Geometric Mixture Models for Electrolyte Conductivity Prediction
Authors: Anyi Li
Jiacheng Cen
Songyou Li
Mingze Li
YANG YU
Wenbing Huang
Venue: NeurIPS 2025 poster ()
Keywords: Geometric Graph Neural Networks
Mixture Property Prediction
Conductivity Prediction
Primary area: machine_learning_for_sciences
TL;DR: A geometry-aware model for mixture property prediction.

Abstract: Accurate prediction of ionic conductivity in electrolyte systems is crucial for advancing numerous scientific and technological applications. While significant progress has been made, current research faces two fundamental challenges: (1) the lack of high-quality standardized benchmarks, and (2) inadequate modeling of geometric structure and intermolecular interactions in mixture systems. To address these limitations, we first reorganize and enhance the CALiSol and DiffMix electrolyte datasets by incorporating geometric graph representations of molecules. We then propose GeoMix, a novel geometry-aware framework that preserves Set-SE(3) equivariance—an essential but challenging property for mixture systems. At the heart of GeoMix lies the Geometric Interaction Network (GIN), an equivariant module specifically designed for intermolecular geometric message passing. Comprehensive experiments demonstrate that GeoMix consistently outperforms diverse baselines (including MLPs, GNNs, and geometric GNNs) across both datasets, validating the importance of cross-molecular geometric interactions and equivariant message passing for accurate property prediction. This work not only establishes new benchmarks for electrolyte research but also provides a general geometric learning framework that advances modeling of mixture systems in energy materials, pharmaceutical development, and beyond.

Geometric Mixture Models for Electrolyte Conductivity Prediction
Anyi Li3 .Jichheng Cen 23, Songyou Li! Mingze Lil23, Yang Yu' Wenbing Huang" 2
Gaoling School of Artificial Intelligence, Renmin University of China
2 Beijing Key Laboratory of Research on Large Models and Intelligent Governance Engineering Research Center of Next-Generation Intelligent Search and Recommendation, MOE
Hisun Pharm
(li_anyi, jiacc. en, rangyubard)@outloonn com; {songyou_l1, hwenbing) @126. com; 1imingzetony@gail.ccm
Abstract
Accurate prediction of ionic conductivity in electrolyte systems is crucial for ad- vancing numerous scientific and technological applications. While significant progress has been made, current research faces two fundamental challenges: (1) the lack of high-quality standardized benchmarks, and (2) inadequate modeling of geometric structure and intermolecular interactions in mixture systems. To address these limitations, we first reorganize and enhance the CALiSol and DiftMix elec- trolyte datasets by incorporating geometric graph representations of molecules. We then propose GeoMix, novel geometry-awe aware framework that preserves Set-SE(3) equivariance an essential but challenging property for mixture systems. At the heart of GeoMix lies the Geometric Interaction Network (GIN), an equivariant module specifically designed for intermolecular geometric message passing. Com- prehensive experiments demonstrate that GeoMix consistently outperforms diverse baselines (including MIPs, GNNs, and geometric GNNs) across both datasets, validating the importance of cross moos-moleecuarr geometric interactions and equivariant message passing for accurate property prediction. This work not only establishes new benchmarks for electrolyte research but also provides a general geometric learning framework that advances modeling of mixture systems in energy materials, pharmaceutical development, and beyond.
Introduction
Predicting the properties of multi-molecular mixture systems plays central role in advancing wide range of emerging technologies, such as next-generation energy storage, advanced materials design, and pharmaceutical formulation [1-4]. Electrolytes, prototypical example of these systems, typi- cally consist of lithium salts and organic solvents of diverse types. Accurate prediction of electrolyte conductivity key research focus, directly determines critical battery performance metrics including charge transfer efficiency, power density, and operational stability [5; 6]. However, model- requires simultaneously accounting
ing such complex systems presents significant challenges, for both the internal structure of individual molecules and the intricate interactions among different molecules. Classical theoretical approaches- -ranging from the Nernst equation dilute solution models-either neglect explicit intermolecular forces or rely on extrapolations from infinitely dilute regimes, resulting limited accuracy and poor universality for practical applications [7: 8).
Spurred by recent advancements in machine learning, particularly deep learning, researchers have increasingly adopted learning-based approaches model electrolyte systems [9-11], achieving
WWenbing Huang is the corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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significant progress over classical methods. As a data fata-driven paradigm, the development of predictive models for these systems hinges critically on high-quality and standardized benchmarks. However, the field still faces substantial challenges: most existing datasets are semi-structured, requiring labor-intensive manual curation from primary literature due to inconsistent formatting standards. This lack of accessible, well-curated benchmarks has severely hindered systematic comparisons and reproducibility across methodologies.
Another critical limitation in existing research stems from inadequate geometric characterization of mixture systems. Previous works such as Former [12] and MolSets [13] represent molecules in electrolyte systems using SMILES sequences or topological graphs. However, these approaches neglect the geometric structure of molecules in 3D space, which significantly impacts the properties of entire systems [14-17]. While recent methods such as DiffMix [11] and Uni-ELF [18] have incorporated 3D structural information, they aggregate molecular information into global features and simulate intermolecular interactions through comparing these aggregated representations, thereby overlooking fine-grained and full-atom interactions across molecules. A naive solution would involve direct atom-level coordinate transfer between molecules, but this violates the fundamental symmetries of mixture systems. Crucially, any physically meaningful representation must maintain consistency under: 1) independent permutation of each molecule's atoms; 2) independent SE(3) transformations (rotation/rranslation of each molecule's ' local coordinate system; and 3) permutation of molecules within the system. We formalize these requirements Ser-SE(3) equivariance (see Eqs. (2) to (4)), challenging but essential property for developing accurate models of molecular interactions.
To address the above two challenges, we first curate and standardize two public datasets for electrolyte conductivity prediction: CALiSol [19] and DiffMix [11]. Inspired by the powerful expressiveness of geometric information [20-44], we augment the molecular representation by constructing geometric graphs where atoms are characterized by both invariant features (e.g. atomic numbers) and equiv- ariant features (e.g., 3D coordinates), ensuring downstream models can fully exploit 3D structural information. We then propose Gco,ix a geometry-aware framework for mixture system modeling and property prediction. At its core is the Geometric Interaction Network (GIN), a novel equivariant module designed for intermolecular geometric message passing. The fundamental idea involves first creating local coordinate frame for each molecule, then computing learnable coordinate transfor- mations to align atom pairs from different local frames, and finally performing equivariant message passing over the transformed coordinates. Our code and dataset are available at Github?.
In summary, this paper contributes to the following aspects:
We curate and standardize two public datasets, CALiSol and DiffMix, covering a wide range of electrolyte mixture systems. The datasets are carefully processed with geometric graph construction, forming benchmark for electrolyte conductivity prediction.
We propose GeoMix, novel geometric framework for modeling mixture systems. GeoMix maintains Set-SE(3) equivariance while capturing fine-grained geometric relationships between molecules, overcoming the limitations of existing methods.
We compare GeoMix against diverse baseline models built upon MLPs, GNNs, and geomet- ric GNNs, under consistent settings. GeoMix consistently outperforms all baselines both datasets, demonstrating its effectiveness in modeling mixture systems.
2 Related Work
Early-Stage Methods Earliest approaches compute ionic conductivity by applying the Nernst-Einstein equation, which assumes that all ions diffuse independently and intermolecular interactions can be neglected. Under this framework, each ionic species contributes additively the total conductivity. While effective at low concentrations, this framework breaks down in concentrated or strongly interacting systems, where correlated ionic motion becomes significant [8]. With advances in computational chemistry, first-principles methods such as Density Functional Theory (DFT) [45] and Machine Learning Force Fields (MLFFs) [10; 46] became available. These techniques accurately capture interactions between ions and molecules over short timescales, enabling more precise es- timates of ion mobilities under realistic intermolecular forces [10; 47- Despite their accuracy, classical and ab initio MD simulations require modeling every atom in the local system, leading
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2

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to substantial computational expense. Moreover, because conductivity predictions rely on multiple intermediate steps, accumulated numerical errors can offset the benefits of high precision.
Deep Learning Methods Recent deep learning frameworks [11-13; 18; 50; 51] aim to model mixture systems directly from their components and proportions. These methods can be roughly grouped into three categories: (1) SMILES-based methods [12; 50; 51] compute mixture embeddings by weighted summation over molecular embeddings derived from molecular text representations such as SMILES, However, these approaches ignore the influence of molecular geometry on both the representation and interactions within the mixture. (2) Topology-basee -based methods such as MolSets [13] model intermolecular interactions using self-attention over molecular- graph embeddings. While molecular topology is captured, spatial interactions are only implicitly encoded through attention mechanisms. (3) Geometry-based methods further leverage 3D molecular information. DifIMix [11] aggregates molecular embeddings using learned polynomial mixing coefficients, while incorporating some physical priors. Uni-ELF [18] combines microscopic descriptors such as radial distribution functions (RDFs) with learned geometric features to extract invariant intermolecular representations. However, it does not capture equivariant features between molecules. While most studies about equiv- ariant models focus on molecular interactions within a single reference frame [52; 53], our proposed method explicitly models node-level geometric relationships across molecules through equivariant message passing. This approach enables fine- grained, geometry aooar aware interaction modeling at the atomic scale.
3 Method
In this section, we first formulate the regression task for mixture systems and discuss its associated symmetry constraints in 3.1. We then introduce GeoMix, a novel geometric learning framework designed for both scalar (invariant) and vector (equivariant) property prediction, in $ 3.2. Finally, we show a crucial component of GeoMix, GIN, in $ 3.3. All proofs are provided in Appendix A.
3.1 Preliminaries
Notations We employ three-level representations to describe mixture systems in a hicrarchical way. 1) System-level: A mixture system is set of M molecules, i.e., S [ JMM where the geometric graph 9m represents the m-th molecule in the system. The global environment vector e encodes physical conditions (e.g. temperature and pressure) and x denotes the target electrolyte conductivity.
2) Graph-level: Each geometric graph (H,, Xm, Vm, Wm) contains Nn nodes. The quantities
Hm E HNN, X,, E N... denote node scalar features and node 3D coordinates, respectively; Vn E 3R refers to node geometric features, which are usually initialized as zeros or just node coordinates X,, 0 < Wm < 1 is the proportion of molecule G. in the mixture, and the sum of all SmS is equal to 1. All coordinates X,, are centered at the origin beforehand, implying that the model should
wawsslatioiiiviiinltt 3) Node-level: We denote by h (m) RRH. ((m) (m) E R3 the representations of the i-th node in G...
Task Definition Formally, our goal is to learn a mapping &p from system space to property space:
We aim to predict both scalar and geometrie properties of the mixture system S.
:p: (19m (Hm, X Ven wm) M e) + ((Hr,, V M
where H', and V.. respectively represent the predicted scalar and geometric properties of all nodes. These node-level properties can be casily aggregated via readout functions derive graph-level or system-level property such as the electrolyte conductivity.
Symmetries Symmetries are fundamental in the physical world and should be satisfied when mod- eling interactions between physical systems. In mixture systems, symmetries are more complicated than those in single-instance systems. To be specific, the symmetries of the function yp in Eqq (1) divided into two parts: the node-level and the graph-level symmetry constraints.
For node-level symmetries, the output should be permutation-equivarian within each graph, namely,
::: (19m HmmPm Xm Vn Pm, Wm } J.M.i c) ((Hi,, V PMM
(2)
3

