c(f2g)), By NW;),,
to c, the function g also has a limited receptive field. The graph-level representation can then be written the integral form
Ngraph = 1 ((N()))
(6)
We refer to properties that admit such region-based representation as spatial properties.
Definition 3.1. A property is called spatial property if it can be expressed functional of a contribution function g(r S), where S is the material structure. For instance, gas adsorption capacity is spatial property: position r, and the total adsorption capacity is then given by the spatial integral of g(r).
may define g(r) the density of adsorbed gas molecules at
Despite this formulation, spatial properties are not fundamentally different from other properties in terms of their mathematical or neural representability.
Theorem 3.1. If the readout function has limited receptive field, the formulations in equation and equation 6 are equivalent in expressivity. That is, any target property expressible by equation can also be expressed by equation 6 and vice versa.
Theorem ensures that under limited receptive field, reformulating the graph-level representation as spatial integral does not reduce neural expressivity. When f has an infinite receptive field, the graph-level feature general cannot be obtained via simple integral, and neural architecture with global receptive field must be applied directly to the contribution function grr). A detailed discussion given in Appendix A.1.2
proof is provided Appendix
3.3 MODEL ARCHITECTURE
The core idea of our method is to correctly decompose the target property into contributions of discrete spatial regions. To achieve this, we first discretize the spatial property p. which is defined integral over continuous domain, into summation set of discrete regions.
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Traditional: Atom-Decomposable
readout
Target Properties
Σ
atomic isomrrphic graph
graph-loval feature
Ours: Sww.s-ppaacppssabeee
Material Btudure
Transfomer
Adsorption Capacity Pore Limited Diameter
atom-spacce heteragencous graph
Architecture of SpatialRead. The top part represents the traditional method, while the bottom part shows SpatialRead. The material structure is taken input. For clarity, the lattice vector and boundary is drawn. The traditional method constructs an atom-based isomorphic graph for message passing, and then pools the atomic features to obtain material-level features. In contrast, ReatialRead first uniformly sample spatial nodes (the red triangle nodes) within the lattice. The atom nodes (the blue round nodes) and spatial nodes form a heterogencous graph. In the message passing process, messages flow between atom nodes (blue lines) and from atom nodes to spatial nodes (red lines). Note there
Figure 2:
messages from spatial nodes to atom nodes. The heterogeneous graph neural network produce an unordered feature list of atom nodes and ordered feature list of spatial nodes. Position embedding can be added to the feature list of spatial nodes. Finally, taking the atom node features as "memory", , decoder of Transformer are used to process both feature lists (Vaswani al. 2017). An additional [CLS] token is added the sequence of spatial nodes (Devlin et al. 2019 which can be used to predict spatial properties such as adsorption capacity, separation ratio, or other properties like topology type and pore limited diameter.
p= g(rrddrrrr R
Σ g(r, AVV
Here, the continuous space is partitioned into N₂ discrete regions (c.g., voxels), each with volume AV; centered at position r,. To computationally represent these regions, we introduce spatial nodes in addition to original atomic nodes. Each spatial node is placed at specific coordinate ry and represents the corresponding spatial region.
3.3.1 REGION- WISE HETEROGENEOUS MESSAGE PASSING
To model the interactions between atoms and these new spatial regions (i.e. g(r)), we convert the original isomorphic atomic graph into heterogeneous graph composed of both atomic nodes and spatial nodes. We denote the spatial node and its feature vector as sy and h2,- The formulation of our Region-wise Heterogeneous Message Passing is as follows:
Not1 = Ulhh,, (Me Sousy ( (hi (eN())))))))
(8)
hL+1 2ehh (bl Cea,, Co,,,, byy)((y)))
Here, U', and Ue are the update function for spatial and atomic nodes. The new feature vector h+1 aggregates messages not only from nearby atoms (u; E (s;)) but also from adjacent spatial nodes (sk (s;)). In practice, the method for constructing spatial node adjacency nodes is
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actly the same that for atoms, which is based on cutoff and the maximum number of neighbors. This inter-spatial node message passing is optional and depends on the physical nature of the target property. For example, when predicting gas adsorption capacity, message passing between spatial nodes is physically meaningful as it captures the cooperative interactions between guest molecules in adjacent regions. In contrast, for property like accessible volume, where adjacent regions independent, such interactions have no physical meaning and can be omitted. In this work, because the global receptive field of the adopted multi-modal Transformer method already enables interac- tion between spatial nodes (see Sec. 3.3.2, we ignore the message passing process between spatial nodes.
3.3.2 PROPERTY-ADAPTIVE READOUT VIA MULTI-MODAL ATTENTION
After obtaining atom and spatial node representations, one may directly pool the spatial-node fea- tures via
P MLP(hs,),
(10)
which already improves performance on spatial properties (Sec. 4. However, spatial nodes are specifically designed for spatially decomposable properties, whereas many widely used material properties remain atom-decomposable. Using spatial pooling alone introduces a mismatched in- ductive bias and degrades performance on these non-spatial tasks. Thus, an effective readout must adaptively combine atomic and spatial information.
To achieve this, we impose an ordering spatial nodes with positional embeddings and feed the or- dered spatial features together with unordered atomic features into a Transformer decoder (Vaswani et al., 2017). The attention- based decoder selectively integrates the two types of representations, enabling SpatialRead to retain the gains on spatial properties without sacrificing performance on conventional non-spatial tasks. A detailed architecture is provided in App. A.2
4 EXPERIMENTS
Our experiments are designed to address the following three questions: (1) Does SpatialRead outper- form -ecomposcoooeeeeee methods on spatial properties? (2) What types of material features benefit most from SpatialRead? (3) For non-spatial properties, does the spatial inductive bias introduce any degradation?
Datasets. We first construct a dataset of spatial properties to evaluate the effectiveness of Spatial- Read. The dataset mainly consists of four material types: (1) Metal Organic Frameworks (MOFs), (2) Covalent Organic Frameworks (COFs) (3) Porous Polymer Networks (PPNs), (4) zeolites, and two types of properties: (1) Geometric features such as topology type and pore-limiting diameter, computed using zeo++ (Willems et al.) 2012). (2) Gas-related properties such adsorption capac- ity and separation ratio, simulated through molecular dynamics. These materials are specifically selected because most of their properties, including adsorption capacity and separation ratio, inherently spatial. Detailed descriptions of the dataset are provided in App. A.3
Baselines. For spatial property prediction, compare against the following baselines: (1)
CGCNN (Xie & Grossman, 2018). widely used MPNN designed for crystals. (2) Gem-
Net (Gasteiger et al. (021), an advanced invariant graph neural network that incorporates angu- lar information, substantially improving performance over other MPNNs. (3) MOFormer (Cao) al. 2023), a contrastive pre-trained variant of CGCNN using SMILES representations. (4) MOF- Transformer (Kang et al., 2023). a multimodal framework with scage pre-training tailored for MOFs. We use the improved weights from PMTrransformer (Park al. 2023). J. which extends the pre-training dataset and enhances performance. (5) JMP (Shoghi et al. 2024). GemNet-based
foundation model pre-trained on 120 million molecular and material samples. Training details are Another kind of baselines are different readout functions. However, in
provided in App. A.4
most case for material molecule property prediction (Schütt et al.) 2017 Xie & Grossman, 2018
Gasteiger et al. 2020: 2021 2022 Shoghi al. 2024; Wang al. 2024), modern GNNs this field focus on the design of complex message passing process, while adopting the simple sum or mean pooling. Nevertheless, in order ensure the completeness of the work, here compare

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Reat with two most typical readout function GraphTrans (Wu et al. 2021 and GMT et al. 2021).
(Back
Ablations. In the following experiments, consider three types of ablation settings. (1) Backbone GNN with simple sum mean pooling of atom nodes, i.c. Base GNN. (2) Backbone GNN enhanced with Spatial Nodes, the graph feature will be pooled from all spatial nodes, i.e. Base GNN SN (Spatial Node) (3) Backbone GNN processes heterogeneous graph atom node and spatial node. A Multi-Modal Transformer architecture is used process both atom features and spatial node features, i.c. Base GNN + SN + MM (Multi Modal).
Table Performance (R2 Score) of SeatialRead on spatial properties in integral form
MOF CsH,/C3HH sep.
MOF N₂ ads.
MOF CH./Na sep.
COF CH4 ads.
PPN CH4 ads.
zeolite
Model
CH, heat.
