Geometric GNNs: Modeling with Geometry

• Geometric GNNs are advanced neural architectures that integrate graph topology with geometric features like distances and angles to model spatial structure.
• They extend traditional message passing by enforcing invariance and equivariance under group symmetries, ensuring robust performance under transformations.
• These models excel in molecular, materials, and 3D vision tasks, yet face challenges in scalability, expressivity, and adaptive depth.

A Geometric Graph Neural Network (Geometric GNN, sometimes "Geo-GNN") is a graph neural architecture that models not only the graph topology but also node, edge, or higher-order features living in a geometric (often Euclidean or Riemannian manifold) space. Such models systematically encode, propagate, and aggregate information in a manner that leverages the geometry underlying the data, including symmetries, distances, angles, curvatures, and metric constraints. Geometric GNNs are foundational for learning on scientific, molecular, physical, material, and 3D geometric datasets where spatial structure, physical invariance, or equivariance to rigid motions are essential.

1. Geometric Data Structures and Symmetry Foundations

A geometric graph is a tuple $$\mathcal G = (V, E, H, X, E_\text{attr}, G_\text{attr})$$ with nodes $V$, edges $E$, invariant node features $H$, coordinate features $X = \{x_i \in \mathbb{R}^d\}_{i=1}^N$ (embedding nodes in physical or latent space), typically invariant or relative-geometric edge attributes $E_\text{attr}$, and optional global features $G_\text{attr}$ (Han et al., 1 Mar 2024).

The relevant symmetry group $\mathcal G$ (e.g., $\mathrm{E}(d)$ for Euclidean isometries, $\mathrm{SE}(3)$ for rigid transformations, or the full conformal group) acts on coordinates and features: $$g \cdot x_i = R x_i + t,\quad (R \in O(d),\ t \in \mathbb{R}^d)$$ A geometric GNN is called $\mathcal G$-invariant if its output is invariant to $\mathcal G$; it is $\mathcal G$-equivariant if its output transforms under the group in concert with the input. These properties ensure that models respect physical symmetries and produce consistent outputs under coordinate changes (Han et al., 2022).

2. Core Modeling Paradigms and Architectures

2.1. Message Passing with Geometric Awareness

Geometric GNNs extend classical message passing by incorporating geometric inputs (coordinates, angles, distances, curvatures) in the message and update functions: $$\begin{aligned} m_{ij} &= \phi_\text{m}(h_i, h_j, x_i, x_j, e_{ij}) \ h_i' &= \psi_\text{h}\left(h_i,\,\sum_{j \in N(i)} m_{ij}\right) \ x_i' &= \psi_\text{v}\left(x_i,\,\sum_{j \in N(i)} \mathbf{m}_{ij}\right) \end{aligned}$$ Messages can be scalars, vectors, or higher-order tensors, and must be constructed to respect group symmetry (permutation invariance/equivariance and, if required, geometric equivariance under $\mathcal G$) (Han et al., 2022, Han et al., 1 Mar 2024).

2.2. Permutation Invariant and Equivariant Aggregation

Architectures differ in how they implement geometric message passing:

• Invariant models: Aggregate scalar geometric features such as distances and angles, providing rotational, reflectional, and translational invariance but generally limited expressiveness for non-local geometric properties (Joshi et al., 2023).
• Equivariant models: Carry vector and tensor features, with updates designed to be equivariant under physical groups, using steerable bases (e.g., spherical harmonics, Clebsch–Gordan products) to achieve higher expressiveness for spatial patterns (Han et al., 2022).

Representative classes include EGNN (scalarization style, E(n)-equivariant), SE(3)-Transformer (irreducible representation style), SchNet/DimeNet (invariant, distance/angle-aware filters), and higher-order gauge equivariant networks.

