Topological Neural Operators

Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

• The paper introduces TNOs that integrate discrete exterior calculus into neural operators, ensuring topology-aware PDE field predictions.
• It employs fixed geometric operators (d, δ, and Hodge Laplacians) to enforce conservation laws and decouple cross-rank information on cell complexes.
• Empirical results demonstrate that TNOs achieve superior error rates and robustness across varying discretizations and complex geometries.

Topological Neural Operators: A Principled Framework for Operator Learning on Cell Complexes

Overview and Motivation

Topological Neural Operators (TNOs) (2606.09806) introduce an operator learning framework leveraging discrete topology and geometry for physical systems governed by partial differential equations (PDEs). The formulation systematically extends neural operators (NOs) from point- and graph-centric paradigms to the general setting of cell complexes, incorporating rigorous discrete exterior calculus (DEC) to model physical fields as cochains distributed across vertices, edges, faces, and volumes. This approach directly encodes geometric and topological constraints of the domain and the physics, providing a means to enforce conservation and compatibility properties natively within the architecture and efficiently couple physical signals of different geometric types.

Figure 1: Topological Neural Operators operate on cell complexes (i) whose physical signals are cochains at multiple ranks (ii). A TNO layer (iii) couples ranks through fixed DEC operators ($d^k$, $\delta^k$) and learns the rank-wise channel mixing, producing a predicted PDE field on the complex (iv).

Discrete Topology and Physical Modeling Background

Physical quantities relevant for PDEs naturally inhabit different geometric supports: scalar potentials map to vertices, vector fields (circulations) to edges, fluxes to faces, and densities to volumes. Classical NOs collapse all such quantities to node features, thus ignoring the incidence structure and differential relations underpinning the governing physics, degrading their ability to represent and enforce conservation laws, differential identities, and topological effects (e.g., presence of harmonic modes or nontrivial cycles).

TNOs resolve this by encoding solutions as cochains over regular cell complexes (RCCs) and their generalizations (combinatorial or CW-complexes). DEC provides three key operator families:

• Boundary/coboundary (exterior derivative, $d^k$): Implementing discrete gradient, curl, and divergence by aggregating over cell incidences and orientations.
• Hodge star and codifferential ($\delta^k$): Mapping higher-rank cochains downward in degree, parameterized by metric data (cell volume/area).
• Hodge Laplacian ($\Delta_k$): Combining $d^k$ and $\delta^k$ to define quadrature of field dynamics, with explicit channel decomposition.

TNOs turn these operators into architectural scaffolding that prescribes information flow between physical field representations while decoupling where messages flow (which is fixed by the topology and metric) from how features are transformed (which is learned).

Topological Neural Operators: Model Construction

The core TNO model takes as input/output tuples of cochains distributed across arbitrary selected ranks. The architecture arranges topological layers, each defined as a block-structured operator-valued map, combining:

• Cross-rank coupling: via $d$ and $\delta$, mediating exact and coexact channels;
• Same-rank propagation: via Hodge Laplacian's upper ($\Delta_k^\uparrow$) and lower ($\delta^k$0) channels, capturing face/vertex mediated effects;
• Local and harmonic channels: separating physical modes via explicit projection onto the kernel of the Laplacian (harmonic subspace).

Compared to standard message passing or transformers, the support of transport is not learned but prescribed by the topology; only mixing, nonlinear activation, and channel weights are learned. This guarantees discretization transferability: the model can be applied across mesh refinements or entirely novel complexes without retraining, as the structural operators recompute with the domain.

Figure 2: Topological domain hierarchy from graphs, to simplicial, cell, and then combinatorial complexes, increasing modeling expressivity for TNOs.

The TNO generalizes both spectral (e.g., FNO) and kernel graph neural operators (GNO), as proven by specialization to rank-0 (vertex-only) cases.

Hierarchical TNOs

To propagate topology-dependent and long-range information while maintaining computational efficiency, Hierarchical TNOs (HTNOs) are introduced. HTNOs operate on multilevel hierarchies of coarsened complexes, connected by degree-preserving commuting transfer maps that mirror the commuting diagrams of finite element exterior calculus (FEEC) multigrid. The architecture composes fine- and coarse-grid TNOs in a $\delta^k$1-cycle, akin to Hodge-compatible multigrid solvers, further reinforcing cross-scale topological compatibility.

