Geometry-Aware Simplicial Message Passing

• The paper introduces a novel geometry-aware simplicial message passing framework that distinguishes different geometric embeddings of identical topologies.
• It integrates vertex coordinate signals into message passing, employing the Geometric Simplicial Weisfeiler–Lehman test to establish provable expressivity bounds.
• Empirical evaluations confirm that geometry-aware SMP outperforms combinatorial methods on mesh classification and regression tasks with superior accuracy.

Geometry-aware simplicial message passing refers to a class of higher-order neural message passing frameworks on simplicial complexes that directly and provably leverage the geometry—specifically, the embedding of vertices—of the underlying domain. Unlike purely combinatorial approaches, these frameworks allow models to distinguish not just abstract connectivity but also different geometric realizations of the same topology. Key advances include the development of expressive invariants (Geometric Simplicial Weisfeiler–Lehman), architectural integration with the Euler Characteristic Transform, new upper/lower bounds on expressivity, and proof-of-concept empirical demonstrations on classification and regression tasks involving meshes and geometric data (Wang et al., 7 May 2026).

1. Geometric Simplicial Message Passing: Definition and Motivation

Classic message passing neural networks for graphs cannot distinguish between isomorphic but differently embedded structures, since their expressive power is fundamentally tied to topology. This limitation extends to the Simplicial Weisfeiler–Lehman (SWL) framework, which is combinatorial and thus blind to the actual embedding of vertices in Euclidean space. To overcome this, geometry-aware simplicial message passing introduces explicit geometric signals—typically vertex coordinates—into the message passing process at the simplicial (higher-dimensional) level. This approach enables discrimination between meshes with identical abstract complexes but different geometric realizations, which is essential for tasks in mesh analysis, physical modeling, and geometric data science (Wang et al., 7 May 2026).

The fundamental objective is a tight characterization of expressivity in the geometry-aware setting: precisely which geometric distinctions can be made at each message passing depth, and how this relates to classical invariants, is made explicit via formal extensions and bounds.

2. The Geometric Simplicial Weisfeiler–Lehman Test (GSWL)

The GSWL test is a generalization of the classical Weisfeiler–Lehman test, incorporating vertex coordinates into the color refinement process used for distinguishing simplicial complexes (Wang et al., 7 May 2026). For a $d$-dimensional simplicial complex with an injective embedding $x: \mathrm{vert}(K) \to \mathbb{R}^d$, GSWL assigns each $k$-simplex $\sigma$ an initial “color” based explicitly on the Euclidean embedding:

• For vertices (0-simplices): $c^{(0)}_{\sigma} = (0, x_v)$.
• For $k$-simplices ($k \geq 1$): $$c^{(0)}_{\sigma} = (k, \Phi_k(x_{v_0},\dots,x_{v_k}))$$, with $\Phi_k$ a fixed, permutation-invariant function, such as centroid or volume.

Color refinement is iterated. At each round, the color of a simplex is updated by hashing together its own current color, the multiset of boundary neighbor colors, and the multiset of coface neighbor colors. After enough rounds, the color of each simplex recovers the (unordered) set of vertex coordinates spanning it, and thus all geometric information necessary for geometric invariants. Two complexes are GSWL-$L$ equivalent if the multisets of all their $x: \mathrm{vert}(K) \to \mathbb{R}^d$0-th round colors coincide.

A crucial lemma establishes that after sufficiently many rounds, the color of each $x: \mathrm{vert}(K) \to \mathbb{R}^d$1-simplex encodes all vertex coordinates and thus determines, for any direction $x: \mathrm{vert}(K) \to \mathbb{R}^d$2, the entry time $x: \mathrm{vert}(K) \to \mathbb{R}^d$3, forming the critical link to geometric invariants such as the Euler Characteristic Transform (Wang et al., 7 May 2026).

3. Expressivity Bounds for Geometry-Aware Simplicial MP

Upper Bound

For any depth-$x: \mathrm{vert}(K) \to \mathbb{R}^d$4 geometry-aware simplicial message passing (SMP) network using per-simplex feature updates of the form

$x: \mathrm{vert}(K) \to \mathbb{R}^d$5

and a final readout $x: \mathrm{vert}(K) \to \mathbb{R}^d$6, the expressive power is upper bounded by the GSWL: if two embedded complexes are GSWL-$x: \mathrm{vert}(K) \to \mathbb{R}^d$7 equivalent, then no choice of network parameters or deterministic readout can distinguish them via $x: \mathrm{vert}(K) \to \mathbb{R}^d$8 (Wang et al., 7 May 2026).

