Learning Repetition-Invariant Representations for Polymer Informatics
Authors: Yihan Zhu
Gang Liu
Eric Inae
Tengfei Luo
Meng Jiang
Venue: NeurIPS 2025 poster ()
Keywords: Graph Learning
Polymer Informatics
Generalization
Data Augmentation
Primary area: machine_learning_for_sciences

Abstract: Polymers are large macromolecules composed of repeating structural units known as monomers and are widely applied in fields such as energy storage, construction, medicine, and aerospace. However, existing graph neural network methods, though effective for small molecules, only model the single unit of polymers and fail to produce consistent vector representations for the true polymer structure with varying numbers of units. To address this challenge, we introduce Graph Repetition Invariance (GRIN), a novel method to learn polymer representations that are invariant to the number of repeating units in their graph representations. GRIN integrates a graph-based maximum spanning tree alignment with repeat-unit augmentation to ensure structural consistency. We provide theoretical guarantees for repetition‐invariance from both model and data perspectives, demonstrating that three repeating units are the minimal augmentation required for optimal invariant representation learning. GRIN outperforms state-of-the-art baselines on both homopolymer and copolymer benchmarks, learning stable, repetition-invariant representations that generalize effectively to polymer chains of unseen sizes.

Learning Repetition-Invarian Representations for Polymer Informatics
Yihan Zhu University of Notre Dame yzhu250nd. edu
Gang Liu University of Notre Dame gliu70nd.
Eric Inae
University of Notre Dame einaeend. edu
Tengfei Luo
Meng Jiang
University of Notre Dame tluo@nd. edu
University of Notre Dame mj mjiang20nd. edu
Abstract
Polymers are large macromolecules composed of repeating structural units known as monomers and are widely applied in fields such as energy storage, construction, medicine, and aerospace. However, existing graph neural network methods, though effective for small molecules, only model the single unit of polymers and fail to produce consistent vector representations for the true polymer structure with varying numbers of units. To address this challenge, we introduce Graph Repetition Invariance (GRIN), a novel method learn polymer representations that are invari- ant to the number of repeating units in their graph representations. GRIN integrates a graph-based maximum spanning tree alignment with repeat-unit augmentation to ensure structural consistency. We provide theoretical guarantees for repetitionin- variance from both model and data perspectives, demonstrating that three repeating units are the minimal augmentation required for optimal invariant representation learning. GRIN outperforms state-of-the-al baselines on both homopolymer and copolymer benchmarks, learning stable, repetition inveationniivariiant representations that generalize effectively to polymer chains of unseen sizes.
Introduction
Polymers are materials composed macromolecules made up of multiple repeating units (RUs), i.e., monomers with polymerization points. For example, homopolymer consists of a single RU repeated along the chain (c.g., polyethylene), resulting in uniform properties. copolymer interleaves two more distinct RUs (e.g., styrene-butadiene rubber) to combine or tune material characteristics I iillustrates two examples of polymers, which can be encoded as graphs of atoms and bonds. Polymer informatics is emerging discipline the intersection of materials science and machine learning, aiming to accelerate the discovery and design of new polymers through data-driven approaches [2] 6 [8
[. Figure
23 19] such graph representation learning. In this direction, graph neural
networks (GNNs) have been developed for small-molecule tasks 4 13 34 36]; however, their generalization polymers has been limited single repeating unit.
Ideally, different graph representations the same polymer should yield identical or highly similar feature vectors. Effective polymer modeling therefore requires learning representations that capture the underlying chemistry repeating units while remaining invariant to repeat size (i.c., the number of repeats). A similar challenge arises in sentiment analysis, where repeating adjectives, clauses, sentences should not change the overall sentiment of a text. Recent advances in NLP enforce such repetition invariance to promote stable behaviors of language models [26] 33) Likewise, robust graph representations for polymer informatics should be expected to exhibit similar invariance.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Augmentation
Homopolymer
A A
Block Copolymer
x
-A-B-B-
A-B-BBAA-BBAAABB-B-
B-BBB..-A-BBBB
Traditional
Ours
True Polymer
Carbon
Fluorine
Nitrogen
Sulfur
Figure 1: Graph representations homopolymers and block copolymers. Left and right: Prior graph learning methods model polymers as small molecules using a single repeat unit (c.g., -A- or -A-B-B-) which may not capture the long-chain features of polymers. Middle and right: Repeating the unit multiple times better approximates realistic polymer structures and
effective data augmentation strategy.
Teart
Augmentation
Add Augmantation
From GCN MST aligmeet To GRIN
(a) GCN
(b) GRIN
Figure 2: Figure 2: I-SNE visualization of polymer embeddings from GCN and GRIN (ours) across repeat sizes (1-20 RUs) the glass transition task. Points colored by repeat count (light-IRU, dark=2ORU), (a) GCN produces inconsistent embeddings for different repeat sizes of the same polymer, clustering by size (same color) rather than identity; our augmentation introduces light-to-date stripes, indicating improved alignment repeat variants. (b) GRIN learns repeat-inve invar-inn representations: each polymer's variants form tight, ineeenddeennent clusters, further improved with augmentation.
Figure (a) visualizes the t-SNE embeddings generated by a standard two-layer GCN, trained with and without augmentation, evaluated on test sets augmented with up to 20 RUs. For the same polymer, embeddings, where darker colors indicate larger repeat sizes in its graphs, drift across the latent space as the number of repeats increases. This suggests that conventional message passing entangles repeat size with the core chemical structure of the repeating unit, failing produce eepetition-invarian representations. GCN benefits from repeat-unit augmentation during training, as embeddings of the same polymer with smaller repeat sizes begin to coalesce into visible striations. However, the clusters remain dispersed overall, and embeddings for larger repeat sizes are still scattered, indicating that augmentation alone not efficient in achieving true generalization.
Accurate property prediction across different graph representations of the same polymer requires mapping all repeat-size variants to identical or highly similar latent vectors, while preserving the

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chemical semantics in the essential structure. This joint objective is challenging, repeat size scales the graph linearly, disrupting the fine-grained information needed for precise representation learning.
In this work, we propose Graph Repetition INvariance (GRIN), a method that combines algorithm alignment with repeat-unit augmentation to address the challenge of repetition-invariann represen- tation learning. GRIN draws inspiration from the edge-greedy logic from the maximum spanning tree (MST) algorithm: dynamic programming
updates enforce a fixed reasoning proce- dure, enabling intrinsic generalization across graph sizes. The MST criterion further preserves the most informative backbone of the polymer graph. We provide theory guarantees for invariant representation learning both model-wise and data-wise: Model-wise, GRIN's 's max suu-agggegttion with a sparsity penalty heuristically encourages alignment with the MST construction process, inheriting the size generaeallliitonnn guarantees from DP. Data-wise, under the repeating unit contraction (defini- and identify the minimal
tion 3.1p. we formally establish latent eepeition-invariann (proposition 3.2
repeat size required for optimal invariant learning (proposition 3.3
We conduct experiments on four homopolymer and two copolymer datasets. Whereas homopolymer tasks mainly test repeat-size extrapolation, the copolymer datasets probe a harder regime: capturing the synergistic interactions between distinct repeating units. GRIN consistently surpasses strong baselines, e.g., improving homopolymer Density R² by 10-15% and reducing copolymer Ionization Potential RMSE by 24-30% while maintaining uniform accuracy across all repeat sizes.
2 Related Work
2.1
Invariant Representation Learning
Current invariant graph representation learning focuses on identifying features that remain stable across different environments to achieve out-of-distribution (OOD) generalization on graphs (8] Early work extended invariant risk minimization to graphs, seeking features that remain
16
20
predictive across training environments [3). DIR 35 masks environment-specifi subgraphs via
counterfactual interventions, while Liu al. [18] proposed method GREA to learn graph rationales
with environment-based augmentations by swapping the environment subgraphs between different samples. Although effective against covariate and local-structure shifts, these approaches largely ignore distributional changes in graph scale.
Size generalization, a specific instantiation of OOD generalization, aiming learning consistent representations when node or edge counts vary. To this end, SizeShiftReg (SSR) (77] enforces invariance by coarsening graphs during training and aligning embeddings of original and reduced- scale counterparts. Huang et al. [14] proposed a disentanglement loss that separates size-related factors from task speciffe features. However, our method explicitly targets the unique challenge of polymers, where key structural information must be preserved as graph size changes due to varying numbers of repeating units.
2.2 Neural Algorithmic Alignment
Neural algorithmic alignment has recently emerged as a powerful tool for enhancing GNN size extrapolation 2 I 30). In classical algorithms like Bellman-Ford [1O] and Prim (24]. solutions are typically constructed through a series of iterations. Neural algorithmic alignment enforces each message-passing layer to mirror one iteration of known graph algorithm, rather than treating the network as a black-box end-to-end mapper. Although this approach yields impressive OOD performance well-defined algorithmic benchmarks, its scope remains limited. For instance, Nerem et al. (22) demonstrated provable extrapolation for shortest-path problem by aligning a GNN with Follman-Ford. Existing models remain restricted to synthetic benchmarks 31 and do not tackle real-world graph classification or regression tasks.
3 Methodology
In this section we present Graph Repetition Invariance (GRIN), novel framework that i) augments polymer graphs by chaining repeating units and ii) aligns message passing with a maximum-spanninn tree (MST) on that chain. GRIN learns representations invariant to the number of repeating units.
3

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3.1 Problem Definition
Prior GNN approaches typically represent a polymer as a monomer graph with polymerization points, G (Va, EA), i.e., single repeating unit -A-, where VA represents atoms and ÉΛ represent bonds. These methods ignore chain-level connectivity [TS] T8 37.. They apply mean/sum aggregation on this monomer graph and assume the learned embedding ho (G) will generalize to arbitrary repeat size n, which we found fails: Model-wise, mean erases all size information and sum grows unbounded with n. Data-wise, real polymers include both intra-monomer and inter-monomer connections, while the previous monomer graph captures only the former.
To capture inter-monomer connectivity, we equip each RU with two anchor vertices wi o (the polymerization points, denoted by *) and chain them construct polymer graphs with different number of RUs. Formally, the polymer graph with n. repeats is defined as G(m) = (V, E):
V == TUUV 1=1
E E UU
(UE U
(U 1=1
HUoe in (+)))
All graphs in the family { NeW, share the same RU and identical ground-truth property value. Training {G(")) JeE enforces a learning objective in which the model must predict the same property y regardless of the repeat count.
We propose repeat-unit augmentation strategy that generates multiple graph instances of polymer with varying numbers of RUs. Each RU represents either a homopolymer monomer (-A-) or copolymer motif (-A-B-). Real polymers often contain a very large number (10=-10*) of such RUs, far beyond the scope of traditional methods can learn. is also infeasible to directly augment the training set with all possible repeat sizes. Therefore, in Section B.3.2 we analyze the minimal repeat size required to learn invariant representations for this augmentation strategy.
3.2 Message Passing with Max Aggregation
Given polymer graph G = (V, E`) with initial node features {I, Jeev and bond features Eavs GRIN employs a max max-agggegation GNN with L layers. We follow the notation setting in Gilmer et al. [2] The feature at node o in layer l is denoted as (0) where the node representation is updated by:
A 1 (111)) +U()
((ft-1)
max NEN())
M0 ewv))
where
[0) = I M Rd x E + Rd, U X -R
Rd
(+)) and M are multi-layer perceptrons (MLPs) and max is element wise maximum over incoming messages. With previous state h' N'-1, any non-max neighbour contributes zero to U leaves corre- sponding feature dimensions remain unchanged. The (f) function takes the features of neighbor and the edge (u, v) into message vector. (T()) updates the embedding of node v. Max aggregation imposes selection bias toward the strongest neighbor, we add an L1 sparsity penalty on the message and update networks suppress non-dominant pathways (Eq. 3), encouraging sparse, MST-like backbone.
3.3 Theoretical Guarantee of Invariant Representation Learning
In this section, we provide GRIN's theoretical guarantees for repetition-invarial learning. First, we show that with max aggregation and sparsity constraint, GRIN emulates a greedy DP recurrence and inherits its sizegennralizattion properties. Second, we prove that augmenting with three RUs is sufficient reach optimal repetition invariance.
3.3.1 Model-Wise: Algorithm Alignment with Maximum-Spamming-Teee
For Prim's algorithm 24), let xe e {0, 1} indicate whether node has been added to the tree after steps. Choose start node s, the MST is constrcuted as
1,
max
Wavt
I (F+1)
x1(0) V 1(c arg max a!!)
(2)
0,
C =0

