Topological Neural Networks

• Topological neural networks are mathematical models that extend traditional neural architectures by leveraging simplicial and cell complexes to capture complex connectivity and higher-order interactions.
• They employ advanced message passing over multidimensional structures and integrate persistent homology for feature extraction and robust performance.
• TNNs find applications in quantum physics, molecular modeling, and decentralized wireless systems, demonstrating enhanced expressivity and interpretability.

Topological neural networks (TNNs) are mathematical and computational models that integrate ideas and techniques from algebraic topology, combinatorial topology, and deep learning to capture, analyze, and exploit higher-order relational, geometric, and topological information in structured data. While traditional neural networks operate on Euclidean spaces or graphs—modeling interactions as vectors or as pairwise relationships—TNNs process data organized over combinatorial complexes, including simplicial complexes, cell complexes, and more general topological spaces. This expanded framework enables TNNs to represent, disentangle, or process complex data structures characterized by connectivity, cycles, voids, and higher-order interactions, providing expressivity, robustness, and interpretability beyond conventional neural networks.

1. Algebraic and Combinatorial Foundations

Algebraic topology provides TNNs with mathematical constructs for encoding and manipulating relationships beyond pairwise adjacency. Fundamental objects include:

• Simplicial complexes (collections of vertices, edges, triangles, and higher-dimensional faces satisfying closure conditions)
• Cell complexes (more general collections where each cell is homeomorphic to an open ball of some dimension)
• Incidence matrices (maps describing which lower-dimensional cells are faces of higher-dimensional cells)

A central operation is the definition of neighborhoods not only by adjacency but by shared faces or cofaces, which permits message passing along higher-dimensional “routes” rather than just edges. Algebraic tools such as the coboundary operator (δ), sheaf Laplacians (Δ), and copresheaf maps enable multi-scale and directional propagation of information.

Recent generalizations, such as copresheaf topological neural networks (CTNNs), define a network in which each cell receives and transmits features through locally defined vector spaces and learnable directional maps, bringing the flexibility to model locality, anisotropy, and heterophily (Hajij et al., 27 May 2025).

2. Core TNN Architectures and Message Passing

TNN architectures are distinguished from graph neural networks (GNNs) by their ability to perform message passing over higher-order structures. Several key schemes are prevalent:

• Simplicial/cellular message passing: Features are associated with cells of varying dimensions; messages can propagate both “down” (to faces) and “up” (to cofaces). This dual neighborhood structure appears in cell attention networks and is also fundamental to the definition of topological filters in “over-the-air” computation settings (Giusti, 10 Feb 2024, Fiorellino et al., 14 Feb 2025).
• Topological CNNs (TCNNs) extend classical convolution by parameterizing filters or feature maps over manifolds with topological significance (e.g., circles or Klein bottles), enforcing locality not only in physical space but also in intrinsic topological space (Love et al., 2021).
• Copresheaf models replace the global latent space of standard architectures with collections of local stalks and learnable morphisms respecting the directionality and overlap induced by a combinatorial complex (Hajij et al., 27 May 2025).

A common thread is the use of permutation-equivariant operations, which guarantee that outputs depend only on the intrinsic structure of complexes, not on arbitrary indexings of their components.

3. Integration of Persistent Homology and Topological Descriptors

Persistent homology (PH) provides quantitative invariants—such as birth and death times of connected components, cycles, and higher-dimensional holes—that summarize the multi-scale topological structure of data or features. TNN frameworks increasingly integrate PH in several ways:

• Feature extraction: Persistent barcodes or landscapes are computed from activation data, input structures, or learned representations, then digitized for use as input to convolutional or message-passing layers (Cang et al., 2017, Wheeler et al., 2021, Verma et al., 5 Jun 2024).
• Structural augmentation: PH descriptors are incorporated as additional channels or merged with learned features to enhance the expressivity of TNNs. This unified approach, termed TopNets, allows persistent diagrams to be merged with dimension-wise TNN outputs, improving the theoretical power of the model and leading to strong empirical performance on tasks such as molecular property prediction and antibody design (Verma et al., 5 Jun 2024).
• Theoretical guarantees: It has been established that certain PH descriptors can distinguish non-isomorphic complexes that cannot be distinguished by standard message-passing alone, thus strictly increasing TNN expressivity.

4. Expressivity, Depth, and Topological Simplification

A prominent theoretical theme is the connection between network architecture (depth, width, choice of activation) and the network’s ability to “untangle” data and render complex topological configurations linearly separable:

• Topological simplification: As data passes through a network, its Betti numbers (β₀ for components, β₁ for holes, etc.) typically decrease—an empirical and theoretical observation linked to the composition of non-homeomorphic (e.g., ReLU) and affine maps (Naitzat et al., 2020, Ergen et al., 2023). Deep ReLU networks have been shown to increase the number of distinguishable topological regions exponentially with depth, whereas shallow networks do so only polynomially.
• Activation function impact: Non-smooth activations such as ReLU actively change topology, folding and collapsing regions, while smooth functions (e.g., tanh) act as homeomorphisms, preserving the topology and thus sometimes limiting expressive capacity (Pandey, 2023, Naitzat et al., 2020).
• Architectural consequences: The topology of the input data determines the required network capacity (width, depth); early excessive dimension reduction may irreversibly “glue” together topologically distinct regions, impeding separation (Hajij et al., 2020).