CGCNN
0.663
0.760
0.718
0.556
0.692
0.411
Scrutch
GemNet (JMP from scratch)
0.729
0,968
0.924
0.816
0.932
0.836
GemNet SN MM
0.753
0.979
0.921
0.986
0.923
0.881
MOFormer
0.616
0.754
0.698
0,541
0.636
0.388
Pretrain
MOFTransformer
0.817
0.918
0.905
0.967
0,942
0.836
JMP
0.774
0,971
0.908
0.884
0.947
0.874
JMP + SN MM
0,792
0.988
0.941
0,982
0,969
0.945
PaiNN PaiNN + GraphTrans
0.691
0.925
0.867
0.736
0.856
0,791
0.712
0.912
0.870
0.750
0.847
0.801
Scratch
PaiNN GMT PaiNN SN (ours) PaiNN SN MM (ours)
0.740
0.924
0.866
0.742
0.863
0.803
0.794
0.978
0.936
0.979
0.978
0.886
0.784
0.987
0.941
0.987
0.977
0.969
4.1 SPATIAL PROPERTIES
We first evaluate SpatialRead on six representative spatial properties, including: (1) gas adsorption
in MOFs, (2) two gas separation tasks in MOFs, (3) gas adsorption in COFs, (4) gas adsorption in PPNs, and (5) adsorption heat in zeolites. Results are summarized in Table 1 Among the three representative MPNNs (CGCNN, GemNet, PaiNN), the ranking GemNet >> PaiNN >>>>>>>> CGCNN high- lights the critical role of backbone design. Using complex readout functions including GraphTrans and GMT marginally improves the performance. Nonetheless, all pure MPNN approaches underper- form MOFTransformer. While PaiNN lags behind MOFTransformer and JMP, augmenting it with spatial nodes (+SN) yields notable improvements across tasks. Adding the attention mechanism (MultiModal, +MM) does not lead to further consistent gains. This can attributed to the fact that these properties naturally be expressed as the sum of regional contributions. Hence, equation I0 is already physically grounded, and introducing additional global receptive fields has limited phys- ical justification. This stands in contrast the results in Sec. 4.3 where global receptive fields indispensable. Overall, these results highlight the effectiveness of spatial nodes in modeling spatial properties: by decomposing properties into regional contributions, performance can be substantially enhanced without altering the backbone architecture.
4.2 BENEFIT FROM ALREADY-PRE-TRAIEE FOUNDATION MODEL
Despite being designed for general GNNs such as PaiNN, it is also important to assess whether SpatialRead can be directly applied a pre-trained foundation model like JMP (Shoghi et al. 2024). Surprisingly, although JMP is pre-trained with a simple sum- sum-pooling readout, it can be seamlessly enhanced by SpatialRead. As shown in Table I both GemNet (i.e., JMP trained from scratch) and JMP experience substantial performance gains when equipped with SpatialRead, and JMP+SpatialRead further outperforms SemNettSpatialRees training remains beneficial even when the downstream readout differs from that used during pre- training. These results highlight the versatility of SpatialRead with complex multi-bodd body and pre- trained GNNs. We note that JMP-SpatialRead does not always surpass PaiNN+SpatialRert likely due to the higher computational cost of GemNet's s four-body interactions and the resulted limited hyperparameter settings, discussed in Appendix A.5
demonstrating that large-scale pre-

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Table 2: Performance (R2 Score) of SpatialRead geometric properties
Model
ASA
VF
PLD
LCD
CGCNN
0.984
0.883
0.536
0.565
Scratch
GemNet
0.994
0.977
0.586
0.667
MOFormer
0.979
0.894
0.563
0.624
Pretrain
MOTTransformer
0.916
0.989
0.966
0.970
JMP
0,995
0.985
0.585
0.650
PaiNN
0,993
0,951
0.594
0.631
Scratch
PaiNN + SN (ours)
0.974
0.999
0.856
0.913
PaiNN SN + MM (ours)
0.996
0.999
0.965
0.975
4.3 GLOBAL GEOMETRIC PROPERTIES
In the previous section, we demonstrated that for spatial properties that can be naturally expressed in integral form, SpatialRead provides significant improvements by reformulating the readout as summation over spatial regions. However, not all spatial properties admit such a simple integral formulation. To further examine this distinction, we evaluate SpatialRead on several representative geometric properties, as reported Table 2 For accessible surface area (ASA), the property essentially node-decomposable surface area is determined primarily by the atoms located at the boundary. this case, directly introducing spatial nodes provides negative influence. Nevertheless, since our multimodal architecture adaptively balances atom- and space-decomposable inductive bi- ases, SpatialRead still achieves a slight improvement over the pure PaiNN baseline. Void fraction (VF) can be expressed in integral form, but the task itself is relatively simple (Kang et al.) (023), and thus most models already achieve high performance. Here, SeatialRead again yields marginal improvements, confirming its robustness without incurring degradation. The situation is markedly different for pore-limiting diameter (PLD) and largest cavity diameter (LCCD). These descriptors cannot be represented as integrals over local contributions, but are instead better understood func- tionals of the signed distance function (SDF) of the material geometry. Their values depend on the global shape of the pore space rather than additive regional properties. such cases, equation TO is no longer physically meaningful. The introduction of spatial nodes already provides significant multimodal architecture further enhances performance through its global receptive field and expressive capacity. These results suggest that SpatialRead is effective not only for integral-type spatial properties, but also for more complex forms.
gains, while the Transformer-
Table Performance of SpatialRead on MatBench
Task
MODNet
coGN
JMP
JMP SpatialRead
JDFT2D (/atom)
25.55
22.25
20.72
18.17
Phonons (cm -1)
34.77
32.12
26.6
25.8
Dielectric (unitless)
0.169
0.178
0.133
0.133
Log GVRH (log10(GPa)
0.073
0.068
0.06
0.06
Log KVRH (log))(GPa))
0.054
0.052
0.044
0.047
Perovskites (e (eV/niicell)
0.093
0.027
0.029
0.030
MP Gap (eV)
0.215
0.153
0.119
0.107
MP Formation Energy (me V/atom)
40.2
17.4
13.6
15.3
4.4 MATBENCH RESULTS FOR OTHER COMMON MATERIAL PROPERTIES
While the primary strength of SpatialRead lies in modeling spatial properties, it is also important to ensure that does not degrade performance on standard material- benchmarks. As shown in Table the foundation model JMP already surpasses leading baselines the MatBench dataset. Adding SpatialRead maintains comparable performance across most tasks. Two tasks behave differ- ently. For MP-E Eee--FFomm SpatialRead slightly reduces accuracy, whereas for bandgap, SpatialRead leads to an improvement. This difference is consistent with the inherent nature of the two quantitics. Formation energy is atom-decomposable property, making simple summation pooling scheme,
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such the used in JMP, suitable inductive bias. In contrast, the bandgap is defined as the energy difference between the valence band maximum (VBM) and the conduction band minimum (CBM). These frontier electronic states are formed by collective contributions from multiple atoms and cannot be meaningfully represented as a sum or average over atom-wise descriptors. Therefore, simple atom pooling provides an inappropriate inductive bias for bandgap prediction. An attention- based readout, which allows the model to selectively weight atoms according to their relevance to the VBM and CBM, naturally more suitable. This improvement is attributable the attention mechanism rather than the presence of spatial nodes. These results further reflect the importance of designing
readout function. summary, SpatialRead preserves performance on non-spatial tasks even when combined with a strong pre-trained model, confirming its versatility across diverse material-pre proopery settings.
A
B
1200
1200
Spearman
ID: 0.963
0.961
1300
OOD: 0.920
3390
0.757
890
590
600
690
400
400
Target
Target
Figure 3: Working mechanism of Spatial Nodes. (A) Visualization of spatial nodes with high contribution. Those spatial nodes with top 10% contributions are drawn as red cubes. The drawn important spatial nodes are mainly located in the pore regions of the material. (B) Out of dis- tribution generalization for PaINN+SummPolling (C) Out of distribution generalization for PaiNN=SSatialRead. The adsorption capacity data is separated according to the void fraction. We choose 1/7 materials with the most high void fraction as the test set to test the out-of-distt generalization ability of different readout function. PaiNN+SpatialReadl is trained for 3 epochs while PumooinSn is trained for 40 epochs to make sure their precision on in-distribution data is similar. PaiNN SpatialRead maintains better spearman correlation coefficient in out-of- distribution data.