---

where w represents the edge weight, given our message passing design in Eq. we have
M (nl (M D, Cuv)
max (EN(M)
mAX
Wuv
GNN aggregation
The additive residual term h in Eq. I ensures that only the feature selected by max-aggregation at layer é receives an update, while all others satisfy ( Ne f mirroring Prim's 's algorithm where nodes not chosen in the greedy step remain unchanged. Additionally, incorporate & sparsity penalty on model parameters as
Listal Lrusk + (100 lh + 1100111))
encourages many entries of 0/0 6() to become exactly zero This further reinforces the MST alignment by pruning weak connections the learned message scores, so that only the strongest (MST-relevant) edges survive, enhancing both sparsity and interpretability.
Max vs. Mean, Sum. While max-aggregation selects the single most informative incoming message at each node, classical sum and mean-aggregattion mix all neighbor messages [T] In particular, for sum and mean we have:
Sum ( = v(t)
(N
E
(1((11) (1(-)) (ca)))
uEN(v)
Mean h = (0(00(1662)) (1((-1))
Ul6)
Σ
(1 emn)
wEN(v)
Sum-aggregation grows linearly with the node degree, and mean-aggreeation normalizes by degree but still blends every neighbor's contribution. Both operations depend on the size of the neighborhood:
Σ M (W()). (EN((o)
Σ M dilutes extremes as [A(u)] grows.
[N(v]]
uEN(e)
These findings are further corroborated by the experimental results in Table T2 which show that sum- and -aggrggggegttoon degrade substantially in performance the repeat size increases. In contrast, by combining -gggregggtii with an & sparsity penalty, GRIN effectively emulates MST construction and thus inherits its inherent size- gereraaeeliiiiionn capabilities, as aligning to DP recurrences enjoy aatoo--distriuttin generalization across input scales 38
3.3.2 Data- Wise: Minimal Repeat Size for Invariant Representation
One can repeat the polymer unit (-A-) as the augmented data point Pn,, it can be viewed as an undirected graph G = (Va, Eg). Let == Val/IVAl represents the number of repeats (or the degree of polymerization). We analyze the model behavior over an abstract hyperchain level and show that repeat-augmented polymers Pm (m 2 2), the model can apply same update rule at each layer, yielding constant output y irrespective of the repeat size.
Definition 3.1 (Polymer Hyperchain). We contract each repeating unit -A- in the polymer graph to single supernode and neglecting all intra-monomer edges, while retaining only the inter-monomer polymerization edges. P, can be abstracted as hyperchain
Sn {s1, Sp.....sph sp), E,, { [s,, 8++1} | =
1}.
This definition based on the dominant backbone of a polymer chain and enforces me-dimensional DP structure for message passing as Dudzik and Veličković define its neighborhood and hyperdegree as
( For each supernode 8; E S,, we
0, 1,
(s.)) = (sy (s., 8/s E F), degp(s:) = [s(s:)
1, i = 1 or = n,
2, i = 2,..., - for > 2.
In the following analysis, hyperdegree-n and degp(.) denote supernodes with hyperdegree.
5

---

Hyperchain Message Passing. We denote by h E Rd the embedding of supernode s after the t-th layer, and m its aggregated message. Then layer-wise message passing reduces
0,
degp(.) 0, (.) 2 1,
m 82
max Ma (a eg,a), (,,)(
(5)
1(++1) = Un (1 , m.
(6)
Given Eq. (5. we notice that training only on Pi (hyperchain of length 1) activates solely the (.gp(.) = branch, leaving dcg 2 1 parameters unlearned which is essential in extrapolating to longer chains. To address this issue, we supervise on a merge set {P1. P. with shared target y Let loss be written as:
(())
(P1:0) - -- | (Pn::) v-12 11011.
Here h.? is supernode embedding after t layers, and ||0|| is the l term introduced in Eq. 3). Proposition 3.2 (Latent Aopetition-Invarinnne Under definition 3.1 and ler 6" arg min C(0). every test hyperchain Pm. with m 2 2, the prediction is
for (Pm) = y.
Training on {Pi, exposes both structural dases-nodes (-gr(() = 0 and deg 1 and the & term zeros out all other pathways, which means the complete update rule in Eq. is learned.
Hyperdegree-2 Supervision Introducing augmentation Pa. with supernodes satisfying (egr(.) N 1 helps model learn extrapolation across repeating units. A hyperdegree-1 supernode receives exactly one incoming message, so its gradient flows along single branch. In contrast, a hyperdegree-2 supernode has two competing branches, introducing supervision for multi-branch aggregation.
Let 6, aC/0m, Rdd be the back pock-propggated error arriving at hyperdegree-2 supernode with two neighbors 82, 8y- Under classical GNN contraction assumption
[88] each gressage-passingg layer assumed to be T--L.ischitz, i.e., it satisfies f(x) [(x')] L| x'| for all inputs æ. x' and some constant L<1, so any gradient norm is reduced by at most a factor of T. per layer.
Proposition 3.3 (Accumulated Gradient Norm). When supervising Pa with 2 3, the total back procpprrppgated gradient norm at any degree-2 supernode satisfies:
1 n-
2 1l 11
| 112 ||0||2 L
||0||2 = max 1
-
Under the hyperchain abstraction, P3 (the smallest sample containing hyperdegree-2 node) the minimal configuration for GRIN to learn true multi- branch fusion. Py never exposes this dual- message scenario and thus could be insufficient. Furthermore, given the contraction factor L, increasing repeat size beyond 3 yields only geometrically vanishing improvements with additional training cost.
4 Experiments
We conduct experiments for three research questions:
Q1) Effectiveness: Does GRIN make more accurate predictions for homopolymers and copolymers than existing methods?
Q2) Repetition-Innariinee Learning: Does GRIN learn invariant representations and maintain performance across varying repeat sizes?
Q3) Ablation study: What the minimal repeat size required for augmentation? How does the merge ratio (proportion of augmented graphs) impact performance?

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Table I: Statisties of six datasets for property prediction.
Dataset
Property
Graphs
Avg Size
Max # Size
GlassTemp
7,174
20.5
69
MeltingTemp
3,651
18.3
60
Homopolymer
PolyDensity
1,694
15.9
48
O2Perm
595
18.0
49
EA
3,000
17.7
36
Copolymer
IP
3,000
17.7
36
4.1 Experimental Settings
Datasets. We evaluate property prediction task on four homopolymer datasets and two copolymer datasets. The four datasets predict the glass transition temperature (GlasxTemp, ?C), polymer density (PolyDensity, g/em") ), melting temperature (MeltingTemp. 'C) and oxygen permeability (OzPerm, Barrer). The other two about electron affinity (EA, eV) and ionization potential (IP, eV). The dataset statistics are given in Table II in which the Size refers to the diameter graph calculated by dia(G) max VEV (u, u). For every polymer, construct family of polymer graphs {G J==1 as Section 3.1 We GeIN-RepAug to denote model training on G(2) and GRIN for model training with repeat-unit augmentation (G()) Dataset details can be found in the appendix A
Evaluation and Baseline. We evaluate the regression performance using the coefficient of deter- mination (R²) and Root Mean Squared Error (RMSE). For baselines. we compare our GRIN with methods designed specifically for size-generalization: DISGEN, SSR, BFGNN 7 [4 22] and others for general OOD, including GREA, DIR, IRM, RPGNN 3
We both GIN and GCN as graph encoder for all models. Please refer to appendix B for details of implementation and appendix for computational efficiency comparison.
18 35).
4.2 Results on Effectiveness (Q1) and Repetition-Invarian Representation (Q2)
We evaluate on (G() 160 for homopolymer and { 120 for copolymer, graph diameter ranges from 10 to 10a. GRIN consistently achieves best results among all tasks, GRIN-RepAug ranks second, which demonstrate observations hold for both GCN and GIN backbones, confirming that the gain stems from the repeat- aware alignment itself rather than the choice of encoder. More results can found at appendix D
method's effectiveness of repetition-invariant learning. All
Polymer Datasets Table and Table 3 report R² and RMSE on Testl (1 RU) and Test60 (60 RUs). Testl measures property prediction performance, whereas Test60 probes a 60x repeat-size extrapolation.
Effectiveness: Across all four homopolymer tasks, GRIN with/without augmentation achieves top-2 accuracy on all test sets. Compared to the strongest baseline, GRIN achieves a 1-2% improvement on GlassTemp, Melting Temp, and OyPerm, and a remarkable 15% PolyDensity. The advantage widens under repeat-size extrapolation: the improvement exceeds 10% for Test60. Several baselines collapse on Test60 (negative R²). particular, DISGEN almost fails PolyDensity and O,Perm: its augmentation pipeline depends a GNN-Explainer module that can misprioritize nodes (common in the noisy, low-sample O,Perm and PolyDensity datasets), results in training failure.
Repetition Invariance: GRIN's performance gap between Testl and Test60 stays within +3%, confirming that the model indeed learns ippetiiio-innvaiinn representation. Once trained on I-RU and 3-RU graphs, transfers to polymers sixty times larger without loss of accuracy.
Copolymer Datasets Table reports R² and RMSE on Testl (1 RU) and Test20 (20 RUs). Testl measures property prediction performance, whereas Test20 probes 20 repeat-size extrapolation.

