Topological Neural Operators

• Topological Neural Operators are neural operator architectures that leverage discrete exterior calculus to decouple fixed topological transport from learned nonlinear transformations.
• They process data on arbitrary cell complexes by modeling inter-cell interactions via gradient, curl, and divergence routes for cross-dimensional coupling.
• Empirical benchmarks on complex PDEs demonstrate reduced L1 errors and faster convergence compared to classical neural operator methods.

Topological Neural Operators (TNOs) are a class of neural operator architectures designed to generalize operator learning from pointwise- or graph-based settings to arbitrary topological domains modeled by cell complexes. TNOs represent data as features on cells of variable dimension and fundamentally rely on Discrete Exterior Calculus (DEC) to explicitly model inter-cell interactions, enabling principled cross-dimensional coupling (e.g., gradient, curl, and divergence routes). The hallmark of the framework is the decoupling of information transport—dictated by fixed topological operators encoded by the discrete geometry—from information transformation, which is performed by learned nonlinear maps. This structure enforces compatibility with physical conservation and geometric laws, subsumes classical neural operator families as special cases, and demonstrates empirical superiority on partial differential equation (PDE) benchmarks featuring complex geometries and higher-rank data (Bastian et al., 8 Jun 2026).

1. Formal Definition and Mathematical Setting

A TNO is defined on a finite cell complex $K$, a set of cells partitioned by dimension (vertices, edges, faces, volumes, ...), each with a rank $\operatorname{rk}(\sigma)$. Cells are glued along faces such that $B_k B_{k+1} = 0$, with $B_k$ the signed incidence matrix.

The signal space is the graded direct sum $$C^\bullet(K) = \bigoplus_{k=0}^N C^k(K; \mathbb{R}^{d_k})$$, with $k$-cochains as array-valued functions on $k$-cells. A TNO is a map

$$\mathcal{T}_\theta^K: \prod_i C^{k_i}(K; \mathbb{R}^{d_i}) \to \prod_j C^{\ell_j}(K; \mathbb{R}^{r_j}),$$

where:

• The domain is a product of ranks $\{k_i\}$ with channel dimensions $d_i$.
• The architecture depends only on the cell complex incidence and metric data.
• Cochain coupling across ranks leverages the discrete calculus structure.
• Weights are shared across different mesh refinements.

Restriction to $\operatorname{rk}(\sigma)$0-cochains (i.e., per-vertex features) recovers graph- or point-based neural operators (Bastian et al., 8 Jun 2026).

2. Discrete Exterior Calculus in TNOs

TNOs operationalize information transport via DEC, wherein:

• The discrete coboundary $\operatorname{rk}(\sigma)$1 realizes topological derivatives: for 0-cochains, the gradient; for 1-cochains, the curl; for 2-cochains, the divergence.
• A diagonal mass matrix $\operatorname{rk}(\sigma)$2 encodes metric data, defining inner products for $\operatorname{rk}(\sigma)$3-cochains.
• The codifferential $\operatorname{rk}(\sigma)$4 is the discrete dual operator, yielding divergence-type analogs.

The $\operatorname{rk}(\sigma)$5-Hodge Laplacian is

$\operatorname{rk}(\sigma)$6

decomposing into "curl-type" (upward) and "divergence-type" (downward) information flow. The Hodge decomposition expresses all $\operatorname{rk}(\sigma)$7-cochains as exact, harmonic, or coexact components, aligning with structural invariants, e.g., Betti numbers (Bastian et al., 8 Jun 2026).

3. Layer Architecture and Transport-Transformation Decoupling

A TNO layer maintains rank decomposition. For hidden features $\operatorname{rk}(\sigma)$8, per-rank outputs are updated by

$\operatorname{rk}(\sigma)$9

where the $B_k B_{k+1} = 0$0 matrices are learnable channel-mixing weights and $B_k B_{k+1} = 0$1 is a shared nonlinear MLP. The DEC operators are fixed by the topology and encode permissible information paths ("where" data flows), while the channel-mixing and nonlinearity encode the transformation ("how" data is changed). This separation guarantees formal compatibility—e.g., $B_k B_{k+1} = 0$2—and topologically invariant structure (Bastian et al., 8 Jun 2026).