4.5 INTERPRETATION OF SPATIAL NODES
We now provide interpretation of how spatial nodes function within SpatialRead. In sum- mary. those sparse regions with few or no atoms are hard to be described by conventional atom- decomposable methods. However, these regions are critical for some spatial properties. Thus, when a material or graph is highly non-unine non-uniform, one should consider adding spatial nodes to those sparse regions. To illustrate this, we employ SchNet (Schütt al. (017 simple backbone with MLP readout (equation TO). trained to predict gas adsorption capacity. In this setting, the prediction is ob- tained by averaging node outputs, allowing each node's output be interpreted its contribution. We visualize the top 10% most contributing spatial nodes in Fig. ??. These nodes are predominantly located within the pore channels of MOFs, highlighting that SpatialRead effectively identifies the structural regions most relevant to adsorption. Under purely node-decomposable inductive bias, such pore-level contributions will be implicitly learned to be assigned to the nearby atoms, which may increase the difficulty of learning the network and its generalization ability. In addition to the case study, we also conducted statistical analysis in the Appendix A.8
4.6 PHYSICAL-GROUNDEER INDUCTIVE BIAS FOR BETTER GENERALIZATION
A natural way to examine whether model has learned physically meaningful features is to test its behavior under distribution shift. For spatial property, although typical GNN can implicitly infer critical regions from atomic environments, this atom-based decomposition becomes unstable when faced with distribution shift. To validate this, construct out-of-distribution (OOD) split by placing the highest porossiy 1/7 MOFs entirely into the test set. To ensure faimess, we early-stop PaiNN+SSatialReed such that its in-distribution accuracy matches that of the base PaiNN.

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Details of the calculation method, data split, and validation metric can be found in A.7 While both models experience increased error due to the severe shift in porosity, P significantly higher ranking stability (Spearman 0.9200.95 the prediction variance of PaiNN+SpatialRead is smaller than PaINN+SumPooling indicating the robust prediction. This ability is of crucial importance for the screening of materials. Because for high-throughpen screening, people are more concerned about whether the performance ranking of different materials is correct rather than the actual prediction error. In summary, the benefit of Rea lies in guiding the model toward physically plausible solutions within this function class, thereby improving robustness of prediction, especially under distribution shift.
maintains 0.76). Fig. ?? also demonstrates that
4.7 SAMPLING STRATEGY AND COMPUTATIONAL COST
Since the position of the introduction of spatial nodes is not unique. The sampling strategy will sig- nificantly affect the model performance. Besides the sampling strategy based on grid/fractional co- ordinates, another method is to sample based on resolution. The resolution-based method allocates different numbers of spatial nodes for different-sized cells. Resolution-basee better performance on large system like COFs. Detailed experiment can be found in Appendix AA.9 Due to space limitations, computational cost are discussed in Appendix A.5 and A.10 For a typical MOF material containing about 300 atoms, adding SpatialRead adds about 30% computational bur- den. Increasing the number of spatial nodes consistently improve the performance. 8 8 points provide good balance between computational burden and performance.
sampling results in
5 CONCLUSION
In this work, revisited the inductive bias of MPNN readouts, noting that the common assumption of node-decomposability is insufficient for many spatial properties. To address this, we proposed Read, which augments the atomic graph with spatial nodes and employs a multimodal Trans- former to adaptively select between atomic- and ppataal-ecoomoosable representations. Extensive experiments show that SpatialRead substantially improves predictions of spatial properties such as gas adsorption capacity, pore limiting diameter (PLD), etc, while remaining the performance of the backbone MPNN in other non-spatial properties. Contribution analysis demonstrates that spa- tial nodes naturally capture critical regions. These benefits incur only modest computational over- head, establishing SpatialRead as a practical, scalable framework that incorporates spatial inductive bias into graph neural networks. SpatialRead emphasizes the importance of designing physically- grounded readout function for the target property, which is commonly ignored in current MPNNs.
AUTHOR CONTRIBUTIONS
J. J.Z. proposes the main idea, conducts the experiments, and writes the paper. Z.W. and H.Q. help with the experiments. W.T. and B. Y. supervise the progress of the project and provide critical feedback on the paper.
ACKNOWLEDGMENTS
We gratefully acknowledge support for this work provided by National Natural Science Foundation of China (NSFC) (Grant Nos.: 623B1004 (J.Z.), 62472102 (B. Y.)), National Key R&D Program of China (2024 2024YFC3400500 and 2024YFC3406501,, the Natural Science Foundation of Shanghai
(247R1490400 to W. T.).
REFERENCES
Davide Bacciu, Alessio Conte, and Francesco Landolfi. Generalizing downsampling from regular data to graphs. In Proceedings of the AAAI Conference Artificial Intelligence, volume 37, pp. 6718-6727, 2023.
Jinheon Back, Minki Kang, and Sung Ju Hwang. Accurate learning of graph representations with graph multiset pooling. arXiv preprint arXiv:2 1533, 2021.
10

---

Published as a conference paper at ICLR 2026
Filippo Maria Bianchi, Daniele Grattarola, and Cesare Alippi. Spectral clustering with graph neural networks for graph pooling, In International conference on machine learning, pp. 874-883.
PMLR. 2020.
N Scott Bobbitt, Kaihang Shi, Benjamin J Bucior, Haoyuan Chen, Nathaniel Tracy-Amoroso, Zhao Li, Yangzesheng Sun, Julia Merlin, J IIja Siepmann, Daniel W Siderius, et al. Mofx-db: An online database of computational adsorption data for nanoporous materials. Journal of Chemical & Engineering Data, 68(2):483-498, 2023.
Cätälina Cangea, Petar Veličković, Nikola Jovanović, Thomas Kipf, and Pietro Liò. Towards sparse hierarchical graph classifiers. arXiv preprint 1r1iv::811.012250 2018
Zhonglin Cao, Rishikesh Magar, Yuyang Wang, and Amir Barati Farimani. Moformer: self- supervised transformer model for metal-organic framework property prediction. Journal of the American Chemical Society, 145(5):2958-296) 2023.
Yongchul G Chung, Emmanuel Haldoupis, Benjamin J Bucior, Maciej Haranczyk, Seulchan Lee, Hongda Zhang, Konstantinos D Vogiatzis, Marija Milisavljevic, Sanliang Ling. Jeffrey S Camp. et al. Advances, updates, and analyties for the computation-reade experimental metal-organic framework database: Core mof 2019. Journal of Chemical & Engineering Data, 64(12):5985- 5998, 2019.
Shuvo Jit Datta, Chutharat Khumnoon, Zhen Hao Lee, Won Kyung Moon, Son Docao, Thanh Huu Nguyen, In Chul Hwang, Dohyun Moon, Peter Oleynikov, Osamu Terasaki, et al. Co2 capture from humid flue gases and humid atmosphere using a microporous coppersilicate. Science, 350
(6258):302-300) 306, 2015.
Kathryn S Deeg, Daiane Damasceno Borges, Daniele Ongari, Nakul Rampal, Leopold Talirz, Aliak- sandr Yakutovich, Johanna M Huck, and Berend Smit. In silico discovery of covalent organic frameworks for carbon capture. ACS Applied Materials & Interfaces, 12(19):21559-215 21568, 2020.
Jacob Devlin, Ming- Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. Proceedings of the 2019 conference of the North Americun chapter of the association for compulational linguistics: human language technologies, volume (long and short papers). pp. 4171-4188 2019.
Alexander Dunn, Qi Wang, Alex Ganose, Daniel Dopp, and Anubhav Jain. Benchmarking materials property prediction methods: the matbench test set and automatminer reference algorithm. npj Computational Materials, 6(1):138, 2020.
Amy Feng and Melanic Weber. Graph pooling via ricci flow. arXiv preprint arXiv:4407.04236 2024.
Hongyang Gao and Shuiwang Ji. Graph u-nets. In international conference machine learning, pp. 2083-2092. PMLR, 2019.
Johannes Gasteiger, Janek GroB, Hi Günnemann, Stephan, and Hi Hi. Directional message passing for molecular graphs. arXiv preprint arXiv:2003.03123, 2020.
Johannes Gasteiger, Florian Becker, and Stephan Günnemann. Gemnet: Universal directional graph neural networks for molecules. Advances Neural Information Processing Systems, 34:6790- 6802, 2021.
Johannes Gasteiger, Muhammed Shuaibi, Anuroop Sriram, Stephan Günnemann, Zachary Ulissi, C Lawrence Zitnick, and Abhishek Das. Gemnet-oc: developing graph neural networks for large and diverse molecular simulation datasets. arXiv preprint 2004v2224..22782. 2022
Justin Gilmer, Samuel Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In International conference on machine learning, pp- 1263-1272. PMLR, 2017.
Hasan Can Gulbalkan, Alper Uzun, and Seda Keskin. Evaluating ch4/n2 separation performances of hundreds of thousands of real and hypothetical mofs by hamnessing molecular modeling and machine learning. ACS Applied Materials & Interfaces, 2023.