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Represontation
Embedding
GraphiMixture Readout
Optimization Objective
V- (initial zeros!
Geomatric Interaction (GIN)
Self-Attention
MSE Loss
B| [|I Ron
X
F
PCA
Encoder
proportion
X.
lev leat
Molecule Go (in frame F.)
Molecule (in std coord system)
Molecule (in std coord system)
Molecules G.. Go (in frame F...
F
muui (ruu)
F
(learnable)
T
m.
(Eq. (13)
7 (rii
F. R. F. (Eq. (10))
F.e (Eq. (11))
with proportion w = [wm]e JM as input, it first constructs local frames F. via PCA, then applies an equivariant encoder to update Hm, Xm. Intermolecular message passing is performed via GIN, which learns transformations 7 (m.x) and (m..)) which enable equivariant message passing across molecules. The scalar features Hm are finally aggregated along with the environment descriptor for prediction.
Figure 1: Overview of our GeoMix. By taking of molecules (G.. MM
where Pm E NN is permutation matrix acting all nodes in graph Gm. Besides, the output should also be rotation-equivariant on each graph, i.e.,
gp: (19m (Hm, Rm X,, Rm Vnn Wm W + (1 R, V m I=))
where R. E 3333 is rotation matrix acting node coordinates and geometric features in G... Translation equivariance is naturally achieved by centering each graph, and is therefore omitted.
For graph-level symmetries, the output should be equivariant with respect to the order of the graphs:
sp: (19a(m) J c) + ((H'(m)) (m) m==:)))
where denotes any permutation acting all graphs {
1MM
Note that the output property invariant any transformation. We call the two-level symmetries as Set-SE(3) Equivariance in this paper.
3.2 Model Architecture
As illustrated in Fig. 1, we design general equivariant architecture that enables the update of both invariant and equivariant features through intramolecular and intermolecular message passing.
Embedding Each molecule in the mixture system is associated with concentration value Wm. Prior studies [13; 50] often treat each molecule as having a fixed representation regardless of its concentration. In chemical thermodynamics, however, the activity of each molecule depends nonlinearly on its mole fraction and the activity coefficient, reflecting coupling between concentration and intermolecular forces [8]. To capture such context dependence, we concatenate each normalized proportion w, directly into atomic-level features via feature concatenation.
Proportion
Intramolecular Encoding We employ either EGNN [54] or TFN [55] as equivariant encoders to characterize molecular geometries through message passing within individual molecules. The same encoder with shared parameters is used for all solvent molecules, while separate encoder is

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employed for salts due to their distinct structural properties (organic salts versus inorganic solvents). In this stage, directly update node coordinates X by treating them geometric features Vn..
Frame Construction The rotation equivariance described in Eq. (3) arises primarily because each molecule is represented in an independent local coordinate system. This inherent symmetry prevents direct transfer of coordinate information between different molecules. However, are able to derive the local frame of each molecule, we can conduct geometric message passing between different molecules by first transforming coordinate information into standard coordinate space, depicted in Fig. 1. Here, we achieve this purpose based on Principal Component Analysis (PCA)', In particular, we first compute the covariance Cov(X,
X X and then perform cigendecomposition (ov(X, m) = F Am F.. to obtain its orthogonal eigenvectors F.. to form local frame. If det (Fe)())))))))))))))))))) 1, flip the third axis to ensure right-handed coordinate system.
Intermolecular Interaction With the pre-constructed local frames { we design novel component GIN to simulate geometric interactions between different molecules. The details of GIN will be presented in $ 3.3. In this module, the coordinates Xx,, are fixed, the geometric features V.. are initialized as zeros and will be updated throughout GIN.
JM
Readout and Prediction Head The outputs of GIN are node-level features. For mixture property prediction tasks such electrolyte conductivity prediction, we first average the node-level features and read them out as graph-level features, which are then modeled by a self-attention encoder without position encoding. The system-level feature obtained by averaging these graph-level features, and finally, we concatenate with the global environment feature and pass the result through an MLP with an activation function generate the prediction. These operations can be formulated as:
hm
1 Hm. y = MLP 1 No
1111
+
hm
C)
(5)
m==1
where represents the tensor concatenate operation, y E R is the predicted conductivity.
Optimization Objective For invariant prediction tasks, we use Mean Squared Error (MSE) as the loss function. To take full advantage of the equivariant network, we also adopt the Noisy Nodes method [57]. Noisy Nodes method encourages the model to learn the configurations with lower energy, effectively regularizing the learning process. Specifically, our model additionally learns denoising process of the encoder:
(G(Hm. m. X moisy
} + (G(Hm. denoiss M Jmmll
(6)
where X-miss X + NO o² I²),, and o 0.3 is taken for general molecular- tasks. The
auxiliary loss is calculated as:
Ldenoise
ΣM 1m=1
X
denoise
(7)
where || ||r is the Frobenius norm. The optimization objective is the sum of two losses given by:
f = LMSE + Litemoiee
(8)
The implementation details can be found in Appendix B.
3.3 Geometric Interaction Network
In this section, we introduce GIN, novel equivariant message-passing module to facilitate the geometric message passing between molecules using coordinate transformations, while promisingly ensuring (e)- equivariance.
Architecture Overview GIN takes input set of geometric graphs (G Xm, V along with the corresponding local frames (Fim M In each layer, the node features (H, 1M and the geometry features (V., m= are updated, while the coordinates {X ) are unchanged.
m=
To model the interaction from the source molecule G.,. and the target Gn. GIN consists of three key components. 1) Intermolecular Transformation: Establishing a transformation within a
'Since frame construction via traditional PCA is not strictly equivariant owing to the non-unique axis- orientation [56]. use improved, equivariance preserrvig PCA variant (see Appendix B.8).
5

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standard coordinate system based on the features of both the source and target molecules. 2) Message Construction: Computing the message in the source molecule's 's coordinate frame using the learned transformation, then converting it to the target molecule's frame. 3) Aggregation and Update: Aggregating the messages to update the node-level embeddings the target molecule. For brevity, in our following formulation, i E 9m consistently denotes the atom receiving message, while j E 9n represents the atom sending message.
Intermolecular Transtormation Since the two molecules exist in different reference frames, the message passing strategy used in traditional equivariant GNNs no longer suitable. Consequently, we aim to design a transformation that facilitates equivariant message passing across distinct reference frames. For atoms i E 9m. and
(m,,)) to combine the invariant In, we first calculate scalar 2ij
features in both atoms as follows:
(m.n) dinr (har(h," (h(m)) ,h) (m) | |l. || " "
Zij
where Cino is an MLP, and subsequent a functions with different subscripts represent distinct MLPs.
Based 2/j (m, and the frames F... Fn,, introduce learnable matrix "I" (m,a) 3x3x3 to "rotate"
the equivariant message from j Gn. into the reference frame of E Gm:
7m Fm ((m FT.
(m.n) rot (2.g (m.))) R333
R
(10)
((m,n) to model the "translation": Similarly, we also design learnable vector tij
{ (m.m)
Fmitfa (m.))
(m.n) (((2((m))) e R
(11)
It is worth noting that the matrix corresponding to T (m,,)) does not represent strict rotation, as
the determinant of R.j R(pan) is not constrained to be 1. We also explore alternative constructions of
R R(m-") with unit determinant, by using quaternion [58] and 6D vector [59]. However, ablation studies (see $ 4.4) reveal that these alternatives yield less favorable results. In fact, we can give the following theorem explain the expressive power of Eq. (10).
Theorem 3.1 (Expressivity of Intermolecular Transformation Matrix), Given geometric graphs
9m and G.. with F.., Fn being respective frames, any matrix Iun satisfying
I R.,. ESO0(3) R., ImaR. R,J must take the form like Eq. (10) (Fnn(FF.. F.,) R.F.
Message Construction
(m.z) with geometric interaction is similar to the The invariant message mij
construction of (m.n) in Eq. (9). The difference is that in order align the invariant message and 2ij
the equivariant message, additionally introduce the modulus of the translation vector & (m,,)) in Eq. (11). The entire formula is given by:
mig (m.n)
meg (h (m) h (n)
(m) ||o|||)|| |1. ||||] ||! |l || l|t,, (m))))))))))....))) [])-
||
(12)
We now compute cross-reference frame equivariant messages by two steps. First, the atom coordinate F.
(m..) in the reference (n) and the geometric feature 5/T)) are combined to construct the message m
frame of G.. Second, this message is transformed into the reference frame of 9m by applying the (m..))
learnable transformation, i.e. the "rotation matrix" T (m.n) and the "translation vector" b,
resulting in the updated message mij (m,n). T (m.n) mi.
(m,n) in the G., frame:
(m) mi
+ t.,) ((m.)) mi
(m..)
a ( (m.n) yaz(1)) + co(m yo
(m.
(13)
Aggregation and Update After devising the message, we proceed to aggregate them and update the atom features. Each atom is required receive message from all atoms in every other molecule. To this end, we employ the average operator to aggregate both invariant and equivariant message:
N,
N.
m (mn)
Σ
(m.n)
(m) m
1
m
(m)
(14)
N
N m
nofi f=1
where N = MM m=1 Nm denotes the number of all atoms in the system. Finally, we update the invariant feature h (n) and equivariant feature (/m) as follows:
h (m)
(n
(m) m U
(m)) (m)
5 mm)
m
(m)
(15)

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(a)
b
Cfll
DifTMiX
622 SOle
6000 urake
sobeett,
solvents, 3
deetra beanco wilho antiorrs
CALISo|
DiffMix
23 chamical fements
choricca dements
(c)
(d)
Conductivity Distribution CALiSol Conductivity Distribution DiffMix
CALISol: t-SNE
DiffMix: t-SNE
count/ing
count
Conductivity
Conductivity
Figure 2: Overview of the CALiSol and DiffMix datasets. (a) Summary of dataset statistics. (b) Composition diversity in terms of the number of solvents used per sample. CALiSol exhibits more diverse mixture design space, while DiffMix contains greater number of samples with 2 or solvents. (c) Log-sc histograms of conductivity values indicate that CALiSol exhibits broader range and more pronounced long-tail distribution than DiffMix dataset. (d) The t-SNE visualization [60] reveals distinct structural characteristics in each dataset. CALiSol, derived from experimental measurements, exhibits greater diversity with many scattered outliers, reflecting heterogeneous sampling. In contrast, DiffMix shows a more uniform and clustered distribution, consistent with its simulation-based origin.
4 Experiments
In this section, we first introduce the two large-scale electrolyte datasets, CALiSol and DifIMix, covering their composition, property distributions, and geometric graph construction in 4.1, and provide corresponding visualizations in Fig. 2. We then describe our experimental setup, including baseline models, GeoMix implementation, and evaluation metrics in 4.2. The main results in 4.3 highlight GeoMix's 's superior performance, and ablation studies in $ 4.4 assess the impact of key components. Further details are provided in Appendix
4.1 Datasets
CALiSol [19] and DiffMix [11] Datasets These two datasets provide comprehensive resources for electrolyte conductivity analysis. CALiSol contains 13,269 experimentally measured ionic con- ductivity entries (after filtering 556 incomplete points) spanning 13 lithium salts and 38 solvents across carbonate, ether, and ester classes, with broad coverage of salt concentrations and tempera- to reflect experimental diversity. contrast, the simulation-driven DifIMix dataset employs systematic discretization with 24,822 formulations focusing on two salts (LiPF,, LiTFSI) and up to three solvents selected from six carbonate types, quantizing salt concentrations into seven levels and normalizing solvent mass fractions via coarse sampling. Both datasets provide metadata including solvent proportions and measurement conditions. with full parameter ranges detailed Appendix
Geometric Graph Construction To facilitate geometric modeling, we further augmented the dataset with geometric molecular graph representations. We obtained molecular conformations from PubChem [61] when available. For molecules not found in PubChem, we generate 3D structures using RDKit [62] or download conformations from the Materials Project [63]. Each molecule is represented as a geometric graph, where atoms are nodes and edges are formed between atoms within a cutoff distance of 6 Å. Edge weights correspond to the Euclidean distances between atoms. Node features include atomic numbers and one-hot encodings of atom types.
Data Splitting Strategy both CALiSol and DiffMix datasets, we adopt a standard random split of 70%/20%/105 into train, validation, and test sets, respectively. Splitting is performed