A summary formula for a folding ReLU architecture capable of “cutting” complex high-dimensional regions is given by

$$\beta_k(F) = \frac{M^{k-1}}{2^{k-1}} \left(\frac{M}{2} - 1\right) \left\lceil \frac{w_{k-1}}{2} \right\rceil,$$

where $\beta_k$ encodes the number of k-dimensional holes after transformation, and $M, w_{k-1}$ are architectural parameters (Ergen et al., 2023).

5. Practical Applications and Architectures

TNNs have been applied across a diverse range of domains, including:

• Quantum many-body physics: Restricted Boltzmann machines (RBMs) with topologically motivated structure can represent ground states and excited states (including anyonic excitations) of SPT and topologically ordered Hamiltonians exactly and efficiently, using only a linear number of parameters relative to system size (1609.09060).
• Molecular and biomolecular modeling: Topological CNNs leveraging element-specific PH provide state-of-the-art results for protein–ligand binding affinity and mutation stability predictions in TopologyNet, outperforming empirical scoring functions and overcoming limited/noisy data with multitask learning (Cang et al., 2017).
• Structured data and combinatorial complexes: CTNNs and related copresheaf models demonstrate strong empirical improvements on data ranging from meshes to token-labeled graphs and scientific simulations, excelling in settings demanding hierarchical, local, or anisotropic information flow (Hajij et al., 27 May 2025).
• Molecular property prediction and classification: Efficient heat kernel signatures computed from a novel CC Laplacian enable transformer-based models to achieve competitive or state-of-the-art performance on standard molecular datasets, with theoretical guarantees that different non-isomorphic complexes are distinguished (Krahn et al., 16 Jul 2025).

In distributed wireless systems, TNNs have been designed to operate “over the air,” integrating channel impairments (fading, noise) into topological filtering and enabling decentralized, robust implementations applicable to sensor and communication networks (Fiorellino et al., 14 Feb 2025).

6. Challenges and Emerging Directions

Despite their advantages, practical deployment and wide adoption of TNNs face several open challenges:

• Computational scaling: Traditional higher-order message passing can be computationally intensive. Approaches based on unified Laplacians, efficient spectral methods, or nonparametric topological layers are advancing scalability (Krahn et al., 16 Jul 2025, Zhao, 2021).
• Model selection and interpretability: The greater expressivity of TNNs introduces complexity in model design. However, TDA-based summaries (e.g., persistence barcodes, landscapes) offer rigorously interpretable diagnostics for model comparison, generalization assessment, and architecture refinement (Wheeler et al., 2021, Świder, 8 Jul 2024).
• Integration with symmetries and geometry: Adaptation to geometric complexes via E(n)-invariant filtrations allows TNNs to respect spatial symmetries, critical in applications such as protein folding or molecular dynamics (Verma et al., 5 Jun 2024).

Ongoing research is focused on leveraging categorical and sheaf-theoretic frameworks to encode symmetries and structural constraints, continuous-time analogues of TNNs, and principled regularization based on topological obstructions in learning (Giusti, 10 Feb 2024, Barannikov et al., 2020).

7. Summary Table: Principal TNN Approaches

Model or Framework	Underlying Structure	Topological Principle	Domain / Application
RBM-based TNNs	Lattices, spin systems	Short-range neural encoding, area-law entanglement	Topological quantum states (1609.09060)
TopologyNet	Molecules (3D)	Persistent homology, element-specific invariants	Protein–ligand binding, mutation (Cang et al., 2017)
TCNN	Images, videos	Manifold-parametrized convolutions (e.g., circle, Klein bottle)	Vision, video (Love et al., 2021)
Copresheaf TNNs (CTNNs)	Combinatorial complexes	Anisotropic, directional message passing via copresheaf maps	Structured data, simulation (Hajij et al., 27 May 2025)
TopNets	Simplicial complexes	Integration of TNN and persistent homology	Graphs, molecules, dynamics (Verma et al., 5 Jun 2024)
Heat kernel TNNs	CCs (any rank)	Unified Laplacian, HKS descriptors	Molecular property prediction (Krahn et al., 16 Jul 2025)
Over-the-air TNN	Cell complexes	Topological convolution with communication impairments	Wireless sensors (Fiorellino et al., 14 Feb 2025)

Topological neural networks represent an active and rapidly evolving area at the intersection of deep learning and algebraic topology. By marrying rigorously defined topological constructs with flexible neural computation, TNNs offer both powerful theoretical guarantees (on expressivity and universal approximation) and practical performance, particularly for data with rich structural dependencies and in applications where higher-order, geometric, or non-Euclidean relationships are central.