11

---

Published as a conference paper at ICLR 2026
Jinghao Hu, Jianfei Zhao, and Tianying Yan. Methane uptakes in covalent organic frameworks with double halogen substitution. The Journal of Physical Chemistry C. (19(4)2010020144 2015.
Siyuan Huang, Yunchong Song, Jiayue Zhou, and Zhouhan Lin. Cluster-wise graph transformer with guannggayyy kernelized attention. Advances in Neural Information Processing Systems, 2024.
Yeonghun Kang, Hyunsoo Park. Berend Smit, and Jihan Kim. A multi-modal pre-training trans- former for universal transfer learning in metal-organic frameworks. Nature Machine Intelligence, 5(3):309-33) 318, 2023.
Mitchell Keren Taraday, Almog David, and Chaim Baskin. Sequential signal mixing aggregation for message passing graph neural networks, Advances in Neural Information Processing Systems, 37:93985 94021, 2024.
Baekjun Kim, Sangwon Lee, and Jihan Kim. Inverse design of porous materials using artificial neural networks. Science Advances, 6())eeaax9324, 2020.
Boris Knyazev, Graham W Taylor, and Mohamed Amer. Understanding attention and generalization in graph neural networks. Advances in neural information processing systems, 32, 2019.
Richard L Martin, Cory M Simon, Berend Smit, and Maciej Haranczyk. In silico design of porous polymer networks: high-throughput screening for methane storage materials. Journal of the Amer- ican Chemical Society, 136(13):5006--502, 2014.
Rocío Mercado, and Berend Smit. In silico design of 2d and 3d covalent organic frameworks for methane storage applications. Chemistry of Materials, (0(15))5069- -086, 2018
Fu, Aliaksandr V Yakutovich, Leopold Talirz, Maciej Haranczyk,
Patrick Nugent, Youssef Belmabkhout, Stephen D Burd, Amy J Cairns, Ryan Luebke, Katherine Forrest, Tony Pham, Shengqian Ma, Brian Space, Lukasz Wojtas, al. Porous materials with optimal adsorption thermodynamics and kinetics for co2 separation. Nature, 495(7439)880-84, 2013.
Hyunsoo Park, Yeonghun Kang, and Jihan Kim. Enhancing structure-property relationships in porous materials through transfer learning and cross-material few-shot learning. ACS Applied 2023.
Materials & Interfaces, 15(48):56375-566)
Ruijie Quan, Wenguan Wang, Fan Ma, Hehe Fan, and Yi Yang. Clustering for protein representation learning. In Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp. 319-329, 2024.
Andrew S Rosen, Shaelyn M Iyer, Debmalya Ray. Zhempeng Yao, Alan Aspuru-Guzik, Laura Gagliardi, Justin M Notestein, and Randall Q Snurr. Machine learning the quantum-chemical properties of metal-organic frameworks for accelerated materials discovery. Matter, 4(5):1578- 1597, 2021.
Andrew S Rosen, Victor Fung, Patrick Huck, Cody T O'Donnell, Matthew K Horton, Donald G Truhlar, Kristin A Persson, Justin M Notestein, and Randall Q Snurr. High-throughput predictions of orgal-organic framework electronic properties: theoretical challenges, graph neural networks, and data exploration. npj Computational Materials, 8(1):1-10, 2022.
Kristof Schütt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre Tkatchenko, and for modeling quantum interactions. Advances in newral information processing systems, 30, 2017.
Müller. Schnet: A continuous-filtee convolutional neural network
Kristof Schütt,, Oliver Unke, and Michael Gastegger. Equivariant message passing for the prediction of tensorial properties and molecular spectra. In International Conference on Machine Learning, pp. 9377-9388. PMLR, 2021.
Nima Shoghi, Adeesh Kolluru, John R Kitchin, Zachary Ward Ulissi, Lawrence Zitnick, and Bran- don M Wood. From molecules to materials: Pre-training large generalizable models for atomic property prediction. In The Twelfth International Conference on Learning Representations, 2024.
12

---

Published as a conference paper at ICLR 2026
Benjamin ER Snyder, Ari B Turkiewicz, Hiroyasu Furukawa, Maria V Paley, Ever O Velasquez, Matthew N Dods, and Jeffrey R Long. A ligand insertion mechanism for cooperative nh3 capture in metal-organe frameworks. Nature, 613(7943):287-299 2023.
Yunchong Song, Siyuan Huang, Xinbing Wang. Chenghu Zhou, and Zhouhan Lin. Graph parsing networks. arXiv preprint arXiv:2402.14393 2024.
Hongjian Tang. Qisong Xu, Mao Wang. and Jianwen Jiang. Rapid screening of metal orgaanc frameworks for propane/propylene separation by synergizing molecular simulation and machine learning. ACS Applied Materials & Interfaces, 13(45):53454 2021.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances Neural Informa- lion Processing Systems, 30, 2017.
Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence sequence for sets. arXiv preprint arXiv:1511.0623 2015.
Yusong Wang, Tong Wang. Shaoning Li, Xinheng He, Mingyu Li, Zun Wang, Nanning Zheng, Bin Shao, and Tie- Yan Liu. Enhancing geometric representations for molecules with equivariant vector-scalar interactive message passing. Nature Communications, 15(1):313, 2024.
Thomas F Willems, Chris H Rycroft, Michaeel Kazi, Juan C Meza, and Maciej Haranezyk. Al- gorithms and tools for high-throughhut geometry-based analysis of crystalline porous materials. Microporous and Mesoporous Materials, 149(1): 2012.
Christopher E Wilmer, Omar K Farha, Youn-Sang Bac, Joseph T Hupp, and Randall Q Snurr. Structure Structure-property relationships of porous materials for carbon dioxide separation and capture. Energy & Environmental Science, 5(12):9849-98)) 2012.
Junran Wu, Xueyuan Chen, Ke Xu, and Shangzhe Li. Structural entropy guided graph hierarchical pooling. In International conference on machine learning, pp. 24017-24030. PMLR, 2022.
Zhanghao Wu, Paras Jain, Matthew Wright, Azalia Mirhoseini, Joseph Gonzalez, and Ion Stoica. Representing long-range context for graph neural networks with global attention. Advances in neural information procexsing systems, 34:13266-3, 2021.
Tian Xie and Jeffrey C Grossman. Crystal graph convolutional neural networks for an accurate and interpretable prediction of material properties. Physical Review Letters, 120(14)) 2018.
Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural
networks? arXiv preprint arXiv:1810.00822 2018a.
Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In International conference on machine learning, pp. 5453-5462, pmir, 2018b.
Sihai Yang. Junliang Sun, Anibal Ramirez-Cuesta, Samantha K Callear, William IF David, Daniel P Anderson, Ruth Newby, Alexander Blake, Julia E Parker, Chiu C Tang, et al. Se- lectivity and direct visualization of carbon dioxide and sulfur dioxide in decorated porous host. Nature Chemistry, 4(1 (1):8887-894 2012.
Zhitao Ying, Jiaxuan You, Christopher Morris, Xiang Ren, Will Hamilton, and Jure Leskovec. Hier- archical graph representation learning with differentiable pooling. Advances in neural information processing systems, 31, 2018.
Xiang Zhao, Yanxiang Wang, Dong-Sheng Li, Xianhui Bu, and Pingyun Feng. Metal-organic frameworks for separation. Advanced Materials. 30057):170588, 2018.
Sheng Zhou, Osama Shekhah, Adrian Ramírez, Pengbo Lyu, Edy Abou-Hamad, Jiangtao Jia, Jiantang Li, Prashant M Bhatt, Zhiyuan Huang, Hao Jiang, et al. Asymmetric pore windows in mof membranes for natural gas valorization. Nature, 606(7915):706-712, 2022.
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A APPENDIX
A.1 PROOFS
A.1.1 PROOF OF THE EQUIVALENCE BETWEEN LOCAL DESCRIPTION FUNCTION AND CONTRIBUTION FUNCTION
Consider a graph G = (V, E)) with node set V. The target property is assumed to be a function of the nodes, i.e. p = pVV). A message-passing graph neural network (MPNN) can be regarded as consisting of local description function and a readout function f. Each node can only perceive the information within its receptive field rmp. Formally,
heg. = <ff Bgg E V, r(U,, 0z) < kmpp)
(11)
H (heg 0/ V}
(12)
Iggaph = f(H)
(13)
Without loss of generality, the readout function can always be written in node-decomposable form:
hgraph f(H)
Σ f(hy, H))
(14)
WVV
This is because when f has an unlimited receptive field, its output can at least be evenly distributed each node. In practice, for most target properties (such as total energy and most properties in QM9 except dipole moment), f is implemented as an MLP depending only on ha More generally, when f has finite receptive field rread, there always exists an equivalent local description function e' defined on expanded neighborhood. N (pos,)
{v; r(U;, U3) < max(Fmp: Treaa) such that K', = f(h,, H) = c (N(pos))
(15)
ligraph Σ R4,
(16)
VEV
Region-decomposable formulation. over space:
We now reformulate the graph-level feature as an integral
hgraph / g(() d''
(17)
pos g(r) = gx)(r) where M(x) = - pos,Il < rg}
(18)
We will show that equation 16]and equation 17 are equivalent in expressive power, by constructively defining mapping from one form to the other and vice versa.