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Table Results of MSE and Pearson correlation coefficient on CALiSol dataset and DiffMix dataset. Bold values indicate the best performance, while underlined values indicate the second-best.
CALiSol
DirIMix
Models
Symmetrics
MSE 4
Pearson
MSE
Pearson
MLP
3.657
0,906
1.363
0.874
MLP-based
None
MM-Mol Former [12]
5.488
0.825
1.901
0.812
Molsets-Conv [13]
2.230
0,924
1.440
0.868
Topology-based
Permutation
MolSets-SAGE [13]
2.751
0.909
0.708
0.937
EGNN-att [54]
2.666
0.908
0,930
Geometry-based
TFN-att [55]
Sct-SE(3)
1.808
0,946
0.804
0,921
EGNN-linear [54]
1.461
0,951
0.195
0.988
TFN-linear [55]
1.107
0.967
0.285
0.973
GeoMix-EGr EGNN
0.552
0.985
0.088
0,992
Geometry-basee based
Set-SE(3)
GooMix-TFN
0.432
0.987
0.035
0.997
independently for CALiSol and DiffMix with a fixed seed to ensure reproducibility. No electrolyte formulation shared across different subsets.
4.2 Experiment Setup
Previous studies such MM-MoLFormer [12], MolSets [13], and Uni-ELF [18] have proposed individual models and conducted comparisons within limited baselines, but no prior work has provided unified evaluation across modeling families using standardized datasets and metries.
Baselines
We compare with diverse baselines across architectural families: 1) MLP-based: We use a vanilla MLP taking the proportion vector imput, and MM-MoLFormer [12], which combines pretrained molecular embeddings with proportion and environment features. These models do not respect the symmetry constraints discussed in $ 3.1 and exhibit limited generalization.
Topology- We adopt MolSets [13], a recent framework designed for molecular mixture modeling, and select GraphConv and GraphSAGE representative backbones (denoted MolSets- and MolSets- SAGE). 3) Geometry-based: To account for spatial effects such as steric hindrance and solvation structure, employ equivariant GNNs including EGNN [54] and TFN [55]. Each paired with either a composition-weighee yielding four geometric baselines: EGNN-att, TFN-att, EGNN-linear, and TFN-linear. The specific hyperparameters of the model are shown in Appendix B. Other recent geometry-basee methods, such DiffMix [11] and Uni-ELF [18], are included in our baselines due the lack of publicly available implementations.
average (-linear) or a learnable attention -based aggregator (-att),
Implementation For our GeoMix, we test it using two classical equivariant networks, EGNN and TFN, as the backbone for intramolecular embedding. Our method consistent with the baseline in terms of hyperparameters such the number of equivariant graph network layers and MLP layers. For other hyperparameters and other implementation details, please refer to Appendix B. We evaluate model performance using MSE and Pearson correlation coefficient (r) between the predicted and measured conductivity values.
4.3
Main Results
Table summarizes the predictive performance of various baseline models and our proposed GeoMix on the CALiSol and DifiMix datasets. From these results, we observe the following: 1. Overall performance: GeoMix consistently achieves the best performance across all metrics on both datasets, demonstrating its effectiveness in modeling molecular mixtures. 2. Advantage of geometry-based methods: Geometry- based methods outperform both MLP-baser and standard Topology-bai baselines, highlighting the importance of incorporating molecular geometry into the learning process.
3. Benefit of higher-degree features: Among geometric models, those using TFN as the backbone generally surpass those based on EGNN, suggesting that modeling higher-degree geometric features particularly beneficial in mixture-related tasks. We further visualize the regression performance of some selected models in Fig. 3, which clearly illustrates the superiority of GeoMix based methods in both accuracy and error consistency.

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0.003
OLMOT
Eepiimmm Conduciiviy (cm)
Experimeztal Condiontivity (m8
Experimental Conductiviy (mS// /m))
Figure 3: Regression plots for electrolyte conductivity prediction. (a-c) show results the CALiSol dataset, using T'NN-linear, GeoMix-EGNN, and GeoMix-TFN, respectively. (d-f) show corresponding results on the DifIMix dataset with the same model order.
4.4 Ablation Studies
To evaluate the individual contributions of our proposed components, we perform a series of ablation studies the CALiSol dataset. All experiments based the GeoMix-EGNN model, which adopts the EGNN backbone due to its architectural simplicity and suitability for controlled analysis.
Proportion Embedding In the ablation variant denoted as "Multiply" in Table 2, we remove the proportion in- formation from atomic features and instead apply scalar weighting each per- graph embedding before aggregation. This strategy mirrors traditional approaches that treat component ratios as external mixing coefficients rather than intrinsic modulators of molecular behavior. Table 2 reveals that embedding proportions into node fea- tures can effectively improve the effect of the model in modeling mixture systems.
Table 2: Ablations on CALiSol dataset.
CALiSol
MSE |
Pearson /
GeoMix
0.552
0.985
Proportion Embedding
Multiply
3.657
0.906
Transformation Matrix Form
Quaternion
0.702
0.981
6D vector
0.574
0.983
Graph-wis. wise
0.662
0.980
Linear v.s. Attention
The Form of the Transformation Matrix We conduct ablation studies examine how different parameteriza-
GeoMix-lineat
0.851
0.975
Noisy Nodes Loss
tions of the transformation matrix R. affect model performance. We consider three alternative designs: 1)
w/o Noisy Nodes
1.213
0.969
Quaternion: The rotation matrix is derived from a learned unit quaternion, ensuring orthogonality with det m/m..)) parameterized by two learned vectors followed by Gram-Schmidt orthogonalization, providing a
(i.e., rigid transformation without scaling). 2) 6D Vector: The rotation is
continuous and differentiable representation of SO(3) [59], also enforcing det det(R( R( 1. 3) transformations, we learn a single transformation per graph pair by pooling node features. This enforces graph-level rigidity and ignores local structural flexibility. From the results in Table 2, we observe that: (1) Models with non-rigid, nide-wise transformations (GeoMix, Quaternion, 6D vector) outperform the graph-rigid variant (Graph-wise), suggesting the importance of fine-grained local transformations. (2) Among non-rigid variants, Ge- oMix, which does not constrain the transformation matrix to be orthogonal or have unit determinant, achieves the best performance. This indicates that allowing scaling transformations further enhances flexibility and expressivity modeling cross-graph geometry.
Graph-wise: Instead of learning
Graph-Level Readout As shown in Table 1, the simple linear average readout performs com- petitively among baseline models. To investigate effect within our framework, we replace original graph-level readout in GcoMix with the linear average strategy, resulting in an ablation

---

variant denoted as DeoMix-lineal As reported in Table 2. GeoMix-lineau yields lower performance compared to the original GeoMix. This suggests that the full model benefits from the rich, attention- based aggregation mechanism, which we attribute to the GIN. The learned transformation-avee -wwae interactions provide expressive relational cues between molecules, enabling the self- module to generate more informative graph-level embeddings beyond simple averaging.
Noisy Nodes Loss We further evaluate the impact of the Noisy Nodes loss [57] by removing from the training objective, denoted as "w/o Noisy Nodes" in Table 2. The results demonstrate that incorporating this regularization consistently improves the model's performance. We attribute this to its role in enhancing representation robustness by perturbing node features during training.
Conclusion
We presented GeoMix, Set-SE(3) equivariant framework for predicting properties of molecular mixtures. GeoMix leverages equivariant graph neural networks to embed individual species and introduces Geometric Interaction Network (GIN) to perform intermolecular message passing via pre-constructing local frames. To support systematic evaluation, we curated two large-scale datasets with geometric features, CALiSol and DiffMix. focused on electrolyte conductivity. Based these datasets, we established a unified benchmark for electrolyte conductivity prediction. Experimental results demonstrate that GeoMix significantly outperforms existing baselines, highlighting the impor- tance of geometry- modeling in mixture systems. Future work may extend this framework to other multi-component tasks in chemistry and materials science.
6 Limitations
Despite the promising results, this work has several limitations that warrant further investigation. First, the available datasets remain limited in both scope and diversity. In particular, no datasets currently provide molecular dynamics-based microscopic structural information, and there is lack of conductivity datasets covering broader range of chemistries, such as sodium-ion potassium-ion electrolytes. Second, the encoder architecture adopted in GeoMix relatively simple, leaving room for improvement.
In future work, we plan to incorporate richer descriptors and leverage recent advances in graph neural networks to enhance both the accuracy and the generalization ability of the model [64; 65). We also aim curate and release new benchmark datasets that include microscopic structural features and broader compositional coverage, thereby enabling more comprehensive evaluation and fostering further research in mixture property prediction.
Acknowledgement
This work was jointly supported by the following projects: the National Natural Science Foundation of China (No. 62376276); Beijing Nova Program (No. 20230484278); the Fundamental Research Funds for the Central Universities; the Research Funds Renmin University of China (23XNKJ19); Public Computing Cloud, Renmin University of China.
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14

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Contents of Appendix
Theoretical Details
16
A.1
Expressivity of Intermolecular Transformation Matrix
16
A.2 Equivariance/Invaiincee of GeoMix
17
A.2.1
Enhancement of Equivariiance//naarinnee
17
A.2.2
Equivariance of Local Frame ................................ .r.............................. _...............................
17
A.2.3
Equivariance of Learnable Transformations
17
More Experiment Details and Results
18
B.1
Dataset Details
18
B.2
Implementation Details
19
B.3
Training Settings
................................
19
B.4
OOD Evaluation across Conductivity.
19
B.5
OOD Evaluation across Temperature.
20
B.6
Experiment of coordinate perturbation.
20
B.7
Runtime Analysis on CALiSol dataset.
B.8
Evaluation of Different Frame Construction Methods
15