2.3. Explicit Geometric Modules

• Latent-space bi-level aggregation: For example, Geom-GCN introduces node embeddings into $\mathbb{R}^d$ or hyperbolic space using Isomap, Poincaré, or struc2vec methods, then builds dual structural neighborhoods and discrete geometric relationships, applying permutation-invariant, multi-level aggregation over these (Pei et al., 2020).
• Curvature modeling: Bakry–Émery curvature provides a local differential-geometric summary of graph neighborhoods, enabling models such as Depth-Adaptive GNNs to adapt message passing depth per node based on estimated diffusion geometry (Hevapathige et al., 3 Mar 2025).
• Distance geometry: MGNN incorporates an explicit metric matrix and an energy functional inspired by the Distance Geometry Problem (DGP), treating edges as springs and propagating embeddings via iterative geometric stress minimization, which handles both homophilic and heterophilic structure (Cui et al., 2022).
• Geometric scattering transforms: Models like GeoScatt-GNN extract stable, multi-scale scattering coefficients using graph wavelets or spectral filters, then inject them into GNN layers for hybrid learning (Zoubir et al., 22 Nov 2024).
• Kolmogorov–Arnold Networks: KA-GNN replaces standard MLPs with Fourier-expandable functional bases, enabling highly expressive, geometry-adaptive nonlinear transformation at all network levels (Li et al., 15 Oct 2024).

3. Expressive Power, Invariance/Equivariance, and Theoretical Guarantees

The geometric Weisfeiler–Leman (GWL) test (Joshi et al., 2023) formalizes the expressivity of geometric GNNs, relating them to universal function approximation over geometric graphs invariant or equivariant to permutations and physical groups. Key results:

• Invariant GNNs (distance/angle-based) cannot distinguish “1-hop identical” graphs—those with identical local point clouds up to isometry. Non-local tasks (e.g., perimeter, centroid distance, dihedral angles) are inexpressible.
• Equivariant GNNs (carrying vector/tensor features) propagate orientation and can distinguish a strictly larger class, up to the distinguishability of the full GWL procedure. Sufficient layer depth and tensor order are required for certain hard cases (e.g., symmetric molecule configurations).
• Higher-order aggregation (3-body, 4-body...) increases power, as shown in systematic counterexamples where body order limits discrimination capability (Joshi et al., 2023).

Generalization on geometric graphs over manifolds is governed by convergence rates that depend polynomially on the number of sampled points and exponentially on the intrinsic dimension, with permutation and Lipschitz constraints inherited from the manifold setting (Wang et al., 8 Sep 2024).

4. Empirical Performance and Applications

Geometric GNNs have demonstrated state-of-the-art or highly competitive results across a wide spectrum of domains:

Application Area	Representative Models	Benchmark Datasets	Key Outcomes
Molecular property	KA-GNN, DimeNet, GeoScatt-GNN	MoleculeNet (BACE, Tox21, etc.)	KA-GNN exceeds other GNNs on ROC-AUC and speed (Li et al., 15 Oct 2024). Multiscale hybrid (GeoScatt-GNN+GIN) achieves 0.9812 AUC (Zoubir et al., 22 Nov 2024).
Materials, quantum	NequIP, PaiNN, SE(3)-Transformer	QM9, MD17	Equivariant models reduce MAE by up to 50% (Han et al., 2022).
Protein/RNA	SE(3)-T, TFN, EGNN	CASP, AlphaFoldDB	Capture folding, flexible docking with high fidelity.
3D/vision	EGNN, AdS-GNN	ModelNet40, ShapeNet	AdS-GNN robust to scaling/conformal deformations (Zhdanov et al., 19 May 2025).
Complex graphs	Geom-GCN, MGNN	Assortative/disassortative transductive datasets	Geom-GCN gains up to +18% test accuracy on WebKB/Chameleon (Pei et al., 2020). MGNN wins on both homophilic and heterophilic regimes (Cui et al., 2022).
Glassy/physical	Geo-GNN	Glass simulation, robotics	Angle- and triplet-aware encoding critical for high-frequency/rough signals (Jiang et al., 2022).

Auxiliary strengths include adaptive message depth (Bakry–Émery curvature), robustness to mesh noise (GeGnn, (Pang et al., 2023)), and energy-efficient computation via spiking and manifold embeddings (Geometry-Aware Spiking GNN (Zhang et al., 9 Aug 2025)).

5. Key Challenges, Limitations, and Open Directions

Computational Cost and Scalability

• Irrep-based equivariant models (e.g., Tensor Field Networks) incur cubic cost in tensor order/channel width, which restricts stacking depth or size (Han et al., 2022).
• Scaling equivariant GNNs to 10⁵–10⁶ nodes remains an unsolved practical problem, particularly for molecular assemblies or large biomolecular complexes.