From node-decomposable to region-decomposable. Define using Dirac delta functions:
g(r)
Σ
h1, 8(r pos,)
(19)
E.EVEeLEN((E)
Since / -ooo, 1, we have
/
Σ Nog / 8(1 - )d'
g(() d(r)
W.EV
Σ N/, hgraph
(20)
BEV Thus any node-decomposable form can be expressed region-deccoppsabble form.
From regoos-dec to node-decomposable Assume g(r) has finite receptive field rg- For each position define a normalized weight over nearby nodes:
||r pos; < <Tg
||r ||r-poe,
(21)
0.
otherwise
(r)
w,(F) Σ
(22)
E Ev,ev, (N(r) ()(
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For any have ,ver wr(r) 1. Therefore,
/ ((()) (o(r)dr== ddr= / g(r) Σ wr(r) NEV
Σ g(r)wx(()dd'
g())w,(r) (r) d
(23)
NEEV l|r-pos, <<r
We can thus define local description function for each node as
c(v))
(r)) dr
(24)
This gives a node-decomposable representation that is equivalent to the original region- decomposable one.
Remarks. Note that the mapping from region to node is not unique, because the choice of weights (r(r) is arbitrary. This implies that inductive bias is crucial in practice. For example, when pre- dicting total energy. because node-level labels are unavailable, one could (in principle) assign all the energy of methyl group its carbon atom and zero to hydrogens. Although this rule is learnable, it is physically incorrect. Therefore, additional inductive biases such as bond length, bond angle, or dihedral angle are typically introduced to guide the model.
Conclusion. In summary, any node-decomposable representation can be mapped to a region- decomposable one, and vice versa. Hence, these two formulations are expressively equivalent: they do not change the representational power of GNNs, but merely reflect different inductive biases.
A.1.2 PROOF THE GLOBAL RECEPTIVE FIELD
On realizing a global readout with local encoders. Even when the readout function f possesses global receptive field, the local description function can still be restricted have only local receptive field. This can be achieved by the following construction.
For each node w. let its local descriptor be
he, =
(25)
which depends only on bounded neighborhood of u.. Assign hy, to a small spatial region R, surrounding the node position pos,, and define the spatial field
g(r) Σ hr, XR, (r),
(26)
WEV
where X.R. (r) is the indicator function of region R4.
Because the number of nodes is finite, one always choose regions R; that are sufficiently small and pairwise disjoint. Let the volume of each region be V; = R Then, within R; the field is constant:
g(r) = hwrr
r E Rç-
(27)
The field g(r) can be discretized as finite-resolution 3D image by sampling on a grid with voxel size 8V such that each R, occupies at least one voxel. This yields a tensor representation {g(r,) 1 of finite spatial resolution. Although this representation cannot capture ideal Dirac delta functions (which would require infinite resolution), can losslessly represent the piecewise-constant field constructed above because each R2 is non-overlapping.
Finally, the global readout can be realized as a general function operating on this spatial field:
p = s(h hev) where F can be any architecture with global receptive field (e.g. Transformer or CNN operating on the 3D grid). In this way, the local descriptors c(v;) remain strictly local, while the global dependency is handled solely by the subsequent global network F. This shows that even if the target readout ff is global, it can be implemented by composing a local encoder e (on the graph) with a global readout F (on the discretized spatial field), without violating the locality constraint c.
F((g(r,)))
(28)
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A.2 DETAILED ARCHITECTURE OF SPATIALREAD
Given a material G = (V, E) with atomic positions {ri). atomic types (zi). and lattice vectors L, SpatialRead operates as follows.
We place spatial nodes on a uniform M x M x M grid inside the unit cell, resulting in a total of M spatial nodes. The fractional coordinates of the spatial node indexed by (i k) are given by
i
k
rg ji
i, O...... M -11 1.
(29)
M
M
M
Next, we transform the fractional coordinates to cartesian coordinates according to the lattice vector. This uniform sampling scheme provides full coverage of the 3D domain with spatial resolution controlled by M.
Next, construct a heterogeneous graph composed of types of nodes: atomic nodes (v; V) and spatial nodes (sz). Edges are built based on Euclidean distance with a cutoff radius Truts typically set to 5-8 Å, and respecting a maximum neighbor limit for efficiency. Crucially, we allow two types of edges:
Atom-atom edges: (vt, 05) if ||r; ryyl Tout
Atom-spatial edges: (v,, 85) if ||r; Tsy || I Peut
No edges are allowed from spatial nodes to atomic nodes, enforcing unidirectional information flow: atoms influence space, but not vice versa. Notably, periodic boundary conditions were taken into account, the materials being dealt with in this work are all crystals.
Message passing is performed using a PaiNN-style MPNN (Schütt et al. 2021), which jointly up- dates scalar and vector node features through interactions along edges. The process runs for T layers, updating atomic and spatial node features, without altering the original MPNN backbone. At the final layer, obtain:
An unordered set of atomic feature vectors: (h Ji=1
An ordered list spatial node feature vectors: {h?, 1512
To process these heterogeneous features, we adopt a multi-modal Transformer decoder (Vaswani al. (017)). Since the number of spatial nodes fixed (i.e., M)3 and their order is determined by the i, J. index of each node in equation 29] we can train a learnable position embedding p of shape [M/ F]. We add the learnable positional embedding each spatial node feature:
h. h" + P
(30)
where Pj is learnable embedding encoding the 3D index of the voxel. A [CLS] token with learned initial embedding h[CLs] is prepended to the sequence of spatial node features.
The atomic features are padded to fixed maximum length M (e.g., M 200) handle variable- sized crystals, The full input to the Transformer decoder is:
Input hiCLs), hep .... har12 [he hypp 0, ..., 0]
(31)
The Transformer decoder, consisting of L attention layers, processes this sequence via self-attention and cross-attention mechanisms. Importantly, no causal masking is applied, allowing full interaction among all tokens. After processing, the final state of the [CLS] token is used for prediction:
h out [CLss = DecooorrreeDDeee((([[[e(e
MLP MLP(houts) (houss))
(32)
where p is the predicted property.
In the ablation setting spnode mlp, we bypass the Transformer and instead use simpler readout: each spatial node feature is processed independently by an MLP to yield a scalar output = MLP Ah ) and the final prediction is the average:
p
1 N
Σ
oj
(33)
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When spatial nodes are absent, this reduces averaging atomic outputs equivalent to conventional numeric-level pooling equation 3).
Table 4: Hyperparameters of the PaiNN backbone and Transformer decoder used in SpatialRead.
Component
Parameter
Value
Number of layers
6
Hidden dimension
128
PaiNN
Filter dimension
128
Cutoff radius cutt
6.0
Maximum neighbors
30
Number of layers
Hidden dimension
128
Transformer Decoder
Number of attention heads
8
Feed-forward dimension dropout
512
0.15
Maximum atomic count (padding)
1024
Number of spatial nodes N, Position embedding size
512 (83)
Spatial Nodes
128
Table 5: Training strategy and optimization hyperparameters used in Spatial
Training Parameter
Value
Total epochs
Optimizer
AdamW
Learning rate (initial)
x 10 4 PeduaeL
Learning rate scheduler
Monitor metric
val_loss min 10
Mode
Patience
Factor
0.8
Threshold
1 x 10-4
Minimum learning rate
1 x 10
Weight decay
0.0
Batch size
A.3 DATASETS
The dataset mainly includes four material types:
Metal Organic Frameworks (MOFs): 23157 samples from Chung al. 2019 (2021), Gulbalkan et al. 2023). and Kang et al. (2023).
Tang et al.
Covalent Organic Frameworks (COFs): 7000 samples from Hu et al (2020).
2015 and Deeg et al.
Porous Polymer Networks (PPNs): 7000 samples from Martin et al.
2014
zeolites: 7000 samples from Kim et al. (2020).
Task types include:
Topology Type
Void Fraction
Accessible Surface Area
Pore Limited Diameter
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A
Distribution of Material Types
B
Distribution Task Types
Zooita
Hemyy
PPM
Sep, Rac.