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Table 2: Results on homopolymer datasets (GlassTemp, Melting Temp) with repeating unit (Test1) and 60 repeating units (Test60): GRIN consistently achieves the highest R2 and smallest RMSE.
GlassTemp
MeltingTemp
Model
Test R-1 RMSEL
Test50 R-1
Testl
Test60
RMSE
R-1
RMSEJ
R²1
RMSEL
GCN 15
0.878:0.001
38.740.1
0.81810.009
47.2+1.0
0.69340.002
62.610,2
0.6642100004
65.510.4
DIR 15
0.701=0.054
60.8+5.7
<00
> 100
0.374=0.235
88.2&14,4
<0
> >100
GREA
0.870=0.005
40.0+0.8
<0
> 100
0.700=0.005
61.9=0,5
0
> -100
IRM
3
0.872=0.016
39.7+2.5
<0
100
0.696#0.010
62.3=1.0
0
> 100
RPGNN 21
0.888=0.005
37.1
<0
> 100
0.691#0.017
62.8+1.7
0
> 100
SSR
0.772=0.044 044
22.8+5.2
100
0.693=0.027
62.6=2.8
0
100
DISGEN
0.88520.00S 005
37.6++0.8
0.84640.006 004
43.5+0.6
0.723+0.003
60.1
0.633+0.013
6..621.3
BPGCN
0.88910.003
37.040.55
0.85010.016
43.0a2.2
0.698.000006
62.6
0.68310.013
63.611.3
ARRIN-ReAuug
0.890=0.00 001
36.8+0.1
0 8.66+00011
40 40.6+2.0
0.741=0.007
57.5=0,8
0.707 009
61.1+0.9
GRIN
0.89320.001
36.3+00.2
0.892a0.001
36,520.2
0.745 004
57.020.4
0.746:0.002
56.910.2
37
8822=00003
38. +
O. 0.848=0.001
43.3+n.1
0.697+0.007
62.2+0,7
0.668#0.015
65. 15.5
DIR 851
0.600=0.087
70.1+6.0
<00
100
0.270=0, 124
95,9=8.1
<00
> 100
GREA 18
0. 0.865=0.008
40.7+1.2
<00
> 100
0.705=0.010
61.4210
<0
> 100
IRM 3
0.88110.002 0.0
38.340.3
<00
100
0.69910.009
62.0s0.9
<0
> 100
RPGNN 21
0.88920.000 004
37.0=0.7
<0
> 100
0.702+0.013
61.2#1.3
<0
> 100
SSR
0.83420.053
44.917.0
100
0.61010.138
69.9s12.0
> 100
DISGEN
LE]
0.88==0.002
37.6+0.4
#0.01
42.8
0.726+0.011
19.2+1.2
0.646=00.023
67.2+2.1
BEGIN
22
888=0.002
37.1+0.4
0.842=0.010
44.1=1.4
0.697=0.006
62.2=0.6
0.676=0.016
64.3+1.6
GIRIN-RepAng
O. 894
26.5
0.8764000009
39.0&1.5
0.73640.006
57.920.6
0.705#0.001
61.440.1
GRIN
0.896=0.001
35.7+0.1
0.895=0.001
36.020.1
0.740=0.001
57.8
0.739=0.002
57.7+0.3
Table 3: Results on homopolymer datasets (PolyDensity. OzPerm) with I repeating unit (Testl) and 60 repeating units (Test60): GRIN consistently achieves the highest R2 and smallest RMSE.
PolyDensity
OyPerm
Model
Testl k-t
Tesch0
Testl
Test5l
RMSFL
R²1
RMSEL
k²1
RMSEL
RMSFJ
GCN 15
0.68120.003 0.12540.001
0.66010.035
0.12940.007
0.87040.008
786.7423.7
0.87640.034
76L.7=110.4
0.662=0.021 0.12940.004
>1
0.121+0.078
2045.3492.4
3000
GREA
18
0.715=0.017
N.118+0.004
<0
1
0.921+0.024
609 1+98.8 8
<0
> 3000
IRM
0.68520.010
0,12410.002
0.87410.079
752.74230.6
<0
3000
RPGNN F11
0.662:00004
0.129H0.001
0.096+100012
2075, 24:
<0
3000
SSR
0.37620.1 149
0.17420.021
<00
I
0.82910.070
887.9 187.9
<0
> -000
DISGEN
1E
0.20330. 104
0.19740.011
0.189
0. 0.199=0.014
<0
3000
3000
BRGON
D663:=0.036
0.134+0.00 000
0.631+0.061
0.134=0.011
0.916+0.014
631.5452.6
0,901=0.014
685.2=49.1
GRIN-Rep.Agg
0.720=0.018
0.11740.0000
0.71550.031
0.118:0:007
0.92344.000
604.7431.9
0.910.1 008
655 2426.6
GRIN
0.730=00.01
0.11540.004
0.74720.009
0.111=00002
0.929:0.002
563447.5
0,929:0.002
581.747,2
GIN 32
0.601:0:011
0.123+0.00 002
0.618:0.036
0.137:00000
0 019
836 345334
0.865:0.014
801 42:40.7
DIR 35
0.61920.060
0.13640.011
<00
>1
0.51740.299 297
1452.34527.1
<0
> 3000
GREA
1
0.723=0.008
0.11740.000 .02
0.918+0.024
619.5494.3
3000
IRM
0.688=000008
0.124H0.002
<00
0.858+0.00 0999
796.3+251
<0
3000
RPGNN
H]
0.62920.039
0.13540.007
<0
I
0.09611.012
2075.2+14.2
<0
> 3000
SSR
0.448:0.022
0.16540 003
<00
0.745+0.027
1101.5459.8
3000
DISGEN
0.213=0.053
0.19660.007
0.208-0 09:0..00
007
> 3000
X000
BEGIN 22
0.673:0.007
0.127 001
0.67820.020
0.126:10 004
0.92410.010
601.6138.7
0,91340.009
643.6232.3
GRIN-RepAug
0.715=00005
0.11740 001
0.720-0.013
0.117=0.003
0.93060.011
577.9+5.3
0.91640.000 003
63122-11.2
GRIN
0.73120.004
0.11510.001
0.75220.006
0.110:0.001
0.93010.007
577.2428.5
0.92920.006
580.9:24.9
GRIN ranks first in every setting and GRIN-RepAug the second. Against the strongest baseline, GRIN raises R² by 1.8-3.7% and cuts RMSE by 14-30% on Test1. The margin widens on Test20, reaching 5.2-7.4% (R2) and 24-35% (RMSE). Baselines show the same sharp performance decline on larger graphs as observed in the homopolymer experiments, underscoring the challenge of repeat-size extrapolation.
Effectiveness:
Repetition Invariance: When graph size grows 20x, GRIN's accuracy remains nearly unchanged: R² drops by at most 0.5% and RMSE increases by less than 2%.
4.3 Ablation Study
Minimal Merge Repeat Size In this experiment, we evaluate the impact of different training pairs on MeltingTemmp (homopolymer) and IP (copolymer). Figure 33 presents the results. Training pair {1,3) (with 3-RU augmentation) reaches the best improvement. Further increasing the merge size beyond 3 leads to convergence, with no significant additional gains. These observations are highly compatible with our theory Section 3.3.2 the (1, 2} training set lacks binary-update supervision, and the hyperdegree-2 gradient strength plateaus for merge size n > 3.
8

---

Table 4: Results copolymer datasets (EA, IP) with repeating unit (Testl) and 20 repeating units (Test20);
GRIN consistently achieves the highest R2 and smallest RMSE.
EA
Model
Test
Test20
Text
Test20
R²1
RMSEL
R²1
RMSE,
R
RMSEL
R²1
RMSEL
GCN L51
0.939=00.006
0.146200.006
0.88740.023
0.19910.020
0.91840.004
0.14120.0004
0.909s0.004
0.14940.003
<00
GREA 18
0.939.0.007
0.146200.008
<0
>-
0.90520.069
0,146222053
<00
>
IRM
0.930=0.012
0.157+0.013
0
RPGNN
0.929:0.003
0.158:0:004
0.9
0.144=0.017
0
SSR 7
0.81820.091
0,248201.022
<0
0.8K340.021
0.168220015
<0
>
DISGEN 14
0.163=0.081
0.542=0.027
0,179#0.082
0.537+0.027
0.232400047
0,452=0.013
0.174=0.062
0,447+0.017
BFGCN 22
0.904000144
0.184:0.014
0.88340.018
0.20340.016
0.91620.014
0.137+0.006
0.901
0.1 DOS
GRIN-RepAAgg
0.952-0.005
0.129=0..07
0.942+0.006
D. 0.14340.008
0.946=00.00
0.113=0.013
0.936=0.010
0.125+0.010
GRIN
0,956z0.002
0.125:0.002
0.952+0.061
0.129:0.002
0.952:0.005
0.108:0.0066
0.947:0.005
0.113+0.005
GIN 37
0.918:0.006
0.170:0.006
0.903a0.007
0.185+0.007
0.925:0.004
0.135=0.004
0.91450.004
0.144
DIR 35
0
>
<00
0.091
0.237=0.072
0.918=0.002
0
IRM
R
0.930=0.010
0.157H+0.011
<0
0.934+0.003
0. 0.126=0.000
<00
>
RPGNN
0.929=0.001
0.158=0.002
0.917=00025
0. 0.141=0.021
0
SSR
0.810:0:11
0.245=0: 104
0.86020.067
0, 0822200042
<0
DISGEN
[
1113200.146
0.554g00044
0.10010.110 107
0.56510.035
0.23220.047
0.432200013
0.18110..06. D60
0.44740.00
BEGIN
0.925=0.00 00/
0.163+0.007
0.91840.004
0.170+0.000
0.929=000006
0.131
0.918=0.002
0.141 002
Rep-Agg
0.954+0.000
0.127+0.006
0.947H0.004
0.137+0,005
0.963+0.001
0.095+0.001
0.953=0.003
0.107++
GRIN
0.961=0.001
0.117=00002
0.960200001
0.119
0.965=00.011
0.092=0..01
0.962=00.011
0.196+40.00
Modling temp (hanopolymter
toop-tymer
10
2.0
2.5
Nelate Improsstiee
2.0
AASE Improvement
2.5
0.5
0.0
lospekymer]
(1.2)
(2.3)
(1.2)
(1.3)
(1.4)
Traningg Scheme
Taining Scheme
(a) Test
(b) Test 60
Figure 3: Performance improvement of GRIN over GRIN-RepAug (without augmentation) under different train- ing schemes (GIN-based). The test sets include repeating unit (a) and 60 repeating units (b) for Melting Temp (homopolymer) and IP (copolymer), respectively. In both cases, training on the (1,3} pair reaches the best improvement, while further increasing the merging size leads convergence.
Effect of Merge Ratio We investigate the effect of different merge ratios in repeat-unit augmenta- tion on the training set. Since we have demonstrated both theoretically and empirically that a repeat size of three is optimal, this experiment uses the training set (1, Unsurprisingly, that Table 5 a balanced 1:1 ratio gives the best performance, while reducing the proportion of augmented samples to 1:0.9 and 1:0.8 leads to monotonic decline in accuracy. This observation further validates our analysis: weakening binary-update supervision (from hyperdegree-2 supernodes) slightly diminishes training effectiveness. More ablation studies provided in appendix D.3
can observe
4.4 Visualization
Figure visualizes the t-SNE embeddings generated by GCN and GRIN, trained with and without augmentation, evaluated on test sets augmented up to 20 RUs. As discussed Section for the same polymer, GCN's embeddings remain sensitive to repeat size, even with augmentation. In contrast, as shown in Figure (b), GRIN produces stable embeddings across all test lengths, when combined with 3-RU augmentation, it collapses all repeats of the same polymer into a single cluster. This behavior aligns with the cosine similarity 0.999 between repeat sizes and reported in Table demonstrating repetition invariance.

---

Table 5: Merge-Ratio (IRU vs. 3RUs) Ablation of GRIN: Weakening binary-update supervision (comes from
supernodes with hyperdegree 2) harms performance. The test include 1 RU and 60 RUs.
MeltingTemp
Ratio
Testl
Test60
R² f
RMSE J
R²
RMSE
1:1
0.745=0.004
57.021=0.417
0.746=0.002
56.934=0, 199
GCN
1:0.9
0.736+0.00 005
58.088#0:499
0.734=0.00)
58,273=0. 0.9
1:0.8
0.729
0996
0.729=0.004
58.873=0.387
1:1
0.740=0.00)
57.809=0.038
0.739=0.002
57.716=0.257
GIN
1:0.9
0.733#0.000
58 08.443=0.950
0.733=0.008
58.353+0.880
1:0.8
0.730
58.759:0.43)
0.729=0.009
08.837=0.949
5
Conclusion
In this work, we addressed limitations of prior polymer learning methods that modeled polymers single monomers and thus failed to capture their realistic, repetitive structures. We made the first attempt to achieve repetition-invariant representation learning. To this end, we proposed GRIN, which combined MST-aligned aggregation with repeat-unit-base data augmentation strategy. We provided a theoretical analysis of the minimal augmentation size required to achieve invariance. Experiments both polymer and copolymer datasets showed that GRIN learned embeddings that reliably generalized to polymers with long repeat lengths.
Acknowledgement
This work was partially supported by NSF USS-2142827, IIS-2146761, IIS-2234058, and CBET- 2332270. We also appreciate the support from the Foundation Models and Applications Lab of Lucy Institute and ND-IBM Tech Ethics Lab.
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12

---

A Dataset Details
A.1 Labeled Data
We evaluate our models on four homopolymer property datasets and two copolymer datasets, each split into 60% training, 10% validation, and 30% test sets. To prevent data leakage, we apply repeat-unit augmentation independently within each split: for every dataset, we generate augmented graphs with repeat sizes from 60. On the training set, we benchmark four augmentation schemes- (1,2). [1,3), {1,2,3,, and [hill while the test set is evaluated on graphs spanning all repeat sizes from through 60.
Homopolymer GlassTemp, MeltingTemp, PolyDensity, and OzPerm. Targets are predicting glass transition temperature (GlassTTemp. ²C). polymer density (PolyDensity, g/em*), melting temperature DeeltingTempp 'C) and oxygen permeability (OyPerm, Barrer), respectively. The first three datasets are extracted from Polymer Info 23]. OyPerm compiled from the Membrane Society of Australasia
portal following [29].
The distributions displayed in Figure (a)-(d) indicate that GlassTemp (~7000 samples) and Melt- ing Temp (~3600 samples) exhibit only moderate skew, which corresponds to stable performance across all models. PolyDensity (~1100 samples) is nearly Gaussian but limited its smaller size, causing some baselines to underperform. OyPerm (~6000 samples) is both heavily right-skewed and long-tailed on a very small dataset, leading most methods fail on the segh-permeabilityy cases.
Copolymer EA (Electron Affinity, eV) and IP (Ionization Potential, eV). These two datasets conducted from the same SMILES strings with different properties. Data are obtained and processed as Aldeghi and Coley 22. Figure (e) and (f) show that both properties are well covered and exhibit only moderate skew.
A.2 Unlabeled Data
We leverage 12,764 unlabeled polymers to further evaluate model generalization by generating graph representations, summarized in Table II
13