Expressivity and Universality

• The range of geometric functions covered by message-passing GNNs versus all possible group-equivariant mappings remains an area of active research, with expressivity guaranteed only up to the power of GWL or higher-order k-body features (Joshi et al., 2023).
• Many invariant models oversmooth or lose fine geometric discrimination, especially for long-range dependencies or localized defects.

Foundations, Data, and Practical Modeling

• Curse of dimensionality: Generalization gaps scale as $N^{-1/(d+4)}$, where $d$ is the manifold dimension; practical application thus relies on low-dimensional geometric priors (Wang et al., 8 Sep 2024).
• Manifold/topology mismatches can cause failure on graphs with geometry unlike that seen in training (GeGnn, (Pang et al., 2023)).
• Group generalization: Extension beyond Euclidean/isometry groups (e.g., to conformal, projective, or discrete symmetries) is under development.

Future Research Directions

• Efficient equivariant architectures: Multipole-based sparsification, scalable steerable kernels, hybrid equivariant-invariant networks (Han et al., 2022).
• Higher order, learned bases, and hybrid geometric modules: Fourier/Wavelet KANs, learnable scattering features, chart-wise manifold GNNs (Li et al., 15 Oct 2024, Zoubir et al., 22 Nov 2024).
• Curvature and topology-aware adaptivity: Adaptive depth/message passing, local curvature, or bottleneck detection for robust feature propagation (Hevapathige et al., 3 Mar 2025).
• Integration with language and foundation models: Merging LLM-derived chemistry/biology priors with geometric GNN foundations for science applications (Han et al., 1 Mar 2024).
• Relaxed or learned symmetries: Partial equivariance, learned symmetry discovery, anisotropic or locally adaptive group action (Han et al., 1 Mar 2024, Zhdanov et al., 19 May 2025).

6. Model Families and Illustrative Algorithms

Major Classes of Geometric GNNs

Model	Geometry	Aggregation/Core Operator	Symmetry
SchNet	$\mathbb{R}^3$	RBF on edge length	E(3)-invariant
DimeNet	$\mathbb{R}^3$	Distance + angular Bessel	E(3)-invariant
EGNN	$\mathbb{R}^d$	Scalar and equivariant vector	E(n)-equivariant
SE(3)-Transformer	$\mathbb{R}^3$	Tensor equipartition (sph harmonic)	SE(3)-equivariant
Geom-GCN	latent or embed	Dual neighborhood, bi-level	Inv+permutative
MGNN	metric learning	Spring energy optimization	Congruence-inv.
KA-GNN	graph/molecule	Fourier-KAN functional basis	Perm-inv, potent
Geo-GNN	physical config	Triplet/angle encoding	Rot-inv encoder

Illustrative Aggregation (Geom-GCN)

$$\begin{aligned} m_{(i,r)}^{v,\ell+1} &= p\bigl(\{h_u^\ell : u \in N_i(v),\ \tau(x_v,x_u) = r\}\bigr) \ \tilde{h}_v^{\ell+1} &= q\left( \bigl\{(m_{(i,r)}^{v,\ell+1},(i,r)) : i\in\{g,s\},\ r\in R \bigr\} \right) \ h_v^{\ell+1} &= \sigma\left(W^\ell \tilde{h}_v^{\ell+1}\right) \end{aligned}$$

with permutation invariance, dual neighborhood partition, and bi-level aggregation (Pei et al., 2020).

7. Summary

Geometric Graph Neural Networks generalize message passing to respect manifold structure, metric constraints, group symmetries, and higher-order geometric dependencies. By blending spatial/topological adjacency with explicit geometric constructions—embedding, curvature, distance, angles, scattering, and functional basis expansions—these models realize state-of-the-art performance across scientific, molecular, vision, and physical simulation domains, while revealing persistent challenges in scalability, theoretical universality, and adaptive inductive bias. Future development is converging towards efficient, expressive models that unify explicit geometry, learned bases, adaptive depth, and generalized symmetries for principled machine learning on arbitrary geometric graphs (Pei et al., 2020, Han et al., 2022, Hevapathige et al., 3 Mar 2025, Joshi et al., 2023, Han et al., 1 Mar 2024, Zoubir et al., 22 Nov 2024, Li et al., 15 Oct 2024, Zhdanov et al., 19 May 2025, Jiang et al., 2022, Pang et al., 2023, Zhang et al., 9 Aug 2025, Wang et al., 8 Sep 2024, Cui et al., 2022).