Geometry
5
cDr
2
23157
Ads, Heat
NOT
Ads. Eao
Figure Dataset spatial properties for porous materials. (A) Distribution of material types. The dataset contains four types of porous materials: (I) Metal Organic Frameworks, (2) Covalent Organic Frameworks, (3) Porous Polymer Networks, (4) zeolites. (B) Distribution task types. Tasks mainly include five types: (1) Geometric Features, such as void fraction, accessible surface area etc. (2) Gas adsorption capacity, (3) Separation ratio, (4) adsorption heat, (5) Henry's 's constant.
Largest Cavity Diameter
The geometric features are calculated by Willems et al. (2012). The probe radius set to 0.5. The number of sampling points are set to 2, 000 for surface area and 50, 000 for volume. These datasets All material structures
are randomly split into train, validation and test set according to can be found in Bobbitt et al. (2023), which are collected from Wilmer et al. (2012) and Codng et al. 2019).
The source of the dataset, as well as the sizes of the training, validation and test sets, shown in Table 6
A.4 TRAINING DETAILS OF BASELINE MODELS
CGCNN (ie & Grossman 2018): CGCNN is one of the most typical MPNN for crystal. But as kind of graph convolutional neural network, the performance has lagged behind modern methods. Despite, it still reflect the basic feature of Graph Convolutional Network. The model architecture follows the original CGCNN. We train CGCNN for 300 epochs, the learning rate to 0.01 and decays to 0.001 and 0.0001 in 100 and 200 epochs. Other settings follow the original CGCNN.
contrastive learning in material structure and SMILES code. MOFormer pretrain CGCNN in about 300,000 MOFs. To achieve full training, we fine-tune the pre-trained CGCNN for 60 epochs instead of the default 30 epochs.
MOFormer
(Cao al. 2023): MOFormer is kind of pre-training method, which
JMP (Shoghi al.) 2024): JMP is GemNet (Gasteiger et al. 2021) pre-trained 120 million material and molecule. We fine-tune JMP for 80 epochs. All training strategy and model ar- chitecture follows the default setting. Nevertheless, the default setting of JMP on qMOF Rosen et al.] 2021 2022) dataset is not suitable for our task. Most MOFs in the qMOF dataset is smaller than CoREMOF. In original settings, JMP adopts an adaptive strategy to set cutoff and max_num_ncieee neighbbbo.... The cutoff is fixed to 19.0 Å. But when the number of atoms is larger than 300, the max_num_neighle neighbors is set to 5, which is too small. A large number of MOFs the CoRE- MOF (where most structures real and obtained by experiments) dataset is larger than 300. We fixed the max num neighbos to 8, which achieves balance between performance and computation cost.
GemNet (Gasteiger et al. 2021): Considering that JMP a re-implementation of GemNet, we use the same code of JMP instead of the original implementation. We train the code of JMP without
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Table 6: Details of the dataset
Task
Unit
Training
Test
Topology VF
4,900
1,050
1,050
4,900
1,050
1,050
2023 Willems al.
Kang
Geo
ASA
m²/g
4,900
1,050
1,050
2012
PLD
4,900
1,050
1,050
I.CD
4,900
1,050
1,050
N₂ Ads.
cm CMT(Ss
5,000
1,000
1,000
CoREMOF
Chung et al.
2019
(STPPg
1,000
CH, Henry
mol/kg/Pa
5,000
1,000
1,000
N; Henry
mol/kg/P
5,000
1,000
1,000
CH4/N2
Gulbalkan et al.
2023
CHu/Nz Sel. (0. Ibar)
5,000
1,000
1,000
CH,/N, Sel. (Ibar)
1,000
CH.N: Sel. (10bar)
5,000
1,000
1,000
C:Ho Ads.
mol/kg
1354
170
170
CyHg Ads.
mol/kg
1354
170
170
C,H,/C,H, Sel. (Ibar)
1354
170
170
C3H6/C388
Tang et al.
2021
C-Ho/C.Hg Sel. (infinite)
1354
170
170
TSN (lbur)
1354
170
170
C,H, Henry
log(mol/kg/Pa)
1354
170
170
C.H. Henry (298K)
log(mol/kg/Pa)
1354
170
170
CH. Ads, (65bar)
em? (STP)/ cm²
5,000
1,000
1,000
PPN
Martin et
2014
CH, Ads. (Ibar)
cm / cm3
5,000
1,000
1,000
CH4 Ads. (65bur)
v (STP)/v /v (STP)
5,000
1,000
1,000
Mercado
2018
CH2 (5.8har)
5,000
1,000
COF
Deog al.
0020
CO2 Ads, Heat.
kj/mol
5,000
1,000
1,000
CO: Henry
log(mol/kg/Pa)
5,000
1,000
1,000
CH, Henry
5,000
1,000
1,000
zeolite
Kim et al.
2020
CH, Ads. Heat
kj/mol
5,000
1,000
1,000
loading the pre-trained checkpoint to obtained the results of GemNet. Other strategies the same as JMP.
(Kang et al. 2023 Park al. 2023 MOFTransformer is transformer-baat based multimodal network pre-trained on about million MOFs. The subsequent work PMTransformer further uses 1.9 million porous materials the pre-training dataset. Here the checkpoint of PMTransformer and the original implementation. The only difference is that the default setting finetune the model for 30 epochs, which is insufficient to converge. Thus finetune the model for 60 epochs.
A.5 COMPLEXITY OF SPATIAL NODES AND SCALABILITY TO LARGER SYSTEM
RpatialRead extends conventional MPNNs by introducing spatial nodes and multimodal Trans- former head, raising the question computational overhead. Using SchNet (Schütt et al.) (017)) backbone for controlled experiments, performance until convergence around x 8 x 8 nodes, corresponding to roughly A per node for typical porous materials. Based on this, recommend 512 spatial nodes for materials such as MOFs, COFs, PPNs, and zeolites. 7 shows that adding spatial nodes increases training time by about 30% compared with the baseline PaiNN, while the full SpatialRead remains lightweight (2.9 MB. 2.78 min/epoch), with most overhead arising from message passing rather than the Transformer head. Overall, SpatialRead delivers substantial performance gains with only modest computational cost.
find that increasing spatial node resolution improves
demonstrated in A.2 the graph construction behavior of spatial nodes is similar to atoms. Each spatial node receive message from the neighboring atoms, which are determined by cutoff and max- imum number of neighbors. The complexity of modern GNNs is commonly linearly related to the number of atoms. For example, for GNN like PaiNN (Schütt et al.) 2021) that takes into account the interaction between two bodies. its complexity is O(Nk), where N is the number of atoms and k is the number of neighboring nodes. For model like GemNet that takes into account interactions among up to four bodies, its complexity is approximately O(N13). Therefore, when the number of spatial nodes is fixed constant M, the model complexity is increased O(N + M)k
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Table 7: Computational cost in the CoREMOF dataset
Model
Params (MB)
Training time / epoch (min)
JMP/GemNet
38.5
3.77
PaiNN
1.3
2.01
PaiNN + SN (ours)
1.3
2.61
PaiNN SN + MM (ours)
2.9
2.78
O(N + M) As the system becomes larger, the additional complexity brought about by the in- crease in spatial nodes can be almost negligible. However, it should be noted that larger system typically implies larger space, and therefore may require more spatial nodes to maintain a reliable resolution.
Due to the consideration of three-body and four-body interactions as well as higher embedding and edge feature encoding, the size of the memory usage and the training load increase rapidly with the number neighbors. For actual MOFs like in the CoREMOF (Chung et al. 2019) dataset, the NVIDIA RIX GeForce 4090 (24 GB) only allows to set the maximum value of max num miu_mnmiinaeeee neighbors to 15 (as a comparison, the original setting of JMP in MOFs set the max num wwunnnummmlllla neighbors to 5). Properties such adsorption capacity significantly influenced by the intermolecular interactions, and therefore may be more sensitive to parameters like cutoff and max_ lii__num_neett neighbors. Even under such disadvantage, JMP still achieved performance comparable to that of PaiNN. It can be expected that if the complete 30 maximum neighbors are enabled, the effect of JMP will surpass that of PaiNN. As for PaiNN, due to its simple two-body message passing process, we can allow each atom to have up to 30 neighboring nodes.
When attention brased modules are employed, the dominant cost arises from the O attention over atoms, rather than from the spatial nodes themselves. Since long-sequence attention is not the focus of this work, we evaluate the spatial-node overhead using the Spatial Node MLP de- sign, whose contribution remains size-independent. Specifically, we choose CoREMOF as the base dataset, since the most common dataset used for MOFs. We construct supercell to enlarge the system size and evaluate the computational cost of PaiNN (+ SpNode). We evaluate how long and how much memory are needed to train a PaiNN (with SpNode) in 5,000 MOFs in the NVIDIA GeForce RTX 4090.