---

(a) GlassTemmp
(b) Melting Temp
335
375
PolyDensity
O_Perm
(c) Electron Affinity
( lonization Potential
Figure 4: Property distributions of homopolymer datasets GlassTemp. (b) Melting Temp. PolyDensity, (d) OzPerm copolymer datasets (e) Electron Affinity, (f) lonization Potential. The x-axis denotes property value and the y- y-axis denotes frequency. For clarity, the O.Perm histogram with long tail is truncated to the 0-50 value range.
14

---

B Implementation Details
All experiments were conducted on a 16-core Intel Xeon Gold 6130 CPU (2.1 GHz) with 96 GB RAM and single NVIDIA A6000 GPU (48 GB). Code is implemented in PYTHON 3.11 and PYTORCH 2.1.0+cul18 18, with graph processing using PYG 2.5.1.
Model configuration. Each model is evaluated with two backbone encoders, GIN and GCN. Since both GRIN and BFGNN[2] are based on algorithmic alignment, they share the same hyperparameter configuration for each task.
Layer Number E {2,3);
Learning Rate {1e-2, 1e-3):
Batch = 32,
Hidden Dimension = 300,
1 Regularization Weight = le-3.
Notice that, do add the & sparsity penalty until after 50 epochs during the training process. This is to allow the network to freely explore and capture the most salient patterns in the data without prematurely shrinking important weights zero. This "warm-up" period prevents the model from over-compressing its parameters too early, ensuring it first learns robust representations before enforeing sparsity.
For GREA, which was originally evaluated on polymer datasets, we adopt the authors' default settings. For all other basclines, hyperparameters are automatically optimized using the OPTUNA MI library.
For baselines, we use the official code package from the authors (DIR, GREA RPGNN DISGEN). For IRM we implement its graph version based on its official repository. Since source codes of BFGNN is not publically available, we implement with PYG package. We choose mean as the pooling function for baseline methods for its intrinsic generalization, which performs better larger repeat size compared to Sum. For each task we report meantsd over three random initializations. Each model is trained for up 400 epochs with early stopping (patience=10M) 100), and efficiency was assessed by the epoch at which each model achieved its best performance.
Repeat-unit Augmentation In this paper, GRIN refers unless an alternative training scheme is specified in Figure 3 By default, the augmentation training used for GRIN contains equal mix of 1-RU and 3-RU samples (1:1 merge ratio), except where otherwise noted in Table
method with 3-RU augmentation,
15

---

C Computational Efficiency
We measured training-cost breakdown on the MeltingTemp task using a single NVIDIA A6000 GPU. All models were trained under identical hyperparameter settings and employed the same GCN backbone for fair comparison. Compared other baselines, GRIN scales efficiently as the training set doubles. GRIN maintains moderate GPU memory usage (557 MiB) and training time (approximately 764 s). ranks near the middle the overall efficiency table, faster than multiple baselines such as SSR and only marginally heavier than its lightweight variant, GRIN-RepAug
Table 6: Efficiency comparison across models on MeltingTemp task.
Model
Peak GPU Memory (MiB)
Training Time (s)
GC
455
156.64
IRM
487
250.16
RPGCN 21
611
424.55
GREA
587
652.27
DIR
533
1419.47
SSR
531
1715.97
DISGEN
629
830.42
2FG
565
790.26
BRIN-RepAug
531
438.39
GRIN
557
763.58
16

---

D Result Details
D.1
More Results on Effectiveness (Q1)
More Baselines To evaluate the performance compared to attention- and transformer- models, which provide stronger long-range interaction modeling and global feature mixing capabilities, we conducted additional experiments using GAT [32] and GraphGPS [25] on the MeltingTent and OyPerm datasets. As shown in Table 7 GRIN outperformed both models.
Tables 8 to 10 presents R² and RMSE on Test5 and Test10 in addition to Testl and Test60. We can observe that:
Baseline: All baselines exhibit steady decline in performance as repeat size in- creases- fromses-from Testl through Test5 and Test10 to Test60 demonstrating limited size extrapolation.
Algorithm Alignment: Both BFGNN and GRIN- Repuu exhibit single performance gap between Testl and Test5, after which their results remain stable through Test and Test60. This stability demonstrates the size- pizz-generalization benefit conferred by algorithmic alignment. The higher accuracy of GRIN GRIN-RepAug compared to BFGNN highlights the additional advantage of alignment with MST.
Algorithm Alignment + Augmentation: GRIN-RepAug's one-time gap between Testl and other test sets highlights the crucial role of repeat-unit augmentation in bridging the gap between single RU training and multi-repeat generalization. With augmentation, GRIN maintain near-constant R² and RMSE across Test1, Test5, 10, and Test60.
These results reinforce the effectiveness of GRIN, which yields robust extrapolation across wide range of repeat sizes, while conventional methods struggle as chain length grows.
Table 7: Results on homopolymer datasets (Melting Temp, OgPerm) with repeating unit (Test and 60 repeating units (Test60): GRIN consistently achieves the highest R2 and smallest RMSE.
Temp
O.Pem
Model
Testl R²T RMSEL
Test60 R²T
Testl R²T
Test60 R²t
RMSEL
RMSEL
RMSEL
GAT 32
1680=000028
63,9+288
0.579+0.029
73.3+2.6
0.873+0.056
760.4+206.9
0.866=00080
774.1+243.0
GIN
GraphGPS 25
0.650=0.033
66.843.2
0.56340.013
74.7&1.1
0.81910.168
865. 365.24408.0
0.754g0.068
1074.14153.1
GRIN-RepAug
0.741=0.007
57.540.8
0.707+0.009
61.140,9
0.923+0.008
604.7a31.9
0.910=0.008
655.2428.6
GRIN
0.745=0.004
57.050.4
0.74650.002
56.920.2
0.929:0.002
583.4+7.5
0.929:0.002
581.747.2
Table 8: Results on homopolymer datasets (GlassTemp. MeltingTemp) with repeating units (Test5) and 10 repeating units 10): GRIN consistently achieves the highest R2 and smallest RMSE.
GlassTemp
MeltingTemp
Model
Test5 R2h
Testi0 R²T
Test5 R²t
Test10 R²1
RMSEJ
RMSEL
RMSEJ
RMSEJ
GCN 15
0.830=0.005
45,8+0.7
0.824=0.000 007
46.5+0.9
0.670=0.004
64 91.9=0.4
0.668+0.004
65.1
DIR 35
<00
> 100
<0
100
<0
> 100
> 100
GREA 18
100
100
<0
100
0
100
IRM 3
<0
> 100
<0
100
<0
> 100
0
> 100
GCN
RPGNN
21
100
SSR 7
<0
> 100
<0
100
<0
100
0
> 100
DISGEN
14
0.863+0.004
41.140.6
0.855+0.004
004
42.340.6
0.681
63.900 63.920.7
0.667+0.011
65.241.0
BFGCN
2
0.855=0.011
42.2+1.6
0.852+0.00 013
42.6+1.9
0.686=0.012
63.3+1.2
0.684=0.013
63.5+1.3
GRIN-RepAug
0.868=0.010
40.3+1.6
0.86740.013
40.5
0,712=0.008
60.6a0.9
0.710=00008
60.940.9
GRIN
0.892=0.001
36.5+0.2
0.892+0.001
36.5+0.2
0.746+0.002
56.9=0.2
0.746=0.002
56.9+0.2
0.85240.001
42.940.1
0.85140.000
42.9+0.1
0.673+0.013
64.6+1.3
0.672-0.014
64.7+1.4
DIR 35
<0
> 100
<0
100
<0
100
<0
> 100
GREA
<00
> 100
<0
<0
100
<00
> 100
IRM 3
100
0
100
100
100
GIN
RPGNN M1
<00
> 100
0
> 100
<00
> 100
<00
> 100
100
>
DISGEN
14
0.867=0.007
40.4=1.1
0.860=0. 009
41.6144
0.677=0.018
64.2=1.8
0.662-0.000
2.020
65.7+2.0
BEGIN 22
0.853+0.000
42.6+0.9
0.847440.00 008
43.3+1.1
0.682+0.018
63.7+1.8
0. 0.680+0.00 01.0
64.0&1.7
GRIN-RepAug
0.894=0.003
36.240.5
0.876+0.00 009
39.0+1.5
0.736=0.006
57.9+0.6
0.705=0.001
61.440.1
GRIN
0.896201001
35.740.1
0.895440.001
36.020.11
0.740e0.001
57.840.1
0.739+0.002
$7.740.3
17

---

Table 9: Results on homopolyme datasets (PolyDensity, O2Perm) with 5 repeating units (Test5) and 10 repeating units (Test10): GRIN consistently achieves the highest R2 and smallest RMSE.
PolyDt
O,Porm
Model
Test5 R-
Test10
Test5 R²1
Test10
RMSEL
R-1
RMSEL
RMSE,
R-1
RMSEJ
GCN 15
0.670=0.026
0.12540.001
0.66520.030
0.12820.006
0.876200.06
761.4217.8
0.87620.006
761.5117.8
DIR BS
<00
>
<0
>1
<0
> 3000
<0
> 3000
GREA 18
00
>
<0
<0
3000
<00
3000
IRM
0
>
<0
3000
0
3000
RPGNN
H1
<0
>
<0
1
<00
3000
<00
3000
SSR
3000
3000
DISGEN
[E]
0.198 105
0.19840.013
0.19440.107
0.198:0.014
3000
0
3000
BFGCN
22
0.63820.054
0.13340.010
0.63540.058
0.134:0.011
0,901
685. 14-189
0.90120.014
685 189.1111911
GRIN-RepAug
0.720=0:028
0.117+0.006
0.718=0.030
0.118=00006
0.920+0.007
621.5+27,9
0.915=000100
645,2428.2
GRIN
0.745210.012
0.112:4.003
0.74744.011
0.112:0.002
0.929100002
581887,55
0.92920.002
581.7772
GIN
0.64520.021
0.13240.004
0.632a0.028
0.86510.014 0.13420.005
801.2a:10.7
0.86520.014
801.32/00.7
DIR 3S
<0
>
<0
>1
<00
>> 3000
<0
3000
GREA 1S
<0
<0
3000
<00
3000
IRM
<0
I
3000
O
3000
RPGNN
N1
<0
<0
<00
3000
<.0
3000
SSR
3000
DISGEN
[4
0.154g0:055
0.204
0.13640.015
0.206=0.007
<
> 3000
0
3000
BEGIN
[2
0.68120.016 0.12540.003
0.67920.018
0.125200.044
0,913
643.6455.3
0.913200.009
643.6133.33
GRIN-RepAug
0.223=00011
0.117+0.002
0.721=0.912
0.112=00.02
0.916H0
631,++112
0,916+000033
631,2+1122
GRIN
0.7502000006
0.111:0.001
0.75120.006
0.11020.001
0.929100.006
580.9124.9
0.9292000006
580.9124.9
Table 10: Results copolymer datasets (EA, IP) with repeating units (Test5) and 10 repeating units (Test10); GRIN consistently achieves the highest R2 and smallest RMSE.
EA
IP
Model
Test5
Test10
Test5
Test1O
RMSEL
R-1
RMSFL
R-1
RMSF)
R²t
RMSEJ
GCY
15
0.90640.010
0.18240 009
0.895=0.017
0.19240016
0.909=0.0014
0.148+0.003
0.909+0.004
0.149+0.003
DIR 35
<00
>
<00
<00
GREA 18
<0
>
<00
1
0
>
<00
>
IRM
RPGNN P1
>
<00
<0
SSR 7
<0
>
<0
<
>1
<0
>
DISGEN
14
0.17840.082
0.53840.022
0.1 08.
0.537+0.027
0.196=0.056
0.44240.015
0.182=0.050
0,445#0.015
BEGON
0.883+0.019
0:203+0.016
0,889+0.018 018
0:20340:016
0.914:0.009
0,145+0.007
0:913:0:010
0.145+0.000
GeIN-RepAug
0.942+0.006 006
D. 0.14
0.942=0.006
0.142+0.008
0.937=0.010
0.124+0.010
0.936=0.010
0.124+0.010
GRIN
0.95240.002
0.129:0.002
0.962:0.002
0.129H0.002
0,949=0.003
0.111:0.003
0.949=0.003
0.111:0.004
GIN 37
0.90340.006
0.18550.006
0.903=0.007
0.18540.006
0.915e0.003
0.144+0.003
0.914=0.004
0.144
DIR
35
-00
000
0
+0
<0
IRM
<00
>
-00
1
<0
<0
RPGNN
P11
<0
<00
<0
DISGEN
44
0. 1.1040.114
0.56000.037
0.102+0.11 108
0.56240.034
0.196+0.006
0.442+0.015
0.182=0.060
0.445+0.016
BEGIN
1988+0.004
0,170::0 004
0.918:0.004
0.170:0.004
0.920:00000 002
0.139H0.000
0.99200.022
0,140:0.000 002
ReIN-R
0.94740.004
0.13740.005
0.947=0.004
0.13740.005
0.954
0.106+0.004
0.95240.003
0.107=0.00
GRIN
0.96020.001
0.119=00.02
0.9600=0.001
0.119:00002
0.962=0.001
0.096260001
0.962=00.01
0.096:20.001
D.2 More Results on Repetition-Invarian Representation (Q2)
To further evaluate model generalization beyond limited labeled data, we transferred model check- points trained on the GlassTemp prediction task 12.764 unlabeled polymers and quantified repeat- size invariance by computing the cosine similarity between embeddings at repeat sizes 1 and 60. As shown in Table TT GRIN practically learns iopetition-invarian representation, with a similarity of 0.999 and zero standard deviation.
Table 11: Results evaluating representation similarity between I repeating unit and 60 repeating units the large-scale unlabeled polymer dataset: GRIN achieves 0.999 similarity without standard deviation.
Method
Cosine similarity between repeat size and 60
GCN
15
0.672+0.119
BFGCN 22
0.903=0.006
GRIN
0.999=00000
18