Empirical Results and Ablation Studies

TNOs and HTNOs are validated on a broad set of challenging PDE problems over irregular and complex geometries, including Poisson equations (with Gaussian and complex source terms), mixed elliptic (Darcy) flow with anisotropy, compressible airfoil flow in transonic and supersonic regimes, nonlinear elasticity, 3D wing-surface aerodynamics, and synthetic topology-controlled benchmarks.

Strong claims are supported by the results:

• Superior error rates: TNO and HTNO consistently improve or match state-of-the-art on all established steady-state PDE benchmarks, often with substantial relative gains—e.g., a reduction in relative $\delta^k$2 error exceeding 30% in several cases over RIGNO and GAOT baselines.
• Robustness to discretization and geometry: The models generalize across variable mesh sizes, domain topologies, and nonuniform geometries, directly benefiting from the structural bias towards the underlying physics.
• Separation of effects: Ablation on harmonic and sheaf/cross-rank channels demonstrates that both components are crucial on problems where physical solutions populate nontrivial topological modes or require multi-rank interaction. For instance, DISABLING harmonic channels led to increases in error of up to 8pp for advection-diffusion, confirming the necessity of explicit Hodge decomposition.

  Figure 3: Qualitative results for Anisotropic Darcy with per-face random tensor orientations, requiring cross-rank information handling for physical coherence.

  

Figure 4: Ablation across vanilla MPNN, TNO without harmonics, and TNO with harmonics/copresheaves on Darcy and advection-diffusion; TNO variants explicitly leveraging topological structure exhibit improved physical fidelity and reduced error.

Figure 5: Qualitative field-level comparison (input, true, prediction, absolute error) for TNO, RIGNO, and HTNO on airfoil and elasticity benchmarks—TNO/HTNO display spatially reduced and physically-informed error profiles.

Figure 6: Field visualization on GAOT suite (Poisson with sines, various airfoil geometries) demonstrates the fidelity and adaptability of TNO/HTNO to regime change and high-frequency source information.

Theoretical and Practical Implications

The TNO construction sharply departs from graph message-passing approaches by enforcing the algebraic backbone of discrete physical laws (e.g., $\delta^k$3), thus making conservation, topological invariance, and compatibility exact even on coarse, irregular complexes. This permits:

• Intrinsic operator learning: Generalization across different domains meshes and resolutions without topology-specific retraining.
• Modularity and extensibility: Unified modeling of multi-physics systems, mixed-degree PDEs, and topological effects (e.g., harmonic forms, gauge constraints), positioning TNOs as foundational surrogates for scientific computing at scale.
• Clarity of model interpretability: Explicit control of physically meaningful channels (exact, coexact, harmonic) provides a framework for scientific diagnosis, hybridization with numerical solvers, or certified surrogates.

HTNOs further offer a path towards scalability and multi-scale information integration in operator learning pipelines, supporting adaptive discretizations and long-range information propagation while retaining the correct structure for physical compatibility.

Prospects for AI and Scientific Machine Learning

TNOs suggest a principled avenue for building scientific foundation models that seamlessly unify geometry, topology, and physical rules. Future directions include:

• Systematic scalability to highly dynamic, non-manifold, and adaptive domains;
• Theoretical analysis for approximation, stability, and convergence in operator learning;
• Extension to inverse problems, manifold evolution, and symmetry/gauge-equivariant architectures;
• Potential links to certified machine learning for engineering design and simulation.

Conclusion

Topological Neural Operators represent a significant formalization of operator learning architectures rooted in the algebraic and geometric structure of the governing equations and their discrete representations. By treating cochains as first-class citizens, explicitly respecting where physical quantities live and how they interact, TNOs ensure strong inductive biases, robust generalization, and interpretability across a spectrum of PDE-based applications. This advances both the theory and practice of machine-learning-based scientific computing, narrowing the gap between data-driven models and structure-preserving solvers (2606.09806).