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A Theoretical Details
A.1 Expressivity of Intermolecular Transformation Matrix
Lemma A.1. For any ((n) -qquivvraaa function f(Z), ) it must fall into the subspace spanned by the columns of Z, namely, there exists a function s( 2) satisfying f(Ø) = Zz
Proof. The proof is given by [66]. Essentially, suppose Z1 is the orthogonal complement of the column space of Z. Then there must exit functions s s(Z) and -- satisfying f(Z)
Zz(Z) + 7-s-(Z), We can always find an orthogonal transformation O allowing OΣ = Z while
O -Z-. With this transformation O, we have j(OZ) f(ž) Z + - and
0f Zs(ž) Z g (Z). The equivariance property of f implies Zs(Z) + Z (Z)
2. 2 - Z- (Z), which derives 8 (Z) = 0. Hence, the proof is concluded.
Lemma A.2. If the O(n) eqqivvaiaa function f (z) lies in the subspace spanned by the columns of Z. then there exists a function a satisfying f(Z) = Zo (Z 2 Z).
Proof. The proof is provided by Corollary 2 in [67]. The basic idea is that f(Z) can be transformed to f(Z) = Zn Z) where (Z)) is
According to Lemma 1-2 in [67], ( must be written as 17( 2 of ZT 2), which completes the proof.
Theorem 3.1 (Expressivity of Intermolecular Transformation Matrix), Given geometric graphs Im and Gn. with Fm. Fn being respective frames, any matrix Imn satisfying SO(3) -equiivrrve 4 R.,. R., ESO(3), Rm Rm must take the form like Eq. (10) as In (F F..) F.R.F....
Proof. First, we want to show that, Tm (F Fa)) equivalent to Im & F.)). We know the Kronecker product F.. @ Fn contains x 9 elements, which could be divided into 9 small matrices of 3 x 3, each corresponding to the product of row in F m [ūā; ū2;; ūgz and a column in F, JJj J. J. i.e. Fm & F. ü @ .. Note य. @ U2 is rank-one matrix and ü; 0y are both unit vectors. Therefore, म. x iij corresponds to two possibilities: (ū., oi..) or ū. -J ). By analyzing the submatrices, possible decomposition, namely (Fm.: F.) and (-F... -F,). Note that during the construction, we agreed that det (m) (et(FF,) 1, so there is a direct bijection between F, & F,, and (Fm, Fn). which means Iy [F, F.)) is equivalent to I (F, @ Fn)).
find that F,, & F,, also corresponds to two
Then, we use the vectorization operator vec(-) to prove ' F... @ F.)) must take the form:
vec( Iym Fm & F.)) = vec
(Fr & F..) vec R).
(16)
Note that F,, Fm are constructed frames, which is full-rank. Then we find F, Fm will be also full-rank, spanning the whole space T where veo( Tym (Fm. F2)) lies in. According to Lemma A.2, know veo( Imn) must satisfy:
veel ToeeTTnm Imm) = (F.. & Fm)o F.) F
(17)
Moreover, we have such formula:
(F, + ( (F.
& (T 13x3 & 13x3 19x9,
(18)
which is constant matrix. Thus ((F, F..) of Fm)) = o (Txx)) will be learnable parameters.
which are equivalent to the veo R) here. Up to this point, the matrix 7, appears to exhibit O(3)- equivariance. However, it is crucial to observe that the frame constructions F., and F. are only
S0(3) -quiiaraquva Consequently, the overall transformation reduces to -quivvariiee This
completes the proof.
D
16

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A.2 Equivariance/Ivarianee of GeoMix
A.2.1 Enhancement of Equivariince/Invriianee
In order prove the equivarianne//vvariance of the entire GeoMix model, need strengthen the conditions.
Theorem A.3 (Enhancement of Equivarianne//lvariane)) If each module of GeoMix is equivari- ant/invariant, then the entire GeoMix will also be equivariant/ivvriint..
Proof. Consider a sequence composed functions {dr : X + x2)) N equivariant same group G, the equivariance lead to an interesting property that
Φx ....... db+1 Pxr (g)b .o 0 UN o 6- Pxo (g) ...... ø1,
holds for all i.j 1,2,.... ................................ .N and G. which means that the group elements be freely exchanged in the composite sequence of equivariant functions. In particular, if one the equivariant functions (e.g. Φx)) is replaced by an invariant function, the group element will be absorbed, that means
Φ1 oN holds for all g E G but only = 2....... k. Although @N o ..... Φx is still equivariant, because the group elements must be input starting from Φ1, the overall ΦN .... 01 is still invariant function.
ΦN
04+1 px( (g)Ø:
Φ1-
Specifically, we need to prove the following:
1. -quivvrruiv of local frame.
2.
of learnable transformations.
A.2.2 Equivariance of Local Frame
Theorem A.4 (Equivariance of Local Frame). The construction of local frame 50(3)- equivvariance
Proof. We prove that the local frame Fm is SO(3) -qquivaraat with respect to the target graph 9m- More formally, for any orthogonal matrix Q pax3 the local frame should satisfy:
F
LocalFrame Qm
(19)
Formally, we have
Cov(Qu QX QX = Q
Nm
N
= =CCov (m)))) = =QF FT Q
(20)
=QFmm Am Ammc)) (Q.m....
Concluding we showed that an SO(3) transformation on the original coordinate matrix X,, results in the same transformation the output local frame Fm..
A.2.3 Equivariance of Learnable Transformations
Theorem A.5 (Equivariance of Learnable Transformations). The construction of learnable transfor- mations is (eee)33-quvaaiannce
Proof. We prove the message (m)) and m
(m.n) in Eq. (13) is SO(3) equivariant on the target graph
9m and the source graph Gn respectively, that is, for any orthogonal matrix Qms Qn E 3x3 the message should satisfy:
m (m,n)
-MsssageSource Q,, X, V.V,),
(21)
Q m mig
(m)
-MessageTTrrget(.. (m.
17

---

It is easy to see mi
(m.n)
is invariant both the source and the target graph. So,
( ( Qnzz
(m..))
(n)
+ o (m. o
(m..))
Qm Jay Qno (m. mQ (m.a)
(m.n)
(m) )
)
(22)
which is (3)3) equivariant on the source graph
We have proven that the local frame Fm. is SO(3) equivariant on the target graph, so F, is SO(3) equivariant the source graph. We have:
Qm F Ri (m)
(Qn Fa) Qumin (m.x)
Qm Fm
(m.x)
Q F m R mm)
F Qu Qn mi ())
+ mmt (())
(23)
Qm ((m)))
mij
(m.n)
+ Qm tx)
(m)
(m)
mj
which is SO(3) equivariant on the target graph.
B
More Experiment Details and Results
Our code and dataset are available https://githinn www.////AAACC/GOO/////oo
B.1 Dataset Details
In this part, we provide supplementary details and visualizations of the datasets introduced in the main text. These include the distribution of molecular species, property ranges, and composition coverage, as illustrated in Table 3.
Table Summary statistics of the CALiSol [19] and DiffMix [11] datasets, including the number of samples, solvent and salt types, elemental coverage, maximum number of solvents per formulation, solute concentration ranges, and temperature ranges.
CATiSol
DiffMix
Number of samples
13,269
24,822
Number of unique solvents
38
Solvent list
EC, PC, DMC, DEC, DME, DMSO, AN, MOEMC, TFP, EA. MA, FEC, DOL, 2-McTHE, DMM, Freon 11, Methylene chloride, THE, Toluene, Sulfolane, 2-Glyme, 3-Glyme, 4-Glyme,
EC, PC, FBC, EMC, DEC, DMC
3-Me-2-Oxazzalidinone 3-MeSulfolane, Ethyldiglyme, DMF, Ethylbenzene, Ethylmonoglyme, Benzene, g B-Butroolactone Cumene. Propylsulfone, Pseudocumeme, TEOS,
m-Xylene, o-Xylene
Number of unique salts
13
Salt list
LiPFs. LiBF,, LiFSI, LiTDI, LiPDI, LiTFSI, LICIO4, LiAsF LiBOB, LiCF,SO,, LiBPFPB, LiBMB, LiN(CF:SO:);
LiPF6. LiTESI
Elements covered
H. Li, B.C. F. P.S. As
H.L.C.N...R....
Maximum number of solvents per sample
4
Solute composition range
0.0286-2.37 mol/kg and 0.0771 -0701-4.00 mol/L.
0.025-3.0 mol/kg
Temperature range
194.15-477.42K
243.15K-293.115
18