---

D.3 More Results on Ablation Study
Different Layer-wise Aggregators To isolate the effect of the max- max-aggregator in message passing (see Section we use Gepug (i.e., without repeat-unit unit augmentation) to compare three layer- wise aggregation functions- sum, mean, and max-on GlassTemp and MeltingTemp with I repeating unit (Testl) and 60 repeating units (Test60). As shown in Table 12 the max-aggregator consistently delivers the best performance on all test sets. Sum outperforms mean on the Test for most cases, while it suffers much larger performance drop at Test60, underscoring that aax-aggregttion best preserves generaenealiioooonn in the MST-aligned architecture.
Table 12:
Results on GRIN- Lugueg with different layer-wiss aggregators: Max aggregation consistently achieves the highest R2 and smallest RMSE
GlassTemp (Testl)
GlassTemp (Test60)
MeltingTemp (Testl)
Melting Temm (Test60)
Encoder
Aggregator
R²1
RMSE.
R²t
RMSE.
R²t
RMSE
R²
RMSE
GCN
Max
0.890=0.001
36.8=0.1
0.86620.014
40.6+2.0
0.741=0.007
57,5=0,8
0.707=0.009
61.1#0,9
Mean
0.87520.005
39.320.7
0.83610.012
44.9=1.6
0.69410.0066
62.540.7
0.67410.002
64.520.2
Sum
0.884&0.002
37.7a0.2
0.81840.003
48.940.4
0.70740.009
61.240.9
0.551#00055
75.714.6
Max
0.894=0.003
36.2+0,5
0.876200,009
39.0=1.5
0.736+00006
$7.9=0.6
0.705=0.001
61.4+0.1
GIN
Mean
0.881=0.003
38.3=0.5
0.838+0.011
44.6+1.5
0.697=0.007
62.2+0.7
0.666+0.008
65.3+0.8
Sum
0.87610.004
39.0.0.6
0.81510.005
48.340.8
0,70740.007
61.240.8
0.60610.009
70.940.8
Repeat-unit Augmentation over Baselines To assess the effect of augmentation, we apply RepAug with repeating units to the GCN and GIN baselines and test on GlassTemp and MeltingTemp with 1 repeating unit (Testl) and 60 repeating units (Test60). The results summarized in Tables [3] and 14 show that RepAug yields consistent improvements: on Test both GCN+RepAug and GeN+RepAug achieve higher R² and lower RMSE compared to their nog-angeetted counterparts. The improvements are more pronounced on Test demonstrating that repeat-unit augmentation significantly enhances size-extrapolation performance not only on our GRIN architecture. Figure (a) visualizes the polymer latent representations from GCN and GCN+RepAug with t-SNE.
Table 13: Results on GCN baseline with 3-RU augmentation: RepAug improves performance on both repeating unit (Testl) and 10 repeating units (Test10) for Glass Temp and MeltingTemp.
Testl
Test10
Task
Model
R²1
RMSE 4
R²
RMSE +
GCN
0.878#0.00 000
38.7+0.1
0.824+0.007
46.5+0.9
GlassTemp
GCN+RepAug
0.884+0.004
37.8+0.6
0.87220.004
39.8+0.5
GCN
0.693+0.002
62.640.2
0.668+0.004
65. 1+0.4
MeltingTemp
GCN+RepAug
0.69420.003
62.5+0.3
0.683+0.002
63.6+0.2
Table 14: Results on GIN baseline with 3-RU augmentation: RepAug improves performance on both 1 repeating unit (Testl) and 10 repeating units (Test10) for GlassTemmp and Melting'Temp.
Test1
Test10
Task
Model
R² 1
RMSE J
R²
RMSE J
G3
0.882+0.003
38.1+0.5
0.851#0.00)
42.9+0.1
GlassTemp
GIN+RepAug
0.887+0.004
37.2+0.6
0.882+0.003
38.020.5
37
0.697+0.007
62.2+0.7
0.672+0.014
64.7=1.4
MeltingTemp
ReN+RepAug
0.701
61.8+0.9
0.69440.006
62.5+0.7
19

---

E
Future Work
The scale of labeled datasets used in our experiments remains limited, as collecting large, high-quality polymer datasets is inherently challenging. For example, researchers have spent nearly 70 years compiling only about 600 polymers with experimentally measured oxygen permeabil- ity in the Polymer Gas Separation Membrane Database [29]. We only partially address the problem by evaluating model generalization on a larger unlabeled polymer dataset by computing representation similarity, described in appendix D Another limitation is that incorporating augmented training samples in GRIN leads to modest increase training time and GPU memory consumption, as reported in appendix C
Limitations
While GRIN already establishes strong foundation for learning repetition-inve invariant polymer repre- sentation, several promising directions could further reinforce its capabilities:
Advanced Algorithm Alignment. Building on our MST-aligned method, future work could explore alignment with more sophisticated dynamic- dynamic-programming algorithms, which may enable GRIN to learn more comprehensive intra-monomer connections especially for branched or cross-linked polymer architectures.
Broader Material Domains. Our current copolymer dataset comprises two primary archi- tectures: block copolymers, with large contiguous runs of the same monomer, and alternating copolymers, featuring -A-B- repetition at first, each generating distinct structural motifs that affect material properties. Future work could incorporate random copolymers, whose monomer order lacks a fixed pattern, and apply the GRIN framework to each architecture, uncovering architecture-specine invariance and informing tailored representation strategies.
20