The empirical results in Table g confirm that the runtime and memory overhead of spatial nodes remain effectively constant when the number of spatial nodes is fixed. As the system size grows, the relative impact of this overhead diminishes. Nevertheless, for sufficiently large systems, additional spatial nodes may be required maintain spatial resolution, in which case the total cost may scale proportionally with the number of inserted spatial nodes.
Table 8: Training time and memory overhead introduced by spatial nodes at different system sizes.
System Size
Model
Time epoch (min)
Memory (MB)
294
PaiNN
4.07
484
294
PaiNN SpNode
5.21
546
634
PaiNN
4.86
752
634
PaiNN SpNode
5.83
819
3092
PaiNN
10.7
2475
3092
PaiNN SpNode
11.3
2520
8476
PaiNN
29.2
5857
8476
PaiNN SpNode
26.4
5896
A.6 CALCULATION DETAIL OF THE DISTANCE BETWEEN ATOM AND PORE
To quantify the distance of each atom to the nearest pore region, we first identify materials that contain sufficiently large pores. For each crystal structure, uniformly sample 32 x 32 x 32 points within the lattice using the same grid construction described in Appendix A.2 For every sampled point, compute its minimum distance to the surruunding atoms. Points whose nearest-
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Published as a conference paper at ICLR 2026
atom distance exceeds the pore threshold 'pooe 2.0 Å are designated as pore points, representing regions of locally low atomic density.
Once the set of pore points is obtained, compute for each atom its minimum Euclidean distance to this pore point set. This value used as the atom's 's distance-to to-pore metric, reflecting how deeply the atom is embedded within dense regions of the structure.
A.7 DETAILS OUT-OF-DISTRIRUTIOO VALIDATION
To evaluate the out-of-distribution (OOD) performance of SpatialRead, we reorganized the N2 ad- sorption dataset from CoRE-MOF (Chung et al. 2019). For each structure, we computed its void fraction using zeo++ (Willems et al.) 2012). Materials in the top one-seventh of void fraction were selected as the OOD test set. The remaining six-sevenths were randomly split into training and validation subsets with a 5:1 ratio, ensuring that the training and validation sets share the same underlying distribution. This setup allows the in-distribution (ID) performance of the model to be assessed independently from its ability to generalize to high-porosity, distribution-shifted MOFs.
Table 9: OOD evaluation on high-porosity MOFs.
SpatialRead preserves ranking stability under
distribution shift.
Model
MAE
R²
Spearman
PaiNN (40 epoch, ID)
18.7
0.907
0.961
PaiNN+SpatialRead (3 epoch, ID)
21.5
0.904
0.963
PaiNN+SpatialRead (40 epoch, ID)
11.8
0.967
0.983
PaiNN (40 epoch, OOD)
151
-0,019
0.757
PaiNN++SpttialRcad (3 epoch, OOD)
169
-0.138
0.920
Pai PaiNN--SpaiiaReead (40 epoch, OOD)
96.0
0.526
0.951
As shown in Table g the 3-epoch SpatialRead-enhameed PaiNN model attains comparable validation-set accuracy to the 40-epoch baseline PaiNN model, reflecting the low training cost of spatial node augmentation. However, both models experience substantial degradation the OOD test set, as high-porosity structures are absent from the training distribution. Because adsorp- tion capacity is positively correlated with void fraction, the models systematically underestimate adsorption for highly porous materials, leading to large drops in MAE and Ri2
Despite this distribution shift, both PaiNN and SpatialRead-enhancee PaiNN preserve the relative ordering of materials, achieving Spearman correlation coefficients of 0.76 and 0.92, respectively. This indicates that SpatialRead substantially improves ranking stability under OOD conditions, even when absolute prediction accuracy deteriorates.
A.8 STATISTICAL ANALYSIS OF THE INTERPRETATION OF SPATIAL NODES
Fig. 5 compares node contributions with the number of atoms contained in their regions. Strikingly, regions with few atoms exhibit the largest contributions. This observation is physically con- sistent: gas molccules cannot adsorbed into dense atomic regions but are much more likely to be stored in sparse pore regions.
A.9 EFFECT OF SAMPLE STRATEGY
Table 10: Adaptive sampling of spatial nodes on COFs. Resolution-basee sampling uses average of 462 spatial nodes, compared to 512 the fixed-grid setting.
Sampling Strategy
MAE
MSE
R²
Fixed Grid
3.52
31.0
0.979
Resolution-based
2.31
18.6
0.987
To evaluate whether adaptive sampling improves the representation of spatial regions, conducted additional experiments on covalent organic frameworks (COFs), which have the largest unit-cell volumes in our dataset and therefore serve appropriate test case. Spatial nodes were sampled
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Contribution Distribution vs Atom Count
60
12
10
8
Contribution
6
4
00 CED
2
0
-2
1
3
Atom Count
Figure 5:
Atom count (in each region) v.s. contribution. We counted the number of atoms contained in the area occupied by each spatial node, well the contribution of that spatial node.
A
B
Model Performance:
Unit Cell Volume Distribution
1.00
Dataset
50-
MAE
45
R2
0.95
1.50
coremoo
megnett
0.90
MAE
2
0.85
0.75
30
0.50
25-
0.75
10
10'
10a
10*
10²
2
6
8
10
12
Volume (Ar)
Grid Size
Figure 6: Effect on sampling ratio. (A) Performance for different number of spatial nodes. We test number of spatial nodes from 2 2 to 12 12 12. SchNet is adopted to test the best setting of spatial nodes. The corresponding MAE and R2 are drawn in the line plot. (B) Distribution of volume for different materials. The volume of single cell is drawn.
using a resolution-based strategy with a spatial resolution of 4.03 A, resulting in an average of 462 nodes per structure, compared the 512 nodes used in the fixed-grid scheme. Despite using fewer nodes, the resolution-based sampling achieves better predictive performance on COFs, indicating that adaptively allocating spatial nodes according to lattice size can improve spatial coverage. We also observe that resolution- based sampling assigns more spatial nodes to larger unit cells, which increases peak memory usage relative the fixed fixed-grid approach.
A. EFFECT ON SAMPLING RATIO
We considered different numbers of Spatial Node sampling points, ranging from 2+2+2 to 12+12*12 12. The results are presented Fig. 6 As the number of sampling points increases, the model accuracy gradually improves.