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B.2
Implementation Details
In this part, we describe the implementation details. Each model architecture remains the same in both datasets.
MLP: A 3-layer multilayer perceptron is used, with each hidden layer having 64 dimensions.
MM-MoL.Former We follow settings [12]. MM-MolFormer uses pre-trained MoL- Former model [68]. obtained from https: petts://ibnn ://ibm....m/ ent. box. com/c /ttp////MMoooooeeree data. The molecular embeddings are concatenated with the corresponding proportion and temper- ature features, and then passed through a 2-layer MLP with hidden size 64 to generate the final prediction.
MolSets:
For both MolSets- and MolSets- SAGE, we retain the original hyperparameter settings from [13], except that the hidden dimension of each graph neural network layer is increased to 64.
EGNN-att: This variant employs a 4-layer EGNN backbone with 64 hidden dimensions per layer. The attention module takes 8 molecular embeddings as input, with 4 attention heads and 3 stacked attention layers. The resulting representation is concatenated with temperature T and salt concentration and then passed through a 3-layer MLP for prediction.
TFN-att:
TFN-att shares the same configuration as EGNN-att except for the backbone, which is replaced with TFN. The TFN backbone uses maximum angular momentum quantum number of max L = 2, and each irreducible representation (irrep) has 8 channels.
This model uses the same EGNN backbone as EGNN-att. Instead of an attention module, a simple non-learnable linear average (scatter) operation is applied to obtain the mixture representation
EGNN-linear:
TFN-linear: This model adopts the same TFN backbone TFN-att, and replaces the attention module with a non-laarable linear average operation, as in EGNN-linear.
GeoMix-EGNN: GeoMix-EGNN adopts a multi-channel EGNN as the backbone, with equivariant channels. Other hyperparameters follow the EGNN-att setting. The GIN consists of layers. The Noisy Nodes loss [57] is applied with hyperparameter 7 = 128.
GeoMix-TFN: GeoMix-TEN uses the same TFN backbone as TFN-att. The GIN also consists of layers. The Noisy Nodes loss is applied with y 128, in GeoMix-EGNN.
B.3 Training Settings
All models are trained using the Adam optimizer with a learning rate of 10-5, weight decay of 1 x 10-12, and a maximum of 500 training epochs. The default batch size is 1024, except for GeoMix models, which use a reduced batch size 128 due to GPU memory constraints. fix the random seed to 7 for all experiments to ensure reproducibility.
We run our methods mainly on A100 80GB GPUs.
B.4 OOD Evaluation across Conductivity.
In practical chemical research, is often necessary to identify substances with high ionic conductivity based on prior knowledge of low-conductivity samples, which naturally poses auttof-distribution (OOD) generalization challenge. However, most existing machine learning models struggle to generalize reliably under such distribution shifts. Here, we specifically consider OOD split where iow-conductivity samples are used for training and validation, while high-conductivity samples are reserved for testing.
Experiment Setup. To evaluate our model's robustness this setting, we selected samples with conductivity < 10 mS/cm as the training and validation sets, and those with conductivity 10 mS/em as the test set on the CALiSol dataset. The training and validation sets were split randomly in a 4:1 ratio. The high-conductivity test set contains 1, 1,244 samples, ensuring that this split is statistically meaningful. Here, we evaluate only EGNN-linear and TFN-linear, the two best rest-performing models
'The salt concentrations are discretized into seven levels: (0.025, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0) molal. For each formulation. the solvent mass fractions vary from 0 I in increments of 0.2 and are normalized to sum
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Table 4: Results of OOD evaluation across conductivity on CALiSol dataset. Bold values indicate the best performance, while underlined values indicate the second-best.
Models
MSE
MAE +
Pearson r
Spearman r
EGNN-linear [54]
32.720
4.457
0.253
0.341
TFN-linear [55]
24.218
3.585
0.397
0.508
GeoMix-EGNN GeoMix-TFN
17.132
2.853
0.579
0.571
19.427
2.925
0.436
0.609
in 4.2, baselines. During training, observed that attention- based aggregation of graph-level information tended to overfit. To mitigate this, we applied a dropout rate of 0.1 to the attention layer and set the attention temperature 2.0. All other settings remained unchanged.
Results. The results this OOD split based on conductivity values are summarized in Table 4. Both baseline models, EGNN-linear and TFN-linear, exhibit limited generalization. Our proposed models, GeoMix-EGNN and GeoMix-TFN, show significantly lower MSE and higher correlation metrics compared to the basclines, demonstrating their effectiveness in capturing trends in this OOD setting. These results highlight that our approach improves predictive performance while enhancing OOD generalization under conductivity-baase distribution shifts.
B.5 OOD Evaluation across Temperature.
Because the free diffusion rate most materials at ambient temperature is low, it is common to perform molecular dynamics simulations at elevated temperatures and then extrapolate to lower temperatures using the Arrhenius empirical relation [8]. Therefore, we explored the consistency of model predictions with respect to temperature.
Experiment Setup. empirically assess this, we performed an OOD generalization experiment where we trained the model data with temperature < 320 K (using train/valid split of 4:1), and evaluated on samples with temperature 320 K. other settings remained unchanged.
Results. Results in Table 5 show that GeoMix achieves significantly better predictive performance on this challenging temperature extrapolation task compared to other baseline models, suggesting that the learned representations effectively capture physically meaningful temperature trends without requiring hard-coded constraints.
Table 5: Results of OOD evaluation across temperature CALiSol dataset. Bold values indicate the
best performance, while underlined values indicate the second-best.
Models
MSE
MAE 1
Pearson
Spearman r
MLP
32.432
3.857
0.432
0.556 0.560
MM-MoLForrmer [12]
22.935
2.837
0.512
MolSets-Conv [13]
9.930
2.196
0.839
0.845
MolSets-SAGE [13]
9.193
2.135
0.784
0.802
EGNN-att [54]
8.134
1.594
0.813
0.861
TFN-att [55]
6.516
1.383
0.853
0.887
EGNN-linear [54]
7.675
1.616
0.827
0.847
TFN-linear [55]
4.780
1,286
0.914
0.930
GeoMix-EGNN
2.366
0.883
0.950
0.948
GeoMix-TFN
2.354
0.917
0.952
0.945
B.6 Experiment of coordinate perturbation.
For complex molecular systems, the initial conformations used in modeling may not always be fully accurate. To assess the robustness of our model perturbations in the input geometries, we conducted an additional robustness evaluation.
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Experiment Setup. All other settings remained unchanged, except that Gaussian noise with standard deviation of = 0.3 Á was added to the atomic coordinates in the test set to simulate conformational uncertainty.
Results. As shown in Table 6, most models inclulignn GeoMix-EGNN exhibit only a marginal performance drop, indicating stability under moderate geometric perturbations. Notably, models based on the TFN encoder show much larger performance drop, which may stem from their reliance on higher-degree equivariant features (e.g., spherical harmonics) that could be sensitive atomic coordinate perturbations.
Table 6: Results of coordinate perturbation CALiSol dataset. Bold values indicate the best performance.
Models
MSE
MAE 1
Pearson r
Spearman r
MolSets-Conv [13]
1.326
0.703
0.960
0.965
MolSets-SAGE [13]
1.748
0.794
0.946
0.958
EGNN-att [54]
1.919
0.587
0.942
0.965
TFN-att [55]
2.294
0.760
0.929
0,954
EGNN-linear [54]
1.114
0.555
0.967
0,976
TFN-linear [55]
2.393
0,937
0.929
0.944
GeoMix-EGi EGNN
0.631
0.388
0.982
0.983
GeoMix-TEM
4.517
1.324
0.859
0.886
B.7 Runtime Analysis on CALiSol dataset.
Results in Table shows the runtime analysis comparing our methods with other main baselines on the CALiSol dataset.
Table 7: Runtime Analysis on CALiSol dataset.
Inference Time
Memory Usage (GB)
Models
MSE
Pearson r
(x 10
EGNN-linear [54]
4.41
2.20
1,461
0,951
TFN-linear [55]
24.13
5.91
1.107
0,967
EGNN-at att [54]
23.40
2.22
2.666
0.908
TFN-att [55]
43.20
5,91
1.808
0,946
GeoMix-
34.04
25.15
0.552
0.985
GeoMix-TFN
53.64
45.18
0.432
0.987
B.8 Evaluation of Different Frame Construction Methods
As described 3.2, the local frame Fn, for each molecule is constructed via degeeodecooooiitionn of the covariance matrix Cov(X,)) However, the axis orientation of F,, obtained from standard PCA is not unique, simultancous sign flips of cigenvectors can lead instability in local frame alignment. To alleviate this ambiguity, we additionally test deterministic version of PCA by enumerating all eight possible axis-orientatio combinations of the eigenbasis and selecting the with the largest coordinate sum, i.e.,
argmax 1 F,,1
(24)
Fir EF(F...)
where (F(F.)) denotes the set of all sign permutations of the three principal axes. This procedure ensures a consistent frame orientation across molecules.
Performance Comparison. Table 8 summarizes the predictive performance of different frame construction methods under same experimental setup as described in $ 4.2. The results show that the deterministic (strict) PCA achieves performance comparable to the standard PCA, though with slightly higher errors in some cases. According to [56], the standard PCA- -Csed frame construction, while not strictly equivariant, remains reliable and effective for defining local coordinate systems in
21