Reviews:
[{"review_id": "gVbYEqpSnF", "reviewer": "Reviewer_y6nT", "summary": "The authors introduce a method to learn polymer representations that are invariant to the number of repeating units in their graph representations. The proposed method outperforms state-of-the-art baselines on both homopolymer and copolymer benchmarks.", "questions": "see weaknesses", "limitations": "NA", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "##Strengths\n- The problem is well-motivated and is important problem.\n- The paper is clear and the method is described well.\n\n## Weaknesses\n- Although authors compare their method to variants of GIN and GCN, but I would like to see how this behaves with Graph Transformers.\n- Experiments can be extended to larger datasets.", "was_revised": true, "final_justification": "Thanks for addressing the questions, I increased my score to borderline accept.", "rebuttal": "## W1: Graph transformers\n>Although authors compare their method to variants of GIN and GCN, but I would like to see how this behaves with Graph Transformers.\n\n### GRIN outperformed the graph transformer-based GraphGPS\n\nThanks for your comments. We conducted new experiments using GAT [1] and GraphGPS [2] on MeltingTemp and O$_2$Perm. The table below showed that GRIN outperformed graph transformer models such as GraphGPS [2].\n\n#### MeltingTemp\n| Model            | Test1 R²↑   | Test1 RMSE↓ | Test60 R²↑  | Test60 RMSE↓ |\n| ---------------- | ----------- | ----------- | ----------- | ------------ |\n| GAT              | 0.680±0.028 | 63.9±2.8    | 0.579±0.029 | 73.3±2.6     |\n| GraphGPS         | 0.650±0.033 | 66.8±3.2    | 0.563±0.013 | 74.7±1.1     |\n| GRIN-RepAug(GCN) | 0.741±0.007 | 57.5±0.8    | 0.707±0.009 | 61.1±0.9     |\n| GRIN(GCN)        | **0.745±0.004** | **57.0±0.4**    | **0.746±0.002** | **56.9±0.2**     |\n\n#### O$_2$\n\n| Model            | Test1 R²↑   | Test1 RMSE↓ | Test60 R²↑  | Test60 RMSE↓ |\n| ---------------- | ----------- | ----------- | ----------- | ------------ |\n| GAT              | 0.873±0.066 | 760.4±206.9 | 0.866±0.080 | 774.1±243.0  |\n| GraphGPS         | 0.819±0.168 | 865.2±408.0 | 0.754±0.068 | 1074.1±153.1 |\n| GRIN-RepAug(GCN) | 0.923±0.008 | 604.7±31.9  | 0.910±0.008 | 655.2±28.6   |\n| GRIN(GCN)        | **0.929±0.002** | **583.4±7.5**   | **0.929±0.002** | **581.7±7.2**    |\n\n## W2: Larger datasets\n> Experiments can be extended to larger datasets.\n\n### GRIN generalizes better on larger unlabeled datasets for repeat-size invariance.\n\nThank you for the suggestions. Collecting large-scale labeled polymer datasets is challenging. For example, researchers have spent nearly 70 years compiling only about 600 polymers with experimentally measured oxygen permeability in the Polymer Gas Separation Membrane Database [3].\n\nTo address your concern, we used a larger unlabeled polymer dataset from [4] to evaluate the effectiveness of our method. Specifically, we applied model checkpoints from the GlassTemp task to 12,764 unlabeled polymers and measured repeat-size invariance using cosine similarity between embeddings at repeat sizes 1 and 60. The results below indicate that GRIN generalizes better to large repeat sizes than other GNNs on this large-scale dataset.\n\n| Method       | Cosine similarity between repeat size 1 and repeat size 60 |\n| ----------- | --------------- |\n| GCN         | 0.672± 0.119    |\n| BFGNN       | 0.903± 0.060    |\n| GRIN        | **0.999±0.000** |\n\n\n## References\n\n[1] Graph Attention Networks, ICLR 2018.\n\n[2] Recipe for a General, Powerful, Scalable Graph Transformer, NeurIPS 2022.\n\n[3] A Thornton, L Robeson, B Freeman, and D Uhlmann. 2012. Polymer Gas Separation Membrane Database.\n\n[4] Otsuka S, Kuwajima I, Hosoya J, Xu Y, Yamazaki M. PoLyInfo: Polymer database for polymeric materials design. International Conference on Emerging Intelligent Data and Web Technologies, 2011."}, {"review_id": "WT9YCLTI4h", "reviewer": "Reviewer_oUBZ", "summary": "This paper proposes GRIN, a novel framework for learning repetition-invariant graph representations of polymers. By combining maximum spanning tree alignment with repeat-unit-based graph augmentation, GRIN enables the model to capture consistent structural patterns across different polymer lengths. Experimental results show that GRIN outperforms strong baselines and generalizes well to unseen long-chain polymers.", "questions": "1. The method is designed for strictly repetitive polymers, but many real-world polymers exhibit branching or block copolymer patterns that deviate from perfect repetition. Have the authors explored how GRIN behaves on polymers with irregular or partially repeating structures?\n\n2. While the paper claims that three RUs are sufficient, it would be helpful to evaluate whether this number is optimal or if performance significantly changes with more or fewer units.", "limitations": "1. Baseline models are only trained on single-RU graphs, while GRIN uses both single- and multi-RU graphs, potentially conflating input richness with model capability.\n\n2. The framework does not include an explicit mechanism (e.g., regularization or constraints) to guarantee invariance to repeat unit count, which may limit interpretability and control.\n\n3. Although GRIN performs well on 60-RU graphs, the paper does not provide runtime or memory benchmarks, which are critical for real-world deployment.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "strengths:\n1. The paper addresses a fundamental limitation in existing graph-based polymer modeling: the inability to handle varying repeat unit numbers. This is a practically relevant problem, as real-world polymers vary widely in chain length and structure.\n2. The experiment result looks good to me. It shows consistently strong performance on Test60, even though training only involved 1 or 3 RUs. This validates the method’s core claim of repetition-invariance and scalability.\n\nWeakness:\n1. All baseline models are trained only on 1-RU graphs, while GRIN is trained on both 1-RU and 3-RU graphs. This makes it unclear whether GRIN's performance gain stems from model design or richer input structures.\n2. The model does not enforce repetition-invariance via explicit constraints or loss terms. Instead, it relies on structural heuristics like max-pooling and augmented training data, which may not generalize to more complex repeat structures.\n3. Although GRIN generalizes well to long chains, the paper does not evaluate training or inference cost as the RU number increases. This may raise concerns about efficiency in industrial-scale deployment.", "was_revised": false, "final_justification": "", "rebuttal": "## W1 & L1: Baseline models on multiple RUs\n>All baseline models are trained only on 1-RU graphs, while GRIN is trained on both 1-RU and 3-RU graphs. This makes it unclear whether GRIN's performance gain stems from model design or richer input structures.\n\n### GRIN outperformed the GCN and GIN baselines with 1RU+3RU augmentation\n\nThank you for your question. As described in Appendix C.2 (Lines 425–432, Tables 10 and 11), repeat-unit augmentation (1RU+3RU) was already applied to both GCN and GIN. While this augmentation improves the performance of all models, GRIN consistently achieves the best performance. A subset of the results on the GlassTemp task is provided below for reference:\n\n| Model       | R² (repeat size 1) | R² (repeat size 10) |\n| ----------- | ------------------ | ------------------- |\n| GIN+RepAug  | 0.887 ± 0.004      | 0.882 ± 0.003       |\n| GCN+RepAug  | 0.884 ± 0.004      | 0.872 ± 0.004       |\n| GRIN (best) | 0.896 ± 0.001      | 0.896 ± 0.001       |\n\n### Ablation studies showned that both data augmentation and model design improved the performance\n\nAppendix C.2 shows the effectiveness of data augmentation.\n\nIn Tables 2–4, GRIN-RepAug represents the variant using only the new model design without data augmentation. This variant still outperformed other baselines. In summary, both the model design and richer input structures contribute to the performance improvement.\n\n## W2 & L1 & Q1: Generalization to more complex structures.\n> The model does not enforce repetition-invariance via explicit constraints or loss terms. Instead, it relies on structural heuristics like max-pooling and augmented training data, which may not generalize to more complex repeat structures.\n\n> The method is designed for strictly repetitive polymers, but many real-world polymers exhibit branching or block copolymer patterns that deviate from perfect repetition. Have the authors explored how GRIN behaves on polymers with irregular or partially repeating structures?\n### GRIN generalizes to complex structures such as copolymers and larger datasets\n\nThanks for your comments. We have evaluated GRIN on both homopolymers (Tables 2 and 3) and copolymers, including block and alternating structures (Table 4), demonstrating that the approach is not limited to single-unit structures and can handle more complex architectures. \n\nWe further evaluated GRIN on a larger dataset by testing model checkpoints from the GlassTemp task on approximately 13K unlabeled polymers from [1]. For each polymer, we computed the cosine similarity between embeddings at repeat sizes 1 and 60. As shown below, GRIN achieved a cosine similarity of 0.999 ± 0.000, demonstrating its ability to generalize to polymers with more repeat patterns. \n\n| Method       | Cosine similarity between repeat size 1 and repeat size 60 |\n| ----------- | --------------- |\n| GCN         | 0.672± 0.119    |\n| BFGNN       | 0.903± 0.060    |\n| GRIN        | **0.999±0.000** |\n\n## W3 & L3: Efficiency\n> Although GRIN generalizes well to long chains, the paper does not evaluate training or inference cost as the RU number increases. This may raise concerns about efficiency in industrial-scale deployment.\n\n### GRIN scales efficiently, outperforming several baselines in speed and memory.\n\nWe measured training‐cost breakdown using the MeltingTemp task on a single NVIDIA A6000 GPU. All models share the same hyperparameter configuration and GCN backbone for fairness.\n| Model              | Peak GPU Memory (MiB)  | Training Time (s) | \n|--------------------------------|-----------|----------|\n| GCN |455| 156.64|\n| IRM | 487 |250.16|\n| RPGCN |611 | 424.55 |\n| GREA| 587 |652.27 |\n| DIR | 533  | 1419.47 |\n| SSR| 531| 1715.97 |\n| DISGEN | 629 | 830.42| \n| BFGCN | 565 |790.26|\n| GRIN-RepAug| 531 |438.39 |\n| GRIN| 557 |763.58|\n\nEven as the training set doubles, GRIN maintains moderate memory usage (557 MiB) and training time (approximately 764 s), ranking near the middle of the efficiency table, faster than many baselines (e.g., SSR, DIR) and only marginally heavier than GRIN RepAug. We will update Section 4.3 to include the new results. \n\n## Q2: Evaluation on multiple RepAug schemes\n>While the paper claims that three RUs are sufficient, it would be helpful to evaluate whether this number is optimal or if performance significantly changes with more or fewer units.\n\n### GRIN with 1RU+3RU augmentation consistently performs best across tasks.\nThank you for the suggestion. This question was already discussed in Section 4.3 (Lines 234–239) and visualized in Figure 3 for the MeltingTemp (homopolymer) and IP (copolymer) tasks. We plotted the figure by varying the number of repeating units and observed a common trend: the {1, 3} scheme outperforms the others. We report the results on the Density task below for reference.\n\n| | Test1 R²↑  | Test1 RMSE↓  | Test60 R²↑ | Test60 RMSE↓  |\n|-----------|-----------|----------|-----------|-----------|\n| 1RU+2RU |0.725±0.003 | 0.117±0.001 | 0.738±0.007 | 0.113±0.002  |\n| 1RU+3RU | **0.730±0.017**| **0.115±0.004**| **0.747±0.009**| **0.111±0.002**|\n| 1RU+4RU| 0.727±0.004|0.116±0.001|0.745±0.005|0.111±0.003|\n| 1RU+5RU| 0.725±0.005|0.116±0.001|0.744±0.003|0.112±0.003|\n\n\n## References\n\n[1] Otsuka S, Kuwajima I, Hosoya J, Xu Y, Yamazaki M. PoLyInfo: Polymer database for polymeric materials design. International Conference on Emerging Intelligent Data and Web Technologies, 2011."}, {"review_id": "lTDYzGozO4", "reviewer": "Reviewer_vHcv", "summary": "The paper considers the problem of applying GNNs to polymers, which are composed of repeated structures. They propose a method that (1) encodes polymers as a graph, (2) uses max aggregation GNNs, (3) augments the training data with polymers of different numbers of repeating units, and (4) encourages sparsity with L_1 regularisation. They provide some theoretical results about the correspondence of their model to the MST algorithm, and about how many repeated units need to be added during data augmentation. They perform experiments showing model performance across 6 datasets against a number of baselines, and also perform an ablation study on the repeat size and proportion of augmented graphs.", "questions": "1. L51 \"The MST criterion further preserves the most informative backbone of the polymer graph.\" -> what does this mean? It is likewise unclear to me what \"dynamic programming(DP)-style updates enforce a fixed reasoning procedure\" means.\n2. L78 What do you mean by \"causal\" information?\n3. L120 - L121 \"With residual connection h, any node not chosen by max-aggregation simply skips the update and retains its previous embedding unchanged.\" -> What does this mean? And how is it reflected in the above equation?\n4. L149 -> I can see a correspondence between MST and max aggregation, but where here is the \"theoretical guarantee\" that was spoken of?\n5. L257 \"MST-aligned max aggregation\" -> is it not just max aggregation, which itself is \"MST-aligned\"? This phrasing is a bit confusing to me.\n\nMinor comments:\n1. Figure 2 is not explained well in the caption. Also, the figure doesn't show explicitly which polymers belong to the same class (i.e. are variants of one another with a different number of repeating units)\n2. L41 -> could be nice to quantify this, rather than just motivating visually\n3. L104 -> the graph encoding would benefit from a visualisation\n4. L127 \"sparsity constraint\" -> this is the first time the sparsity constraint is mentioned. It is only defined later\n5. L132 -> typo", "limitations": "Limitations of the work are not discussed at all. Future work is included in the appendix.\n\nThe authors need to seriously consider their paper and the NeurIPS requirements to discuss limitations, as stated in the checklist item 2.\nThe bullet points under the guidelines provide a thorough explanation of many things the authors could have included in their paper as limitations. As one simple example: the paper should have considered the limitation of using soft invariance vs guaranteed invariance.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. Strong model performance - outperformed the baselines in thorough experiments.\n2. Nice motivation for the solution: invariance to the number of repeating units is a very fitting invariance to encode for the problem being solved.\n3. Generally technically sound - terms were mostly well defined and explained.\n4. Writing was clear, concise, and readable.\n\nWeaknesses:\n1. The claims of the paper are often overstated and do not reflect the actual contribution. This is the primary reason I have assigned a score of 1 for Quality and my biggest reason to reject the paper. My current perspective is that large portions of the paper need to be re-written with the true contributions in mind. To expand on this:\n\n1.1. The paper constantly refers to their embeddings as \"repeat-invariant\". However, nothing about their model guarantees this, and the embeddings shown in Figure 2 show that GRIN is not actually invariant, just approximately so. E.g. on L252, the phrase \"demonstrating true repetition invariance\" is used. By contrast, GCNs are truly invariant to node permutation, and other GNNs are also invariant to operators such as translation and rotation. In the paper, there needs to be a much clearer distinction between true invariance and approximate invariance.\n\n1.2 The paper claims to \"align the model with the MST construction process\" - this is very vague terminology and the actual theory presented in section 3.3.1 does not correspond to the language used. Furthermore, the actual \"alignment\" is between any GNN using max aggregation and a sparsity constraint, so is not particular to the GRIN model. It is also not clear why the \"alignment\" with MST is actually useful to the problem of repetition invariance - this needs to be expanded on.\n\n1.3 The paper phrases their contribution as a new model (GRIN). However, this can be separated into (1) some common choices for GNN architecture (max aggregation and sparsity constraints), (2) a graph encoding scheme, and (3) data augmentation. (2) and (3) seem to be the main novelty of the paper to me, so the messaging should be centred around them, rather than pitching it as a new model.\n\n2. Some of the theory could be made clearer, and the impact of the results more clearly stated up-front. For example, section 3.3.2 was dense and hard to understand - it should be expanded, and the explanation given on L174 of why the theory is useful explained better up-front, so the motivation is clear. At other times, terms are not defined or cited (e.g. \"L-Lipschitz\" on L181).\n\n3. Some motivation is lacking, and could be better shown in the experiments. For example, on L99, it is stated that \"sum grows unbounded\" as motivation for using max aggregation. However, it is not clear why why is this a problem. In practice, the repeat size will not become infinitely large. This argument also applies to any graph as the size increases. This claim could be demonstrated empirically by using sum aggregation vs max in the GRIN framework and showing the benefits of max.\n\n4. Some of the experimental results are underwhelming. For example, the performance gain of GRIN over other models such as BFGIN seems quite marginal. This seems to undercut the thesis of the necessity of repetition invariance.", "was_revised": true, "final_justification": "I have increase my score to a borderline accept. The authors did well to address my concerns in rebuttal.\nHowever, the paper does need a lot of changes, particularly the ones that the the authors agreed to in my review and over the course of our conversation: the paper messaging around its contribution (and many other areas) are not clear in the version submitted to the conference.\n\nIf these changes are addressed, I think the paper should be accepted: the work is a nice application, has strong results, and uses some neat theoretical insights. However, given the number and extent of the required changes, I cannot increase my score beyond a borderline accept, and I somewhat lean in the direction of recommending that the authors substantially edit the paper and then resubmit.", "rebuttal": "## W1.1: Invariance\n>The paper constantly refers to their embeddings as \"repeat-invariant\". ...\n\n#### GRIN achieves a cosine similarity of 0.999 ± 0.000 between repeat sizes 1 and 60 on ~13K unlabeled polymer SMILES.\nThank you for pointing out the distinction between exact mathematical invariance and the practical invariance we observe. We appreciate this suggestion and have revised the phrasing to ensure greater rigor. Our quantitative analysis shows that the embedding similarity between repeat sizes 1 and 60 is very close to 1, indicating strong repeat invariance.\n\nSpecifically, we have updated the manuscript as follows:\n* **L252:** “…demonstrating true repetition invariance.” → to: “…demonstrating repetition invariance.”\n* We re-scanned the manuscript and found no other instances of absolute phrases such as “true repetition invariance.” To avoid ambiguity, we will add the following clarifying sentence in the Introduction:\n  *“Throughout this work, we use ‘repetition-invariant’ to denote practical invariance, defined as a latent representation similarity above 0.99 across repeat sizes.”*\n\nTo quantify repetition invariance, we evaluated the model checkpoints from the GlassTemp task on ~13K unlabeled polymer SMILES [1]. We computed the cosine similarity between embeddings at repeat sizes 1 and 60. As shown below, GRIN achieved a cosine similarity of 0.999 ± 0.000.\n| Method       | Cosine similarity between repeat size 1 and 60 |\n| ----------- | --------------- |\n| GCN         | 0.672± 0.119    |\n| BFGNN       | 0.903± 0.060    |\n| GRIN        | **0.999±0.000** |\n\n>and the embeddings shown in Figure 2 show that GRIN is not actually invariant, just approximately so.\n\nWe calculated the mean cosine similarity between embeddings at repeat sizes 1 and 60 on the GlassTemp test set. GRIN achieves 0.998 (± 0.000).  (Full table please refer to Minor 2)\n\n## W1.2: MST alignment\n>The paper claims to \"align the model with the MST construction process\" - very vague terminology \n\n>  \"dynamic programming(DP)-style updates enforce a fixed reasoning procedure\" means.\n\nWe follow the term \"neural algorithmic alignment\" [4], aiming to design neural architectures that align structurally with specific DP paradigms.\n* **Algorithmic paradigm:** MST algorithms build a tree by greedily selecting sparse, non-redundant edges.\n* **Neural architecture:** GRIN’s Max aggregation selects the strongest incoming message, while the sparsity penalty suppresses less informative edges. This encourages sparse decision paths and avoids uniform mixing of neighbor information, resulting in a tree-like information flow qualitatively aligned with MST-style connectivity.\n\nGRIN's update scheme mimics DP-style reasoning: a fixed local rule is applied layer-wise, resulting in deterministic update paths. It avoids arbitrary neighbor mixing, as seen in mean or soft attention aggregation.\n\nWe clarified the phrasing as:\n* **L49:** \"GRIN aligns message passing with the edge-greedy logic\" → to \"GRIN draws inspiration from the edge-greedy logic\"\n* **L53-54:** \"aligns it with the MST construction process.\" → to \"heuristically encourages alignment with ...\"\n\n>the actual \"alignment\" is not particular to the GRIN model. \n\nMost GNNs adopt sum or mean aggregation [2,3]. The combination of max aggregation with an L1 sparsity penalty is underexplored and, to our knowledge, has not been theoretically or empirically studied for polymer property prediction. This architectural choice, together with repeat-unit augmentation, distinguishes GRIN from prior work.\n\n>  why the \"alignment\" with MST is actually useful  \n\n>L51 \"The MST criterion further preserves the most informative backbone of the polymer graph.\"\n\nThe inductive bias of MST is particularly beneficial for repetition invariance. When a polymer graph is expanded with repeated units, many edges are redundant. MST-style selection prevents uniform mixing of the duplicates and concentrates information flow along maximal, consistent paths that capture the core structural features independent of graph size. \n\n## W1.3 GRIN phrases\nThank you for the suggestion. The main text reflects this point in Lines 47–49: “In this work, we propose Graph Repetition INvariance (GRIN), a method that combines algorithm-alignment with repeat-unit augmentation to address the challenge of repetition-invariant representation learning.”\n\n## W2: Clarity of theory\nThanks for your suggestion. We revised and moved **L174:** \"With repeat-augmented polymers $P_m$ ($m$ ≥ 2), the model can apply same update rule at each layer, yielding a constant output $y^⋆$, irrespective of the repeat size.\" to the beginning of Section 3.3.2 (after L153). \n\nWe revised the sentence on **L181:** \"...is $L$-Lipschitz with $L$<1...\" \n→ to \"... is assumed to be $L$-Lipschitz, i.e., it satisfies $|f(x)−f(x′)|≤ L|x−x′|$ for all inputs $x$, $x′$ and some constant $L$<1...\"\"\n\n## W3: Motivation for max aggregation\n\nIn Appendix C.2 (Lines 417–424, Table 9), we already evaluated different aggregation schemes within the GRIN-RepAug (GCN/GIN backbone). Both sum and mean aggregators lead to varying degrees of performance degradation. \n\n## W4: Experimental Results\n\n### GRIN consistently outperforms baselines across repeat sizes, with especially strong generalization on Test60 where many baselines fail.\n\nTesting on single repeating units (Test1), GRIN consistently outperforms BFGNN across homopolymer tasks such as MeltingTemp (\\~5%) and density (\\~6%), and more challenging copolymer tasks EA (\\~5%) and IP (\\~5%).\n\nOn Test60, GRIN’s advantages become more pronounced as it maintains strong performance across all tasks. While 5/8 baselines have negative $R^2$.\n\n## Q2: L78 \n\n“Causal information” refers to task-relevant structural features that remain consistent regardless of the repeat count. We acknowledge that “causal” may suggest a formal treatment (e.g., counterfactuals) which we do not pursue here. To avoid confusion, we revised **L78:** \"causal information” → to “key structural information”\n\n## Q3: L120 - L121 \n\nThank you for catching this. In our design, skip arises implicitly: max aggregation selects one dominant neighbor per dimension, and L1 sparsity suppresses the rest, making their contributions negligible—functionally approximating an identity mapping for non-winning paths.\nWe revised **L121:** \"With residual connection h, any node not chosen by max-aggregation simply skips the update...\" → to \"With previous state $h^{l-1}$, any non‑max neighbour contributes zero to $U^l$ leaves corresponding feature dimensions remain unchanged...\"\n\n## Q4: L149\nThe guarantees are stated in Section 3.3:\n* Proposition 3.2: Training on {$P_1$, $P_n$} with L1 sparsity yields predictions invariant to chain length ($m ≥ 2$) under the hyperchain abstraction, as all node types are exposed and non-dominant paths are suppressed.\n* Proposition 3.3: $n = 3$ is the minimal augmentation introducing a degree‑2 node and enabling true two-branch competition; larger $n$ yields diminishing returns under the $L$‑Lipschitz contraction, consistent with observed performance saturation.\n\n## Q5: L257\nWe intended to emphasize that the max-aggregation was inspired by MST, but saying “MST-aligned max aggregation” can be redundant. We simplified this to avoid confusion:\n* **L257:** “... MST-aligned max aggregation ...” \n→ to “... MST-aligned aggregation ...”\n\n## Minor 1: Figure 2\n> Figure 2 is not explained well in the caption. \n\nThe caption is revised as: \"Figure 2: ... glass transition task. Points are colored by repeat count (light = 1RU, dark = 20RU). (a) GCN produces inconsistent embeddings for different repeat sizes of the same polymer, clustering by size (same color) rather than identity; our augmentation introduces light-to-dark stripes, indicating improved alignment of repeat variants. (b) ...\"\n> Also, the figure doesn't show explicitly which polymers belong to the same class...\n\nThough Figure 2 does not label polymer identities, it clearly visualizes improved repeat-size invariance across models—supported by Minor 2’s scores:\n- **GCN**: Embeddings cluster by repeat count—e.g., all dark points (20RU) group together, showing low invariance (sim ~ 0.73).\n- **GCN + RepAug**:  Light-to-dark stripes emerge, showing variants of same polymers begin to align, shifting clustering from repeat size to identity (sim ~ 0.85).\n- **GRIN w/wo RepAug**: Each polymer’s embeddings form tight, size-independent clusters, indicating strong invariance (sim > 0.98).\n\n## Minor 2: L41\nThanks for your comment. We computed embedding similarity between repeat sizes 1 and 60 on GlassTemp and found that RepAug improves GCN stability, while GRIN variants achieves consistency.\n| Model       | Cosine similarity between repeat size 1 and 60 |\n| ----------- | ------------------------------------ |\n| GCN         | 0.731±0.154  |\n| GCN+RepAug  | 0.849±0.014  |\n| GRIN-RepAug | 0.980±0.000  |\n| GRIN        | **0.998±0.000**|\n\n## Minor 3: L104 \nThanks for your comment. We added a visualization for this part. According to the conference's protocol, we are not allowed to attach figure or link here.\n\n## Minor 4: L127\nThanks for your comment, we defined it at first mention: **Revised (L124–L131):** “Max aggregation imposes a selection bias toward the strongest neighbor, we add an L1 sparsity penalty on the message and update networks to suppress non‑dominant pathways (Eq. 3), encouraging a sparse, MST‑like backbone.”\n\n## Minor 5: L132\nWe corrected “constrcuted” → “constructed” in the description.\n\n## References\n\n[1] PoLyInfo: Polymer database for polymeric materials design. International Conference on Emerging Intelligent Data and Web Technologies,  2011.\n\n[2] How Powerful Are Graph Neural Networks? (ICLR 2019)\n\n[3] Physical Pooling Functions in Graph Neural Networks for Molecular Property Prediction, 2022. \n\n[4] Graph neural networks extrapolate out-of-distribution for shortest paths, 2025."}, {"review_id": "j4Hgb8QFZF", "reviewer": "Reviewer_W2de", "summary": "This paper addresses the challenge that conventional graph neural networks, which model polymers using only a single repeating unit, produce embeddings that drift as the number of monomer repeats varies. To overcome this, the authors propose Graph Repetition Invariance, which augments polymer graphs with multiple repeat units and aligns GNN message passing to a maximum spanning tree via max‐aggregation and a sparsity penalty—ensuring that only the most informative edges drive representation updates. Three‐unit augmentations suffice for learning truly repeat‐invariant embeddings and demonstrate, across both homopolymer and copolymer benchmarks, that GRIN yields stable, size‐agnostic feature vectors that outperform existing baselines in property prediction tasks", "questions": "I don’t quite understand the example from lines 27–33. Could you provide a concrete illustration? Moreover, do large language models exhibit this problem? If not, perhaps we can handle such invariance naturally via augmentations present in real data.\n\nWhat is the meaning of Figure 2? How should we interpret the right-hand panel?\n\nI’d like to confirm the overall pipeline: we start with a set of polymers, extract their monomer units, generate augmented data from those monomers, append it to the training set, and then train a GRIN model. Is that correct?\n\nBeyond these geometric deep learning baselines, what about kernel-based methods? I believe repeating patterns could be addressed using domain knowledge—perhaps via motif-inspired embeddings. Could we design (deep) graph kernels to tackle this problem?\n\nHow would models like Residual Gated Graph ConvNets, which have an inherent sequential design, perform on this task?\n\nThe improvement of the GRIN variant over the baseline—and of training on G₁+G₃ versus G₁ alone—is not very pronounced given the limited training data. Could you explain this? Does it imply that encoding such inductive biases as hard constraints is unnecessary when a large amount of training data is available?", "limitations": "N/A", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper is generally well written.\n2. The experimental section is comprehensive\n3. It poses an interesting, timely question in the graph-learning domain that could inspire future research. Despite minor issues, the work has the potential to benefit the broader community.\n\nWeakness:\n1. I think there are some points not clearly stated. I will check with them in the question part.\n2. The theoretical part is more akin to intuition than rigorous evidence. \n3. The scale of datasets seems limited", "was_revised": true, "final_justification": "I support the acceptance", "rebuttal": "## W1 & Q1: Example from lines 27-33\n> I don’t quite understand the example from lines 27–33. Could you provide a concrete illustration? \n\nFor L27-33, we draw an analogy between repeated phrases in language and repeated monomer units in polymers. For instance, in sentiment analysis, phrases like “very fair”, “very very very fair”, \"very fair very fair\" typically express the same sentiment. The sentiment model should recognize that the core meaning remains unchanged despite repeated modifiers.\n\nSimilarly, a polymer with one repeating unit (1RU) and many RUs should be assigned highly similar feature representations since their chemical identity remains unchanged. \n\n> Moreover, do large language models exhibit this problem? If not, perhaps we can handle such invariance naturally via augmentations present in real data.\n\n### Real-world augmentations benefit language models but transfer poorly to graph domains.\n* **In NLP, repetition is ubiquitous and cross-domain transferable.**\nNatural language corpora consistently exhibit repeated structures across domains (e.g., “very very good”) such as news, literature, and scientific writing [3]. Allowing LLMs to learn repeat-invariant patterns through large-scale pretraining without task-specific augmentation.\n* **In graph domains, repetition patterns are highly domain-specific.**\nAs explicitly noted by Chen et al., 2022 [4], repetition in polymer graphs (see Figure 1) is highly domain-specific, distinguishing them from small molecules and social graphs.\n* **Polymer data scarcity**\nAs we discussed in W3, labeled polymer data is limited due to the high cost and complexity of obtaining reliable property labels.\n\n## Q2: Figure 2\n>What is the meaning of Figure 2? \n\nFigure 2 visualizes how repeat invariance improves progressively—from GCN’s repeat-size-based clustering, to GRIN’s repeat-invariant latent space. \n\nIt shows a t‑SNE projection of polymer embeddings on the GlassTemp task for four models. We visualize 50 polymers, each represented by 20 embeddings (repeat sizes 1–20), colored ight (1RU) to dark (20RU).\n\n* **GCN (leftmost):** Embeddings group by repeat size (e.g., all 20RU points, darker one, cluster together), not polymer identity, indicating poor invariance (sim ~ 0.73).\n* **GCN+RepAug:** Light-to-dark stripes emerge—indicating that repeat variants of the same polymer start grouping together, as RepAug promotes consistency across repeat sizes (sim ~ 0.85).\n* **GRIN-RepAug and GRIN (right panels):** Each polymer’s 20 repeat variants collapse into a tight, size-invariant cluster (sim > 0.98).\n\nThis trend is also quantified:\n\n| Model       | Cosine similarity between repeat size 1 and 60 |\n| ----------- | ------------------------------------ |\n| GCN         | 0.731±0.154  |\n| GCN+RepAug  | 0.849±0.014  |\n| GRIN-RepAug | 0.980±0.000  |\n| GRIN        | **0.998±0.000**|\n\nTo make it clearer, we revised the figure caption as:\n\n- \"Figure 2: t‑SNE visualization of polymer embeddings from GCN and GRIN (ours) across repeat sizes (1–20 RUs) on the glass transition task.\nPoints are colored by repeat count (light = 1RU, dark = 20RU). (a) GCN produces inconsistent embeddings for different repeat sizes of the same polymer, clustering by size (same color) rather than identity; our augmentation introduces light-to-dark stripes, indicating improved alignment of repeat variants. (b) GRIN learns repeat‑invariant representations: each polymer’s variants form tight, size‑independent clusters, further improved with augmentation.\"\n\n>How should we interpret the right-hand panel?\n\nThe last two panels show GRIN-RepAug and GRIN, where each polymer’s 20 variants (across repeat sizes) form tight, color-mixed clusters. This indicates that the model outputs nearly identical embeddings for different repeat sizes.\n\n## Q3: Overall pipeline\n> I’d like to confirm the overall pipeline:...\n\nThe reviewer’s understanding of the pipeline is generally correct for model training. For inference, the trained model can take polymers with arbitrary repeat lengths and produce invariant representations. \n\n## Q4: Kernel-based methods\n>Beyond these geometric deep learning baselines, what about kernel-based methods? I believe repeating patterns could be addressed using domain knowledge—perhaps via motif-inspired embeddings. \n\nWe agree that domain-specific methods like graph kernels or motif-counting approaches are promising directions for polymer representation, especially since repeating patterns are core to polymer structure.\n> Could we design (deep) graph kernels to tackle this problem?\n\nHowever, designing a polymer-aware kernel or motif library is non-trivial. The key structural features often span both intra- and inter-monomer interactions, are sensitive to anchor positions within the chain, and vary with copolymer patterns (e.g., block, alternating, or random). Additionally, to support size invariance, such motifs must be periodic and invariant to repeat count—a challenge for traditional motif extraction.\n\nIn this work, we focused on GNN-based methods to learn such patterns automatically, without requiring an explicit library of motifs. \n\n## Q5: Residual Gated Graph ConvNets\n>How would models like Residual Gated Graph ConvNets, which have an inherent sequential design, perform on this task?\n\n### GRIN outperformed ResGatedGCN\nWe conducted new experiments on ResGatedGCN [5] and a pure sequence baseline, the SMILES-based LSTM [6]. GRIN outperformed ResGatedGCN [5] as follows:\n\n#### MeltingTemp\n\n| Model            | Test1 R²↑   | Test1 RMSE↓ | Test60 R²↑  | Test60 RMSE↓ |\n| ---------------- | ----------- | ----------- | ----------- | ------------ |\n| ResGatedGCN      | 0.710±0.003 | 60.8±0.3    | 0.454±0.048 | 83.5±3.7     |\n| LSTM             | 0.610±0.032 | 70.6±2.8    | <0          | >3000        |\n| GRIN-RepAug(GCN) | 0.741±0.007 | 57.5±0.8    | 0.707±0.009 | 61.1±0.9     |\n| GRIN(GCN)        | 0.745±0.004 | 57.0±0.4    | 0.746±0.002 | 56.9±0.2     |\n\n## Q6: The improvement of GRIN\n\n>The improvement of the GRIN variant over the baseline—and of training on G₁+G₃ versus G₁ alone—is not very pronounced given the limited training data. Could you explain this? \n\nAcross all tasks, GRIN consistently ranks among the top and remains stable, while baseline models often fluctuate—some peak on specific datasets and degrade significantly at larger repeat sizes.\n* Testing on a single repeating unit (Test1), GRIN consistently outperforms BFGNN across homopolymer tasks such as MeltingTemp (\\~5%) and density (\\~6–7%), as well as more challenging copolymer tasks such as EA (\\~5%) and IP (\\~5%).\n* Testing on 60 repeating units (Test60), GRIN’s advantages become more pronounced as it maintains strong performance across all tasks. Five out of eight baselines have negative $R^2$, contrasting GRIN’s generalization to different repeating sizes.\n\nGRIN-RepAug is an ablated variant of GRIN without repetition augmentation. Its strong performance demonstrates the effectiveness of the new neural architecture under limited training data, but it does not imply that the inductive bias becomes unnecessary with larger datasets. From Tables 3, 4, 10 and 11, we found the new architecture remains important even when data augmentation is applied.\n\n>Does it imply that encoding such inductive biases as hard constraints is unnecessary when a large amount of training data is available?\n\nAs discussed in W3 and Q1, polymer datasets are limited and costly to label. GRIN addresses this by embedding repetition invariance as an architectural prior, enabling generalization under realistic, small-data regimes.\n\n## W2: Theory part\n>The theoretical part is more akin to intuition than rigorous evidence.\n\nThe theoretical analysis in Section 3.3 provides an algorithmic insight aiming to explain how specific architectural choices—such as max aggregation and L1 sparsity—promote consistent reasoning across varying graph sizes.\n\nDiscussion at L189–L204 show that, under these design choices and augmentation schemes, the model produces stable outputs regardless of repeat length. This aligns with our goal: to show that GRIN encourages repeat-invariant behavior.\n\n## W3: Limited dataset scale\n\n### GRIN generalizes better on larger unlabeled datasets for repeat-size invariance.\n\nCollecting large-scale labeled polymer datasets is challenging. For example, researchers have spent nearly 70 years compiling only about 600 polymers with experimentally measured oxygen permeability in the Polymer Gas Separation Membrane Database [1].\n\nTo address your concern, we used a larger unlabeled polymer dataset from [2] to evaluate the effectiveness of our method. We applied model checkpoints from the GlassTemp task to ~13k unlabeled polymers and measured repeat-size invariance using cosine similarity between embeddings at repeat sizes 1 and 60. The results below indicate that GRIN generalizes better to large repeat sizes than other GNNs on this large-scale dataset.\n\n| Method       | Cosine similarity between repeat size 1 and 60 |\n| ----------- | --------------- |\n| GCN         | 0.672± 0.119    |\n| BFGNN       | 0.903± 0.060    |\n| GRIN        | **0.999±0.000** |\n\n## References\n\n[1] A Thornton, L Robeson, B Freeman, and D Uhlmann. 2012. Polymer Gas Separation Membrane Database.\n\n[2] Otsuka S, Kuwajima I, Hosoya J, Xu Y, Yamazaki M. PoLyInfo: Polymer database for polymeric materials design. International Conference on Emerging Intelligent Data and Web Technologies, 2011.\n\n[3] Don't Stop Pretraining: Adapt Language Models to Domains and Tasks. (ACL 2020)\n\n[4] Graph Representation Learning for Polymer Property Prediction. (NeurIPS 2022)\n\n[5] Residual Gated Graph ConvNets.\n[6] Chen, G., Tao, L., & Li, Y. (2021). Predicting Polymers' Glass Transition Temperature by a Chemical Language Processing Model. Polymers, 13(11), 1898."}]