4. Hierarchical Extension: HTNOs

To incorporate global, long-range, and strongly topology-dependent effects, Hierarchical TNOs (HTNOs) stack TNO layers across scales via a learned hierarchy of coarse complexes. Levels $B_k B_{k+1} = 0$3 are connected by restriction ($B_k B_{k+1} = 0$4) and prolongation ($B_k B_{k+1} = 0$5) operators. A standard two-grid HTNO block (a learned multigrid V-cycle analog) comprises pre-smoothing on $B_k B_{k+1} = 0$6, restriction, coarse update on $B_k B_{k+1} = 0$7, prolongation/correction, and post-smoothing. Discretization coarsening can be precomputed (e.g., $B_k B_{k+1} = 0$8-means) or optimized end-to-end using learnable soft Voronoi partitions (Bastian et al., 8 Jun 2026).

5. Operator Learning on Topological Vector Spaces

A related thread is the generalization of operator neural architectures from Banach spaces to arbitrary Hausdorff locally convex spaces, as instantiated in topological DeepONets. Here, input functions $B_k B_{k+1} = 0$9 are generalized to elements in a locally convex topological vector space $B_k$0, with data accessed via continuous linear functionals $B_k$1 rather than point evaluations. The Topological DeepONet architecture employs:

• A branch network on $B_k$2, with neurons computing $B_k$3 for $B_k$4.
• A trunk network parameterized over the output Euclidean domain.

The universality theorem establishes that for any continuous operator $B_k$5, where $B_k$6 and $B_k$7 is a compact set, one can uniformly approximate $B_k$8 by such a separable neural operator; this holds regardless of normability or Banach-space structures (Ismailov, 12 Mar 2026). This framework admits arbitrary topological vector spaces, including sequence spaces, function spaces (e.g., distributions), and non-normable Fréchet spaces as valid input domains.

6. Empirical Benchmarks and Inductive Biases

TNO and HTNO architectures have been systematically benchmarked against classical neural operators, including MeshGraphNet, Geo-FNO, PointNet, and others, on PDE suites involving irregular geometries and higher-rank features. Reported $B_k$9 errors are consistently lower for TNOs/HTNOs, e.g.:

• Poisson-Gauss: TNO 1.03%, HTNO 1.30% vs. Geo-FNO 8.16%
• Elasticity: HTNO 1.70% vs. Geo-FNO 5.53%
• Surface RANS (EmmiWing): HTNO 2.41%, outpacing PointNet, Transformer, and others and converging 6× faster than RIGNO.

Higher-rank ingestion (e.g., face tensors in Darcy flow) further reduces error by ~0.5 percentage points compared to per-vertex architectures, providing quantitative advantage. Ablation studies confirm the synergistic effect of including harmonic components and sheaf (copresheaf) transport, e.g., for Darcy/Advection–Diffusion, TNOs with both biases achieve <5.2% error, while omitting them doubles the error rate. Explicit DEC-induced transport and Hodge-theoretic decomposition are empirically essential for performance on topologically rich domains (Bastian et al., 8 Jun 2026).

7. Limitations, Theoretical Extensions, and Open Problems

TNOs require explicit knowledge of the cell complex and precompute all DEC operators, which may incur overhead for highly dynamic or unstructured settings. The universality proof for topological DeepONets is existential; it guarantees uniform approximation on compact input/output domains but does not provide convergence rates or sample-complexity bounds. The extension to $$C^\bullet(K) = \bigoplus_{k=0}^N C^k(K; \mathbb{R}^{d_k})$$0 or probabilistic error metrics can relax compactness and continuity assumptions but weakens error control. One open direction is to further exploit the flexibility of functionals or global measurements in the branch network—potentially capturing global or distributional features beyond pointwise sampling (Ismailov, 12 Mar 2026). Another is to derive complexity and expressivity results, as current proofs do not yield explicit rate guarantees for parameter efficiency or sample scaling. A final open area is the integration of DEC-compatible architectures into continuous-time operator learning and generative modeling frameworks.

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Summary Table: Key Distinctions of TNO Framework

Aspect	TNO/HTNO	Classical Neural Operator
Input domain	Cell complex (any rank)	Points/graphs
Information coupling	DEC (grad/curl/div)	Ad-hoc/pointwise
Handles conservation & compatibility	Yes	Typically not
Hierarchical (multi-scale) extension	HTNO (coarse complexes)	Not native
Explicit Hodge decomposition	Yes	No

TNOs constitute a unification and strict generalization of neural operator architectures, particularly effective for cases with complex topological structure and higher-order physical conservation constraints (Bastian et al., 8 Jun 2026).