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molecular modeling. The reduced flexibility of the strictly equivariant variant may account for its marginally lower performance. addition, the methods in [16; 17] can also be used as alternatives, which we will study in future work.
Table 8: Results on the CALiSol dataset. FC Methods denotes the frame construction methods. The strict PCA refers to the deterministic variant defined in Eq. (24).
Models
GeoMix-EGNN FC Methods
MSE
Pearson r 1
PCA
0.552
0.985
GeoMix-TEN
PCA
0.432
0.987
GeoMix-EGNN
strict PCA
0.628
0.981
ReoMix-TFN
strict PCA
0.460
0.986
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Reviews:
[{"review_id": "qDSJuPujjs", "reviewer": "Reviewer_x1YU", "summary": "This paper proposes GeoMix, an equivariant neural network model for predicting the electrical conductivity of electrolyte solutions. A systematic comparison of GeoMix with various neural network models, including MLP-based, topology-based, and geometry-based architectures, is conducted on the task of electrical conductivity prediction. Furthermore, the study supplements the CALiSol-23 and DiffMix datasets with relevant information, enhancing the future evaluation of predictive capabilities on these datasets.", "questions": "See weaknesses.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\nThis work introduces a novel equivariant 3D graph neural network model for predicting the electrical conductivity of mixed molecular systems. The proposed model achieves state-of-the-art (SOTA) prediction accuracy on both the CALiSol-23 and DiffMix datasets.\n\n## Weaknesses\n\n**Major issues**\n\n1. The evaluation of electrical conductivity prediction accuracy in this work relies solely on random splitting within the CALiSol-23 and DiffMix datasets. This approach risks yielding overly optimistic results, as evidenced by the Pearson r exceeding 0.98 for GeoMix and generally above 0.95 for baseline models in Table 1. In practical electrical conductivity prediction (and molecular property prediction in general), models frequently encounter out-of-sample scenarios, such as predicting the conductivity of: unknown electrolytes in known solvents, known electrolytes in unknown solvents, or novel combinations of both unknown electrolytes and solvents. Therefore, it is recommended that the authors explicitly designate specific electrolyte and solvent molecules to be held out exclusively in test set for evaluating the model's predictive performance under these out-of-sample conditions.\n\n2. Predicting electrolytes with higher conductivity is a critical scientific objective. To assess this capability, it is recommended that the authors partition a subset of high-conductivity data as an independent test set, utilizing only low-conductivity data for training and validation. Specifically, for the CALiSol-23 dataset, data points with conductivity >20 mS/cm could be reserved solely for testing. Similarly, for the DiffMix dataset, data points with conductivity >10 mS/cm could be reserved for testing. This strategy would provide a meaningful assessment of the model's extrapolation performance.\n\n3. Upon reviewing the provided data and code, it was observed that for all lithium salts, only the 3D conformations of their anionic moieties were generated. While this approach might be adequate for the current dataset containing solely lithium cations, its applicability becomes significantly limited when datasets incorporate other cations, thereby severely constraining the model's generalization ability. Although generating accurate 3D structures for complete ionic compounds (containing both cation and anion) presents challenges, the authors should nevertheless consider and address how to effectively integrate the structural features of diverse cations into the model.\n\n**Minor issues**\n\n1. The term \"DiffMix\" is used to refer to both a dataset and a model in the manuscript, which may cause confusion. It is recommended to distinguish them by adding prefixes or suffixes\n\n2. The authors' expansion of the CALiSol-23 and DiffMix datasets appears limited to supplementing 3D molecular coordinates (without DFT optimization and excluding cation structures for ionic compounds). If accurate, the claims regarding dataset contributions should be moderated.\n\n3. Although no implementation is available for the DiffMix model, its original publication reports a conductivity prediction MAE of 0.044 mS/cm, a result comparable to or potentially better than the best performance in this work. The authors should compute and report GeoMix's MAE on the DiffMix dataset to enable direct comparison.\n\n4. The page number of reference 10 is incorrect, and the issue number of reference 11 is redundant ('(1)' should be deleted). Please check the accuracy of all references thoroughly.\n\n5. In the Checklist section, the author seems to have provided two code repositories, https://anonymous.4open.science/r/GeoMix-74AC/ and https://anonymous.4open.science/r/GeoMix-73AC/. The former is an invalid address. Please correct it.", "was_revised": true, "final_justification": "My primary concern stemmed from the experiments in the paper, which the authors have since addressed by supplementing additional experiments in the rebuttal. The writing and experimental evaluation of this paper meet my threshold for acceptance. However, I must emphasize that the reported Pearson R of 0.58 is not practically useful for scientists, which limits the practical applicability of this work.", "rebuttal": "Thank you for your thoughtful comments and questions.  We truly appreciate the time and effort you invested in reviewing our work.  Below, we summarize and respond to your points in detail.\n\n> **R1 (Major Issue 1). The necessity of out-of-sample evaluation.**\n\nThank you very much for your insightful comments regarding the evaluation protocol and the importance of testing the model’s performance under out-of-sample scenarios. We fully agree that for practical electrical conductivity prediction tasks, it is critical to assess how well a model generalizes to completely new combinations of electrolytes and solvents, as well as to previously unseen individual molecules.\n\nIndeed, we recognize that extrapolating to completely novel electrolytes or solvents remains a significant challenge, particularly given the limited size and chemical diversity of current datasets. In realistic deployment scenarios, it is often more plausible to assume that the model has previously encountered the individual constituent molecules (both solvents and electrolytes) during training, though possibly only in combinations of different concentration ratios. Our current experimental design adopts this assumption and primarily focuses on interpolation within relevant composition spaces, which is crucial for accurately modeling mixture behaviors.\n\nNonetheless, we fully acknowledge the importance of evaluating model robustness beyond random data splits. In response to the reviewer’s valuable suggestion, we have now conducted an additional evaluation in which the dataset is partitioned based on conductivity ranges, rather than through random sampling. This protocol provides a meaningful proxy for assessing out-of-sample generalization, particularly with respect to extrapolation from low- to high-conductivity regimes. The detailed results and analysis of this experiment are presented in our response **R2** below.\n\n> **R2 (Major Issue 2). Out-of-sample evaluation of high-conductivity electrolytes.**\n\nThank you for your recommendation to hold out high-conductivity electrolytes as a more meaningful out-of-sample test set. In line with your suggestion, we have performed new experiments on the CALiSol-23 dataset, where we use only samples with conductivity ≤10 mS/cm for training and validation (random split of 8:2), and reserved all samples with conductivity >10 mS/cm exclusively for testing.\n\nWe would like to note that while we initially consider using the >20 mS/cm threshold as you proposed, there are only 64 samples in this range, which is insufficient for statistically robust evaluation. To ensure a meaningful and reliable assessment, we therefore adopt the >10 mS/cm threshold, resulting in a high-conductivity test set comprising 1,244 data points. This experimental protocol thus enables a direct and statistically sound evaluation of the model’s capability to extrapolate to high-conductivity out-of-sample regimes.\n\n**Table S1: Results of out-of-sample evaluation on high-conductivity test set.**\n\n|Models|MSE|MAE|Pearson r|Spearman r|\n|--|--|--|--|--|\n|EGNN_linear|32.72010|4.45704|0.25309|0.34116|\n|TFN_linear|24.21760|3.58482|0.39704|0.50828|\n|GeoMix_EGNN|**17.13204**|**2.85305**|**0.57858**|0.57091|\n|GeoMix_TFN|19.42731|2.92452|0.43568|**0.60949**|\n\nAs shown in the table, performance across all models decreases notably in this challenging out-of-sample setting, which underscores the difficulty of extrapolating to high-conductivity regimes. Importantly, our proposed GeoMix model, particularly with the EGNN encoder, demonstrates clear improvements over baseline approaches in all metrics. These results and discussions will be incorporated into our revised manuscript.\n\n> **R3 (Major Issue 3). Discussion when datasets incorporate other cations.**\n\nWe appreciate your close reading of our code and data, and your attention to the implications for model generalization when datasets include cations other than lithium. We apologize for not explaining the relevant chemical background more clearly in the manuscript.\nIn this work, we generate 3D conformations only for the anionic moieties, treating Li⁺ as a point charge without explicit geometry. This approach reflects the chemical reality of our dataset: the salts are fully dissociated, with cations and anions freely distributed, making it inappropriate to assign a fixed 3D structure to the complete ionic compound.\nMoreover, our model is general by design and can accommodate diverse cation types. While the current dataset only includes lithium salts (Li⁺), the proposed framework can readily integrate other cations (such as Na⁺, K⁺, or multivalent ions) by explicitly incorporating their structural and physicochemical features into the model input.  We will clarify this point in the revised manuscript.\n\n> **R4 (Minor Issue 2). Clarification of our dataset contributions.**\n\nWe appreciate the reviewer’s careful reading.  Our contributions to the CALiSol and DiffMix datasets primarily consist of the following three aspects: (1) supplementing 3D molecular coordinates using standardized procedures such as RDKit conformer generation and PubChem retrieval to enable spatial graph learning;  (2) performing data cleaning by removing invalid or inconsistent samples to improve dataset quality and robustness;  and (3) establishing a standardized data split protocol to facilitate fair and reproducible benchmarking. \n\n> **R5 (Minor Issue 3). Clarifying DiffMix baseline comparison.**\n\nThank you very much for your insightful comments regarding the comparison between GeoMix and DiffMix, especially in relation to evaluation metrics and dataset split strategies.\n\nFirst, we would like to clarify that a direct, numerical comparison of performance between GeoMix and DiffMix is not straightforward, primarily due to fundamental differences in task formulation and dataset construction. Specifically, the two works differ in the distributions of target values, normalization procedures, and data split schemes. Additionally, DiffMix incorporates supplementary features such as excess molar enthalpy and volume—features that are not used in GeoMix—which may contribute to their reported regression performance.\n\nSecond, with respect to dataset splits, although the DiffMix repository provides an overall 8:1:1 split ratio (train:validation:test), it does not release the specific split indices or the individual 3D molecular structures used. As a result, it is challenging to reproduce their data partitions and baseline models without access to these intermediate processed datasets. In contrast, our study employs a consistent and standardized 7:2:1 split for both the CALiSol-23 and DiffMix datasets, facilitating fair within-study comparison and robust cross-dataset validation. We believe this approach ensures methodological consistency throughout our work.\n\nFor these reasons, metric-to-metric comparisons are inherently limited due to disparities in data preprocessing, feature design, and splitting strategies. Moving forward, we support community-driven efforts to unify benchmark standards and facilitate the sharing of intermediate representations, which will enable more reliable and consistent comparisons across different models.\n\nThank you again for highlighting these important considerations.\n\n> **R6 (Minor Issues 1, 4 & 5).**\n\nWe sincerely thank the reviewer for the detailed and constructive feedback. Below we respond to each point individually:\n- **Minor Issue 1. Clarification of the term \"DiffMix\":** We agree that using \"DiffMix\" to refer to both the model and the dataset may cause confusion. In the revised version, we will clearly distinguish between the two by using \"DiffMix-dataset\" for the data and \"DiffMix\" for the baseline method throughout the manuscript and figures.\n- **Minor Issue 4. Reference formatting issues:** We have carefully checked all reference entries and will correct these formatting issues in the revised version.\n- **Minor Issue 5. Invalid code repository link:** We will remove the incorrect link and ensure only the valid repository is listed in the Checklist and manuscript.\n\n***\nWe sincerely appreciate your valuable suggestions and constructive feedback. We will incorporate the requested clarifications and additional results in the revised manuscript. We hope that our responses and the improvements made will allow you to reconsider the significance of our contributions. Thank you again for your time and thoughtful review."}, {"review_id": "rLPEm3kPEO", "reviewer": "Reviewer_1upL", "summary": "This paper introduces GeoMix, a novel geometric framework for predicting properties of molecular mixtures, specifically focusing on electrolyte conductivity. The authors address two key challenges: (1) the lack of standardized benchmarks in the field, and (2) inadequate modeling of geometric interactions between molecules in mixture systems. They curate two datasets (CALiSol and DiffMix) and propose a Geometric Interaction Network (GIN) that maintains Set-SE(3) equivariance while enabling fine-grained intermolecular geometric message passing through learnable coordinate transformations.", "questions": "1. Why use PCA for frame construction? Have you tried other methods (e.g., based on molecular moments of inertia)? How does the model behave for e.g. spherical molecules (PC) where PCA axes are arbitrary? \n2. Could you visualize some learned transformation matrices to see if they capture meaningful patterns? Does those transformations correlate with MD‑derived relative orientations?\n3. Can the model extrapolate to mixtures with more components than training data?\n4. Can you report “train on DiffMix, test on CALiSol” and vice‑versa, or a leave‑one‑solvent‑family‑out split?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength**\n\n1. **Benchmark Standardization**: The effort to curate and standardize the CALiSol and DiffMix datasets is commendable and provides significant value to the community. The careful preprocessing with geometric graph construction addresses a real need in the field.\n\n2. **Novel Geometric Framework**: The learnable transformation approach between molecular coordinate systems is innovative. The mathematical formulation of Set-SE(3) equivariance for mixture systems is rigorous and well-motivated.