Lower Bound (Finite-Family Realizability)

Given a finite family of embedded complexes and a fixed $x: \mathrm{vert}(K) \to \mathbb{R}^d$9, there exists a choice of network parameters (with suitably large hidden dimension and the finite interpolation property, e.g., with wide enough ReLU MLPs) such that the model matches the GSWL test exactly. Explicitly, one-hot encodings and parameterized transitions can realize the GSWL color refinement process, yielding a perfect correspondence between GSWL equivalence and the learned representations on the finite family (Wang et al., 7 May 2026). This establishes a strong expressivity hierarchy:

Model Class	Expressivity w.r.t. Geometry
Combinatorial SMP	Topology only (SWL/L)
1-skeleton GNN (GCN/GIN)	Node/edge-level geometry
Point-cloud MLP/DeepSets	Geometry, but not higher-simplice-wise
Geometry-aware SMP (GSWL)	Full geometric information

4. Relation to the Euler Characteristic Transform

The Euler Characteristic Transform (ECT) is a complete invariant for geometric simplicial complexes (fixed abstract structure). ECT associates a function $k$0, measuring the Euler characteristic of the sublevel sets defined by projections in various directions:

$k$1

where $k$2.

When combined with GSWL, geometry-aware SMP can recover ECT up to any finite sampling. On compact classes of embeddings, approximation is arbitrarily good by ensuring that vertex coordinates are carried through as skip connections and final representations are read out via sufficiently powerful MLPs, making the method stable with respect to perturbations in vertex positions (Wang et al., 7 May 2026).

This linkage shows that geometry-aware SMP is not only maximally expressive (in the GSWL sense) but is also theoretically complete for tasks involving geometric distinctions in simplicial complexes, up to the resolution dictated by ECT sampling.

5. Practical Implementation and Empirical Hierarchy

Geometry-aware SMP schemes initialize simplex features with geometric quantities, propagate information via boundary and coface relations, and aggregate these using permutation-invariant functions or MLPs. Notable cases include implementations using geometric invariants (volumes, centroids), custom message functions parameterized by geometry, or direct coordinate propagation with skip channels (Wang et al., 7 May 2026).

Empirical studies underline the theoretical predictions:

• On synthetic mesh deformation datasets, combinatorial SMP achieves only chance-level classification ($k$30.24), while geometry-aware SMP achieves perfect accuracy (1.00); MLPs on raw coordinates perform slightly worse (0.987).
• In ECT regression, geometry-aware SMP achieves lower MSE than combinatorial SMP or even direct coordinate MLPs.
• On FAUST meshes (human body pose classification), geometry-aware SMP outperforms DeepSets, GCN+coords, GIN+coords, and combinatorial SMP.
• Downward flow (from higher to lower-dimensional simplices) is necessary for tasks such as curvature estimation, confirmed by a major drop in MSE when the coboundary flow is omitted.
• Experiments on different manifolds and triangulations confirm strict expressivity: only geometry-aware SMP separates deformation classes (Wang et al., 7 May 2026).

6. Connections to Broader Geometry-Aware Frameworks

The GSWL-based geometry-aware SMP can be contrasted with other geometric/topological message passing architectures:

• Clifford group equivariant SMP networks use Clifford algebras to encode steerable, equivariant simplex features, generalizing geometric products and capturing volumes/areas at the algebraic level. These layers commute with orthogonal transformations, and in practice outperform less expressive geometric or simplicial architectures across volume estimation and motion prediction tasks (Liu et al., 2024).
• E(n)-equivariant SMP networks combine simplicial message passing with E(n)-invariant geometric kernels (volumes, distances, angles). These networks, by design, preserve equivariance under orthogonal transformations and improve over Hodge-based or node/edge-only approaches, especially by reducing over-smoothing in higher-order structures (Eijkelboom et al., 2023).
• Sheafified higher-order message passing reframes geometry-awareness in terms of cellular sheaves, where restriction maps encode local data dependencies and the resulting sheaf Laplacian defines a geometry-aware diffusion process tied to sheaf cohomology, not just singular cohomology. Learned or handcrafted restriction maps enable anisotropic, data-driven harmonic modes unattainable by Hodge-based diffusion alone (Hume et al., 27 Sep 2025).
• Generic geometric message passing frameworks (e.g., AMPS) allow arbitrary geometric features (lengths, normals, centroids) to be included in the messages, but without the formal expressivity characterization of GSWL-based schemes (Hajij et al., 2021).

Geometry-aware SMP thus encompasses both theoretically optimal invariant characterizations (GSWL/ECT) and a broad family of implementation strategies, including algebraically structured equivariant networks and sheaf-diffusion variants.

7. Theoretical and Empirical Impact

The introduction of the GSWL test and its associated SMP realizability/expressivity bounds constitutes a rigorous geometric counterpart to the classical SWL characterization of combinatorial message passing (Wang et al., 7 May 2026). When combined with complete geometric invariants such as ECT, this approach provides both finite and infinite class guarantees—exactness for finite families and uniform approximation for bounded embedding classes.

Empirically, geometry-aware SMP defines a clear, experimentally validated expressivity hierarchy:

$k$4

with ablation studies confirming the necessity of downward coboundary flow for intrinsic geometric tasks and superiority of geometry-aware architectures on synthetic and real mesh datasets.

Geometry-aware simplicial message passing thus establishes both the foundational and practical route for fully geometric, higher-order neural message passing on simplicial complexes, enabling advances in mesh analysis, geometric machine learning, and topological deep learning (Wang et al., 7 May 2026).