Author rebuttal:

Meta-review: The paper proposes a framework for learning repetition-invariant representations for polymers, addressing the challenge that repeating unit structures cause redundancy and inconsistency in existing polymer informatics models. The method employs a repetition-invariant encoder that normalizes input representations and leverages contrastive learning to enforce consistency across repeat counts. Extensive experiments on multiple polymer property prediction tasks demonstrate that the approach outperforms strong baselines, improving both accuracy and robustness. 

Reviewers highlight several strengths.
- The paper tackles an important and underexplored problem in materials informatics.
- The proposed formulation of repetition invariance is novel and well motivated, and the methodology is clear and technically sound.
- The empirical results are convincing across diverse datasets. Ablation studies further support the contribution of the repetition-invariant encoder and contrastive objectives.

Some concerns were noted, including limited exploration of scalability to very large polymer systems, a need for more theoretical justification of the invariance principle, and a heavier reliance on curated datasets. However, these issues were viewed as incremental and did not undermine the central contribution. The rebuttal provided clarifications on dataset design, additional ablations, and insights into possible extensions, which reviewers found satisfactory.

We recommend acceptance, with the following suggestions for the camera-ready: (1) expanding the limitations section, providing a clearer discussion on computational efficiency and scalability, (2) outlining potential applications to real-world polymer design scenarios, and (3) including the experiments added during the rebuttal.
Decision: Accept (poster)