\n\n3. **Strong Empirical Results**: GeoMix consistently outperforms diverse baselines, achieving significant improvements (e.g., MSE of 0.432 vs 1.107 on CALiSol).\n\n4. **Extensive Comparison on Learnable Transformation** Comparison with different learnable transformation (quaternions etc.) is interesting and commendable.\n\n5. **Thorough ablations & theory appendix**: Clarifies why non‑orthogonal matrices increase expressivity and proves equivariance.\n\n**Weakness**\n1. **Limited Physical Interpretability**:  The physical meaning of  the learnable transformations is unclear. The paper doesn't discuss what these transformations represent  in terms of actual molecular interactions.\n\n2. **Computational Complexity**: The method requires computing transformations for all atom pairs across molecules, which could be computationally expensive for large systems. The paper doesn't discuss scalability or computational costs. Furthermore, no runtime analysis or comparison with baselines.\n\n3. **Dataset Limitations**: Generalisation outside Li‑ion solvents untested. Both datasets are Li‑salt carbonates/ethers; performance on, e.g., Na‑ion or ionic‑liquid systems is unknown.", "was_revised": true, "final_justification": "Most of my concerns have been addressed, and the paper is well-written. However, given the limited technical novelty—the architecture is not new, and neither is the dataset—I would assign this paper a borderline accept score.\nTo strengthen the paper, I encourage the authors to include a visualization of the learned transformations, which represents one of the few technical novelties in this work. Additionally, the dataset limitations remain unaddressed. I suggest the authors include more diverse systems to substantiate their claim that \"the GeoMix framework is designed to be chemically general and data-driven\"", "rebuttal": "Thank you very much for your valuable and constructive feedback. We greatly appreciate your careful reading, positive remarks, and thoughtful questions. Please find our detailed responses below:\n\n> **R1 (W1 & Q2). Limited physical interpretability & visualization.**\n\nThank you for your thoughtful question about the physical meaning of our model's learnable transformations. We appreciate the opportunity to better clarify their connection to molecular interactions.\n\nOur goal of introducing these transformation matrices and translation vectors is to model the relative geometry between interacting molecules in a flexible, data-driven way. This design reflects the fact that real electrolyte interactions depend on spatial arrangement, orientation, and solvation effects, which are often more complex than rigid SO(3) rotations. Allowing more general transformations lets the model learn subtle geometric patterns—such as solvation shell adjustment and local flexibility—that rigid constraints might miss. This is supported by our ablation studies (Section 4.4), which show that strictly enforcing SO(3) rotation reduces prediction performance.\n\nWe agree that direct visualization of the learned transformations would enhance physical interpretability, especially in revealing correlations with molecular dynamics (MD)-derived orientations. While the current NeurIPS format limits the inclusion of visualizations during the discussion, we plan to incorporate such analysis in the revised version.\n\n> **R2 (W2). Computational complexity.**\n\nThank you for raising the important point regarding computational complexity and scalability. You are correct that our method computes interactions between all intermolecular atom pairs, resulting in a complexity of $\\mathcal{O}(N^2)$, where $N$ is the total number of atoms from one copy of each component molecule. In our datasets, $N$ is generally below 100, and this approach enables the model to capture detailed pairwise interactions crucial for accurate property prediction in molecular mixtures.\nAs you suggested, we have added a runtime and memory analysis (see **Table S1**) comparing our method with several baselines on the CALiSol dataset. The results show:\n- The inference time of GeoMix is comparable to that of baseline models with attention-based encoders (e.g., EGNN_atten, TFN_atten). In these models, the attention mechanism, rather than our interaction module, is typically the main contributor to the runtime cost.\n- GeoMix’s memory usage is higher due to modeling all pairwise atom-level edges across molecules, but this overhead correlates with a clear improvement in predictive performance.\n\nWe hope this additional analysis clarifies the scalability of our method. We believe the observed performance improvements justify the higher computational costs for most practical scenarios involving moderately sized molecular systems.\n\n**Table S1: Runtime analysis on CALiSol dataset.**\n|Models|Inference Time ($\\times 10^{-2}$s)|Memory (GB)|MSE|Pearson r|\n|--|--|--|--|--|\n|EGNN_linear|4.41|2.20|1.461|0.951|\n|TFN_linear|24.13|5.91|1.107|0.967|\n|EGNN_atten|23.40|2.22|2.666|0.908|\n|TFN_atten|43.20|5.91|1.808|0.946|\n|GeoMix_EGNN|34.04|25.15|0.552|0.985|\n|GeoMix_TFN|53.64|45.18|**0.432**|**0.987**|\n\n> **R3 (W3). Dataset limitations.**\n\nThank you for raising this important point regarding dataset diversity and generalization. We agree that evaluating model performance on other electrolyte systems, such as Na-ion or ionic liquids, would be valuable for understanding broader applicability.\n\nWhile our current study is limited to Li-salt electrolytes (carbonates and ethers), we would like to emphasize that the GeoMix framework is designed to be chemically general and data-driven. It operates on atom-level features and learns intermolecular geometric interactions via equivariant message passing, which should, in principle, capture fundamental physical interactions relevant across diverse electrolyte chemistries.\n\nGeoMix can be readily adapted to systems containing alternative cations (e.g., Na⁺, K⁺, or organic ions) or novel solvents, provided that datasets include consistent conductivity measurements and 3D molecular structures. Even for ionic liquids with more complex ion packing, the model's representational capability remains valid.\n\nHowever, we note that large-scale, well-annotated datasets for non-Li electrolytes with the necessary structural information are currently unavailable to our knowledge. We have explicitly acknowledged this limitation in the revised manuscript and hope our work will stimulate future data collection and benchmarking efforts for broader electrolyte chemistries.\n\n> **R4 (Q1). PCA frame constructions.**\n\nThank you for your thoughtful question regarding our use of PCA for molecular frame construction. In response to your suggestion, we have evaluated an alternative based on the Molecular Moment of Inertia (MMI) tensor, where principal axes are defined as eigenvectors of the inertia matrix. We have now included this variant in our ablation study. The results are as follows:\n\n**Table S2: Results of ablation study about frame constructions.**\n\n|Models|MSE|Pearson r|\n|--|--|--|\n|GeoMix_EGNN (PCA Frame)|0.552|0.985|\n|GeoMix_EGNN (MMI Frame)|0.542|0.984|\n\nAs shown, both methods yield nearly identical performance, suggesting that the model is largely insensitive to the specific frame construction method. This is expected since PCA and inertia-based axes are both computed through spectral decomposition (e.g., SVD) of second-order symmetric tensors (covariance matrix or inertia tensor), and therefore often produce similar or equivalent orthonormal bases for molecular structures.\n\nIt is also important that our model treats molecular frames as fixed inputs determined prior to training. These frames are not learnable parameters and do not depend on model weights or optimization. The model's predictions are robust to the choice of frame construction method, as long as the method is deterministic and consistently applied, due to the equivariant nature of the geometric message-passing operators.\n\nFurthermore, molecules with high symmetry (e.g., spherical molecules) do not affect the behavior of our model. Although the selection of the local coordinate system may be arbitrary, all geometric interactions are computed within a standardized coordinate system, where the transformations $\\boldsymbol{R}\\_{ij}^{m,n},\\boldsymbol{t}\\_{ij}^{m,n}$ are invariant to the selected local frame. In practice, perfectly symmetric molecules are extremely rare except for isolated atoms.\n\nWe hope this clarifies the rationale and robustness of our frame construction strategy. We will include a brief discussion and the additional experimental result in the revised version.\n\n> **R5 (Q3 & Q4). OOD evaluation of different dataset splitations.**\n\nThank you for these valuable questions on the model's ability to extrapolate to mixtures with more components or unseen solvent families, and on cross-dataset generalization.\n\nWhile we have not yet performed the exact \"train on DiffMix, test on CALiSol\" experiments, we have designed a related experiment to directly probe the model's extrapolation to new mixture regimes. Specifically, we trained the model only on samples with conductivity ≤10 mS/cm (80/20 train/validation split), while all high-conductivity samples (>10 mS/cm) were reserved for testing. This protocol compels the model to predict for mixtures and composition spaces not seen during training, including those likely to contain more components, novel combinations, or distinct solvent–electrolyte behaviors.\n\nWe believe this conductivity-based split is a stringent test of the model’s extrapolation to novel mixture compositions and more diverse regimes, beyond mere memorization of training patterns.\n\n**Table S3: Results of OOD evaluation on high-conductivity test set.**\n\n|Models|MSE|MAE|Pearson r|Spearman r|\n|--|--|--|--|--|\n|EGNN_linear|32.72010|4.45704|0.25309|0.34116|\n|TFN_linear|24.21760|3.58482|0.39704|0.50828|\n|GeoMix_EGNN|**17.13204**|**2.85305**|**0.57858**|0.57091|\n|GeoMix_TFN|19.42731|2.92452|0.43568|**0.60949**|\n\nThe results (see **Table S3**) demonstrate that GeoMix, especially with the EGNN encoder, consistently outperforms baseline methods—even under this challenging setting.\n\n***\n\nThank you again for your comprehensive review and high-level comments, which have significantly helped us to improve the clarity, rigor, and impact of our work. We will incorporate all clarifications, additional experiments, and references as discussed in the revised manuscript."}, {"review_id": "DzGTqn0pFa", "reviewer": "Reviewer_FnmS", "summary": "This paper introduces GeoMix, a geometry-aware framework for predicting ionic conductivity in multi-molecule electrolyte mixtures. GeoMix first augments two public datasets (CALiSol and DiffMix) with geometric graphs, then couples an intra-molecular equivariant backbone (EGNN or TFN) with a novel Geometric Interaction Network (GIN) that passes messages between molecules through learned rotation/translation operators that respect Set-SE(3) symmetry. Across both datasets, GeoMix-TFN reduces test-set MSE by a large margin relative to the strongest geometric baselines, establishing new state-of-the-art performance.", "questions": "1. \"For molecules not found in PubChem, we generate 3D structures 246 using RDKit [29] or download conformations from the Materials Project [30].\" -- How do you handle cases where a single molecule has multiple low-energy conformers? Do you average over them, select a representative structure, or treat each conformer as a separate training sample?\n\n2. How does GeoMix scale with mixture size M and total atoms N in practice (time/GB)? What is the asymptotic time/space complexity of the GIN with respect to M and N?\n\n3. In Eq. (3) you state that the symmetry is defined such that each molecule is independently SE(3)-equivariant. Could you clarify why independent equivariance is reasonable? In realistic systems, inter-molecular interactions can couple the components.", "limitations": "Yes, the authors have discussed their limitations.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n+ Strong theoretical underpinning and clear derivations.\n+ Large performance gains on two datasets with code released.\n+ Careful ablations isolate the contribution of each design choice.\n\nWeaknesses\n+ **Dataset Curation and Conformational Noise**. One potential concern is the dataset curation step, which may introduce conformational noise, especially when molecules with multiple low-energy conformers are considered. The paper does not discuss how multiple conformers are treated—whether they are averaged, a representative structure is selected, or each conformer is treated as a separate training sample. Does this affect the quality of the dataset?\n+ **Complexity analysis absent.** The paper provides an in-depth explanation of the network architecture, but there is no discussion of the computational complexity of the GeoMix framework in terms of time and memory consumption. With potentially high-order backbone networks, what is the complexity of the overall proposed method?", "was_revised": true, "final_justification": "The authors' response has satisfactorily addressed my concerns. I believe this work is significant for the NIPS conference and the entire field. I have raised my score accordingly.", "rebuttal": "Thank you very much for your positive feedback and thoughtful, detailed comments. We greatly appreciate your careful reading and constructive suggestions, which will help us improve both the clarity and impact of our paper. We summarize and respond to your questions as follows:\n\n> **R1 (W1 & Q1). Dataset curation and conformational noise.**\n\nThank you for raising this important point regarding conformational noise and dataset curation. In our current pipeline, we assign a single representative 3D structure to each molecule: if available, we retrieve a 3D conformation from PubChem or the Materials Project and optimize it using RDKit’s force field; otherwise, we generate and optimize the structure directly with RDKit. This approach ensures consistency across all data samples. Selecting a single conformer is a common convention in molecular machine learning, but our framework is compatible with alternative protocols, such as including multiple conformers as independent samples, should such data become available in the future.\n\nHere, to better evaluate model robustness to conformational variability, we have conducted an additional experiment: we apply Gaussian perturbations ($\\sigma=0.3$ Å) to atomic positions, simulating conformational diversity. As shown in **Table S1** below, most models—including GeoMix_EGNN—exhibited only a marginal performance drop, indicating stability under moderate geometric deviations. Notably, the TFN encoder derives much worse performance, likely due to its reliance on higher-degree equivariant features (e.g., spherical harmonics) that could be sensitive to atomic position perturbations. We will make this limitation explicit in the revised manuscript.\n\n**Table S1: Results of coordinate perturbation test.**\n|Models|MSE|MAE|Pearson r|Spearman r|\n|--|--|--|--|--|\n|MolSets_conv|1.32605|0.70309|0.95992|0.96488|\n|MolSets_SAGE|1.74835|0.79402|0.94599|0.95805|\n|EGNN_atten|1.91948|0.58744|0.94193|0.96495|\n|TFN_atten|2.29420|0.75980|0.92930|0.95379|\n|EGNN_linear|1.11415|0.55469|0.96660|0.97623|\n|TFN_linear|2.39326|0.93700|0.92892|0.94426|\n|GeoMix_EGNN|**0.63117**|**0.38836**|**0.98174**|**0.98348**|\n|GeoMix_TFN|4.51728|1.32373|0.85851|0.88631|\n\nWe appreciate your suggestion and will include these clarifications and perturbation results in the final version.