22


Reviews:
[{"review_id": "U5aUReW9PJ", "reviewer": "Reviewer_4Vi5", "summary": "The authors present SpatialRead, a new region-based readout function which introduces additional spacial nodes to represent voxelized space that is an important characteristic of some materials. SpatialRead also fuses both atom and spatial nodes passed to Transformer-based readout.\n\nThe empirical evaluation of SpatialRead is performed with several tasks including spatial properties of materials such as gas absorption capacity, showing the superiority of SpatialRead against several existing methods.\n\nThe paper addresses an important problem in AI and materials science. Their approach adds important features on spacial properties which have not been effectively considered by the existing GNN-based approaches. \nIn addition to the evaluation with standard machine learning metrics such as $\\mbox{R}^2$ scores and MAE, the authors also attempt to interpret what happens with the spatial nodes with good illustrations regarding absorption of materials (i.e., Figure 2B).\nI believe this is also an important contribution which reveals the behavior of SpatialRead, especially if/when SpatialRead is used in practical situations to work on materials discovery . \n\nOne comment is that a better strategy which is commonly used in graph embedding for organic molecules can be used easily to further improve the performance. See Equation 4.2 in the following paper: \n\nKeyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. \"How Powerful Are Graph Neural Networks?\", ICLR 2019. \n\nI wonder why the existing approaches as well as SpatialRead do not incorporate this simple, well-known improvement. \n\nThe other minor comment is that Definition 3.1 should be introduced before Equation (8). \n\nAll in all, this is an interesting paper which includes important contributions to the AI and materials informatics research communities.", "questions": "Considering that it is a simple enhancement known to improve the performance for organic molecules, how would the performance fare if all methods presented in the paper incorporate  equation 4.2 in the paper pointed out above?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Introducing an important readout function that can additionally consider spacial capacities of materials\n\nPerformance evaluation of SpacialRead that shows the superiority to other existing methods\n\nInterpretation of SpacialRead regarding the contributions of spacial nodes, which clearly shows strong evidence on why SpacialRead works effectively", "weaknesses": "Comparison/discussion against an approach that enumerates the embedding from iterations 0, 1, .. to T as is done by so-called Graph Isomorphism Network (see equation 4.2 in the paper mentioned above)", "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": "3cukUTUuRq", "reviewer": "Reviewer_LnWH", "summary": "This paper proposes a novel spatial readout mechanism for Graph Neural Networks, with applications to materials science tasks, particularly Metal-Organic Frameworks (MOFs). The authors argue that pooling/readout mechanisms in GNNs still have room for innovation and introduce a method that incorporates additional “spatial” nodes (mostly in void/vacuum) space to improve property predictions.", "questions": "1. Can you provide complete MatBench results for all properties rather than a selective subset?\n2. Can you include an ablation study comparing: (a) base model, (b) base model + void nodes + simple pooling, (c) base model + void nodes + proposed readout?\n3. Why is a new benchmark needed when established ones like mofdscribe exist?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Addresses a relevant problem: readout mechanisms in GNNs remain an area where improvements are possible\n- Applies methodology to practical materials science applications", "weaknesses": "### 1. Weak Empirical Validation\n\nThe experimental evaluation is insufficient to support the claims:\n\n- **Trivial target properties**: The main benchmarks focus on predicting surface area and pore volume for MOFs. From a materials science perspective, these properties are almost directly derivable from the structure itself (via van der Waals sphere calculations), making them poor choices for demonstrating the value of a novel readout mechanism.\n\n- **Selective MatBench reporting**: The paper only reports results on a subset of MatBench properties. This selective reporting raises concerns about the method's general applicability. What happened to the other MatBench tasks?\n\n- **Inconsistent and marginal improvements**: Across benchmarks, the improvements are neither consistent nor substantial enough to justify the added complexity.\n\n### 2. Missing Ablation Studies\n\nThe paper lacks essential ablation studies to disentangle the contributions of different components:\n\n- **Effect of additional void nodes**: The method adds nodes in empty/void space, which allows the model to trivially compute the ratio between occupied and unoccupied space—essentially providing a direct signal for pore accessible volume fraction. This could be the primary driver of any improvements, independent of the readout mechanism.\n\n- **Required ablation**: Compare additional void nodes with standard pooling (max/mean) versus the proposed spatial readout. This would demonstrate whether the gains come from the readout innovation or simply from the additional structural information.\n\n### 3. Added Complexity Without Clear Justification\n\n- The method introduces an additional hyperparameter (sampling degree for void space), increasing model complexity and optimization difficulty\n- Given the weak empirical gains, it's unclear whether this added complexity is warranted\n\n### 4. Questionable Benchmark Choice\n\nThe authors introduce a new benchmark, but the field already has established benchmarks (e.g., mofdscribe for MOFs). The rationale for a new benchmark is not clearly articulated.", "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": "6QnrsjZQQZ", "reviewer": "Reviewer_G88g", "summary": "This paper identifies a key inductive bias limitation in graph neural networks (GNNs) for material property prediction — namely, that existing message passing–readout frameworks assume atom-decomposable properties, which is not valid for spatially defined material properties such as adsorption, separation, or accessible surface area.\n\nTo overcome this, the authors propose SpatialRead, a novel readout function that reformulates the graph-level representation as an integral over space rather than a sum over atoms. The method introduces spatial nodes that voxelize the material domain, forming a heterogeneous atom–space graph with one-way message flow from atoms to spatial nodes. A Transformer-based multimodal readout fuses atomic and spatial representations.\n\nTheoretically, the authors prove that the space-integral and node-summation formulations are equivalent in expressivity when the receptive field is finite. Empirically, SpatialRead achieves state-of-the-art results on 44,157 porous materials and 27 spatial-property prediction tasks, outperforming even large-scale pre-trained foundation models such as JMP (120M samples). The approach also maintains strong performance on conventional (non-spatial) material properties.", "questions": "- Does the voxelization scheme preserve lattice periodicity? If not, predictions might vary under cell replication.\n- While efficient for moderate datasets, its scalability to extremely large systems (>10⁴ atoms) isn’t thoroughly discussed.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The first formal unification of node-decomposable and region-decomposable graph readouts.\n- The directionality (atoms → space) aligns with actual field theory and adsorption physics.\n- Equivalence theorem ensures expressivity preservation.\n- 44k+ samples and 27 tasks, across multiple material systems.\n- ~2.9 MB model surpasses 38 MB foundation models.\n- Bridges discrete atomic graphs and continuous spatial modeling paradigms.", "weaknesses": "- The voxelization scheme (e.g., 8×8×8) is fixed; adaptive resolution or hierarchical partitioning could further improve scalability and accuracy?\n- Although SpatialRead outperforms pretrained models from scratch, it would be interesting to see whether adding pretraining further enhances its capability.\n-  The paper focuses on porous materials and spatial properties; it remains unclear whether the proposed readout function generalizes to conventional datasets like Materials Project or JARVIS, especially since the baseline models (e.g., CGCNN, ALIGNN, Matformer) were originally designed and benchmarked on such conventional crystal datasets.", "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": "d4cWJlRCNH", "reviewer": "Reviewer_DEPu", "summary": "Introduces a new readout function for materials property prediction tasks using GNNs/MPNNs. The idea of the paper is to introduce a grid of additional spatial nodes and readout functions that only use learned node features of these spatial nodes or readout functions that take into account atom and spatial nodes. Experiments are performed on different datasets, and very good performance is demonstrated across different tasks and datasets, compared to fine-tuned models and models trained from scratch.", "questions": "In Section 2, you discuss readout functions used in the GNN/MPNN literature. All references are older than 2019. Please report about more recent developments of readout functions, e.g. the ones discussed in [1] or even completely different approaches such as [2].\n[1] Liu, C., Zhan, Y., Wu, J., Li, C., Du, B., Hu, W., Liu, T. and Tao, D., 2022. Graph pooling for graph neural networks: Progress, challenges, and opportunities. arXiv preprint arXiv:2204.07321.\n[2] Wu, Z., Jain, P., Wright, M., Mirhoseini, A., Gonzalez, J.E. and Stoica, I., 2021. Representing long-range context for graph neural networks with global attention. Advances in neural information processing systems, 34, pp.13266-13279.\n\nIf Theorem 3.1 is true and Eq. 6 and 8 have the same expressivity, what is the advantage of Eq. 8 over Eq. 6?\n\nWhat is the difference between extensive quantities of materials and your definition of spatial properties according to definition 3.1? Are all extensive quantities also spatial quantities?\n\nWhat are the exact definitions of N(s_j) and N(v_j) in Eq. 9 and 10? How many atomic node connections does every spatial node have, and how many spatial node connections? Is this based on cutoff radii as indicated by the phrase \"nearby atoms\"? How are \"adjacent spatial nodes\" defined? The 6 nearest neighbor voxels? Or 26? Or also based on a cutoff radius? Figure 1 does not indicate any s_i to s_j edges. How are the positions of the spatial nodes selected? Figure 1 does not indicate a regular lattice. How is the spatial voxel grid defined for more complex space groups than the one shown in Figure 1? Are distances defined in relative coordinates (within the unit cell) or in absolute coordinates (in real space)? \n\nMulti-Modal Transformer: \"we impose an explicit ordering\" - How is this done, and how does it compare for different types of symmetry groups with different unit cell shapes?\n\nEquation 11: Why does every spatial node only contribute a scalar contribution to the final property? Why do you not use p = MLP_final(Sum(MLP(h))), where MLP(h) is outputting a vector rather than a scalar?\n\nMinor comments: Mistake in Theorem 3.1: \"then is must can be\".", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The experiments show very strong performance of the newly introduced readout functions based on spatial nodes.\n\nThe analysis in Figure 2 and the additional experiments in Figure 3 are very interesting and insightful.", "weaknesses": "The paper claims that current read-out functions introduce an inductive bias that is not suitable for certain tasks and materials, e.g. the prediction of gas adsorption capacity in metal organic frameworks. This claim is not completely unreasonable, but it is also not supported by any quantifiable evidence. An equivariant GNN can learn local geometric atom environments (also around a pore), up to a cutoff radius which is determined by cutoff-hyperparameters and model depth. Thus, in principle, there is no direct reason to claim that it is impossible to learn pore volumes as a function of mean- or sum-pooled atom embeddings. Every atom can learn from its local geometric environment if it is adjacent to a pore or not, and what pore volume is \"attributed\" to this atom, so sum pooling can yield the pore volume.\n\nThe baseline models CGCNN, GemNet, and PaiNN are more than 5 years old. The performance of newer GNN architectures is not compared. What is the performance of the models that are currently leading the MatBench benchmark?\nThe pretrained models are newer, but the authors do not compare the performance of the fine-tuned models to the performance if those models were trained from scratch.\n\nGenerally, a more differentiated comparison to other, more recent readout functions is missing.", "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)