\n\n> **R2 (W2 & Q2). Complexity analysis absent.**\n\nThank you for raising the important issue of computational complexity and scalability.\n\nThe main computational cost in GeoMix comes from two sources: (1) the molecular encoder, which processes individual molecules, and (2) the cross-molecular interaction module (GIN), which performs global message passing between all atoms across different molecules. For a mixture containing a total of $N$ atoms (typically fewer than 100 per sample), these components scale as follows:\n- **Molecular Encoder:** For EGNN, the per-layer time complexity is $\\mathcal{O}(N^2)$. For TFN, the per-layer complexity rises to $\\mathcal{O}(N^2\\cdot L^6)$ with $L$ being the maximum tensor order used, due to the involvement of higher-degree features.\n- **GIN Module:** The cross-molecular interaction is modeled as a fully connected bipartite graph (across molecular pairs), also resulting in an $\\mathcal{O}(N^2)$ time and space complexity per layer, typically dominating total costs.\n\nIn practice, as $N$ is usually modest, the computational cost remains tractable even for deeper networks or high-degree backbones. Although our method scales quadratically with the number of atoms, it remains significantly more efficient than traditional molecular dynamics simulations. \n\nWe have also benchmarked inference time and memory usage on the CALiSol dataset. Results are shown in **Table S2** below. \n- The inference time of GeoMix is comparable to that of baseline models with attention-based encoders (e.g., EGNN_atten, TFN_atten). In these models, the attention mechanism, rather than our interaction module, is typically the main contributor to the runtime cost.\n- GeoMix’s memory usage is higher due to modeling all pairwise atom-level edges across molecules, but this overhead correlates with a clear improvement in predictive performance.\n\n**Table S2: Runtime analysis on CALiSol dataset.**\n\n|Models|Inference Time ($\\times 10^{-2}$s)|Memory (GB)|MSE|Pearson r|\n|--|--|--|--|--|\n|EGNN_linear|4.41|2.20|1.461|0.951|\n|TFN_linear|24.13|5.91|1.107|0.967|\n|EGNN_atten|23.40|2.22|2.666|0.908|\n|TFN_atten|43.20|5.91|1.808|0.946|\n|GeoMix_EGNN|34.04|25.15|0.552|0.985|\n|GeoMix_TFN|53.64|45.18|**0.432**|**0.987**|\n\n> **R3 (Q3). Rationality of independent SE(3)-equivariant.**\n\nThank you for this insightful question regarding independent SE(3)-equivariance.\n\nOur statement refers specifically to the initial geometric encoding stage, where each molecule is provided as an independent input with its own atomic coordinates and frame of reference. This setup makes it mathematically valid to apply arbitrary rigid transformations to each molecule separately, justifying independent equivariant embeddings at this stage.\n\nHowever, our model explicitly accounts for intermolecular interactions in subsequent layers via the GIN module, which passes features and enables attention-based aggregation across molecules. These steps allow the model to capture coupling effects and realistic intermolecular behaviors, while still respecting fundamental symmetries.\n\nWe also acknowledge the limitation that our model assumes molecules are not co-embedded in a shared spatial configuration—a reflection of experimental constraints, since such atomistic details are rarely available for mixtures in solution. Nonetheless, our design balances physical principles with practical considerations, ensuring that meaningful coupling can be learned even with independent geometric encodings. We will clarify this rationale in the revised manuscript.\n\n***\n\nThank you once again for your constructive feedback, which has directly contributed to improving the clarity, empirical evaluation, and theoretical discussion of our work. We will incorporate all clarifications and additional results into the revised version of the manuscript. We sincerely hope that our responses and revisions will address your concerns, and we would be grateful if you might consider these improvements in your evaluation."}, {"review_id": "hEPPMJUVrw", "reviewer": "Reviewer_Syf6", "summary": "The paper proposes GeoMix, a novel geometry-aware framework that preserves Set-SE(3) equivariance for accurate prediction of ionic conductivity in mixture electrolyte systems. The authors curate and standardize data from two existing works, CALiSol and DiffMix, and augment the molecular representation by constructing geometric graphs where atoms are characterized by both invariant features and equivariant features. Then the authors develop GeoMix which maintains Set-SE(3) equivariance while capturing fine-grained geometric relationships between molecules, overcoming the limitations of existing methods. The extensive experimental results demonstrate the superior performance of GeoMix over baseline models, demonstrating its effectiveness in modeling mixture systems.", "questions": "1. GeoMix reports mean squared error (MSE = 0.035), while DiffMix reports mean absolute error (MAE = 0.044), which corresponds to an estimated MSE of around 0.002 by a rough estimation. While both the two paper present near perfect regression based on the true values vs predicted values plot, could this be due to differences in normalization, target scaling, or task formulation, and how should we interpret these metrics fairly across the two papers?\n\n1. The GeoMix paper uses a different data split strategy from the original DiffMix paper, despite DiffMix providing the random seed for reproducibility. Therefore, is any reason for not using the same data split as DiffMix which could have been included as a baseline model?\n\n1. GeoMix generates mixture geometries by randomly translating and rotating molecular conformers, forming mixture-level graphs from a single such placement. Is there any investigation in how sensitive the model is to this sampling strategy?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The paper addresses an important and underexplored problem in machine learning for chemistry: predicting electrolyte properties from complex mixtures with geometric and compositional diversity. By introducing a mixture-level graph that includes atomic interactions across multiple molecules, the authors move beyond traditional molecular featurization approaches that treat molecules as independent graphs.\n\n1. The paper introduces Geometric Interaction Network (GIN) which models the molecular interactions by intermolecular geometric message passing, surpassing previous methods which only learn a global feature vector of molecules while overlooking fine-grained and full-atom interactions across molecules. \n\n1. The denoise loss adds a regularization that encourages robustness to small geometric perturbations, which is especially relevant in modeling systems with inherent thermal fluctuations and conformational noise.\n\nWeaknesses:\n1. Unlike DiffMix, which fits a VFT equation to enforce physically consistent trends across temperature, GeoMix treats temperature as a scalar input without enforcing any physical constraints. As a result, predictions at different temperatures may be inconsistent. \n\n1. GeoMix constructs a mixture-level graph by placing individual molecules randomly in 3D space and connecting atoms based on proximity. However, the relative positions of molecules in a mixture are governed by thermodynamic distributions and local structuring (e.g., solvation shells), which are not captured by this random placement. Without sampling from realistic ensembles, such as those obtained via molecular dynamics, the constructed graphs may not reflect statistically meaningful intermolecular interactions.", "was_revised": true, "final_justification": "After discussion with the authors, their rebuttal has resovled most of my concerns, especially the good generalization performance to temperatures. Meanwhile, they also acknowledged the importance for more physics-consistent data sampling strategies, and they explained why a direct comparison to DiffMix is difficult. Therefore, I raised my score accordingly.", "rebuttal": "Thank you very much for your insightful and constructive feedback. We sincerely appreciate your careful review and the valuable suggestions you have provided. Below, we address your comments point by point.\n\n> **R1 (W1). Generalization at different temperatures.**\n\nWe appreciate your insightful comment. Indeed, GeoMix does not enforce explicit physical constraints such as the VFT or Arrhenius equations during training. These constraints are typically adopted when the goal is to extrapolate conductivity at room temperature from high-temperature measurements, particularly in the context of molecular dynamics simulations. In contrast, our framework is designed as a general-purpose model applicable to various downstream tasks and properties, where hard-coding temperature-dependent functional forms may reduce flexibility or limit generalization.\n\nThat said, we recognize the importance of ensuring physical consistency with respect to temperature. To empirically assess this, we performed an OOD generalization experiment where we trained the model on data with temperature ≤ 320 K (using a train/valid split of 4:1), and evaluated on samples with temperature > 320 K. Results show that GeoMix achieves significantly better predictive performance on this challenging temperature extrapolation task compared to other baseline models, suggesting that the learned representations effectively capture physically meaningful temperature trends without requiring hard-coded constraints.\n\n**Table S1: Results of OOD evaluation on high temperature.**\n\n|Models|MSE|MAE|Pearson r|Spearman r|\n|--|--|--|--|--|\n|MLP|32.43188|3.85718|0.43183|0.55550|\n|MM_MoLFormer|22.93524|2.83730|0.51219|0.56041|\n|MolSets_conv|9.92957|2.19592|0.83933|0.84519|\n|MolSets_SAGE|9.19274|2.13546|0.78423|0.80184|\n|EGNN_atten|8.13370|1.59404|0.81299|0.86055|\n|TFN_atten|6.51550|1.38267|0.85262|0.88674|\n|EGNN_linear|7.67502|1.61563|0.82692|0.84713|\n|TFN_linear|4.77976|1.28618|0.91367|0.92965|\n|GeoMix_EGNN|2.36597|**0.88293**|0.94950|**0.94805**|\n|GeoMix_TFN|**2.35390**|0.91722|**0.95186**|0.94535|\n\n> **R2 (W2). Geometric interactions without MD simulations.**\n\nThank you for this important observation. We fully recognize that Molecular Dynamics (MD) simulations offer physically realistic ensembles that capture crucial thermodynamic distributions and local structuring, such as solvation shells. However, generating large, diverse MD-based datasets is highly challenging due to substantial computational costs and the scarcity of robust force fields for complex electrolyte mixtures. A key motivation of our work is to develop a learning-based alternative capable of capturing meaningful intermolecular interactions without relying on resource-intensive MD simulations.\n\nAlthough our graph construction does not impose physical constraints on initial molecular placement, the model architecture is specifically designed to recover relevant spatial relationships from the data. The Geometry Interaction Network (GIN) enables atomic-level message passing across molecules, allowing the model to learn interaction patterns reminiscent of those observed in MD ensembles. The learnable transformation matrices for each atom pair further serve as flexible approximations of local rearrangements, such as solvation and steric effects. Additionally, the use of Noisy Nodes introduces a form of structural noise, which encourages robustness and helps the model generalize to the kinds of fluctuations typically seen in MD simulations.\n\nWe agree that integrating more physically realistic configurations would further enhance model fidelity, and as part of future work, we plan to explore incorporating MD-derived structures or Radial Distribution Functions (RDFs), potentially as auxiliary supervision. This potential extension will be discussed in the revised manuscript.\n\n> **R3 (Q1 & Q2). Fairness compared to DiffMix.**\n\nThank you very much for your insightful comments regarding the comparison between GeoMix and DiffMix, especially in relation to evaluation metrics and dataset split strategies.\n\nFirst, we would like to clarify that a direct, numerical comparison of performance between GeoMix and DiffMix is not straightforward, primarily due to fundamental differences in task formulation and dataset construction. Specifically, the two works differ in the distributions of target values, normalization procedures, and data split schemes. Additionally, DiffMix incorporates supplementary features such as excess molar enthalpy and volume—features that are not used in GeoMix—which may contribute to their reported regression performance.\n\nSecond, with respect to dataset splits, although the DiffMix repository provides an overall 8:1:1 split ratio (train:validation:test), it does not release the specific split indices or the individual 3D molecular structures used. As a result, it is challenging to reproduce their data partitions and baseline models without access to these intermediate processed datasets. In contrast, our study employs a consistent and standardized 7:2:1 split for both the CALiSol-23 and DiffMix datasets, facilitating fair within-study comparison and robust cross-dataset validation. We believe this approach ensures methodological consistency throughout our work.\n\nFor these reasons, metric-to-metric comparisons are inherently limited due to disparities in data preprocessing, feature design, and splitting strategies. Moving forward, we support community-driven efforts to unify benchmark standards and facilitate the sharing of intermediate representations, which will enable more reliable and consistent comparisons across different models.\n\nThank you again for highlighting these important considerations.\n\n> **R4 (Q3). Misunderstanding of random translation and rotation.**\n\nThank you for highlighting this important aspect. We apologize for the misunderstanding and appreciate the opportunity to clarify this point. Our approach does NOT involve applying any random translations or rotations to molecular conformers. Instead, each molecule is first transformed into a canonical coordinate frame using PCA, after which the hybrid-level geometry is deterministically constructed. Additionally, our model is SE(3)-equivariant with respect to the input molecular conformers. As a result, the outputs are insensitive to random translations and rotations, as mentioned in your question.\n\n***\n\nThank you again for your thoughtful and detailed review. We greatly value your feedback and will address all points raised by incorporating clarifications and new analyses into the revised manuscript. We hope these efforts demonstrate the strength of our work and that you will reassess our contributions."}]

Author rebuttal:

Meta-review: Reviews of this work are positive, with reviewers praising the combination of data-centric (improved curation of CALiSol and DiffMix with geometric properties) and methodological contributions (various ways of incorporating both non-spatial-type molecular features such as atomic numbers, and spatial-type features) which build on the data made available by the authors. Different reviewers focused on different parts of these contributions (Reviewer x1YU and Reviewer Syf6 on the methodological side, Reviewer 1upL and Reviewer FnmS on the data side), but unanimously found the problem to be important and evaluation to be reasonably comprehensive. Most weaknesses raised had to do with computational costs of the methodological work not being discussed in enough detail, which the authors note are similar to other equivariant networks. 

On basis of these reviews, I find the work has merit, and recommend acceptance.
Decision: Accept (poster)