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Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework

Mustafa Hajij, Lennart Bastian, Sarah Osentoski, Hardik Kabaria, John L. Davenport, Sheik Dawood, Balaji Cherukuri, Joseph G. Kocheemoolayil, Nastaran Shahmansouri, Adrian Lew, Theodore Papamarkou, Tolga Birdal

TL;DR

Copresheaf Topological Neural Networks (CTNNs) introduce a unifying framework for deep learning on structured data by equipping each local region (cell) of a combinatorial complex with its own latent space and learning directional, cell-to-cell transport maps $\rho_{y\to x}$. This copresheaf-based message passing generalizes graph neural networks, sheaf neural networks, and topological neural networks, enabling multiscale, anisotropic, and task-specific information flow across diverse domains. The paper develops CNMs, CAMs, CIMs, and a general CMPNN framework, and instantiates architectures such as Copresheaf Transformers, Copresheaf GNNs, and CopresheafConv layers. Theoretical connections to diffusion and quiver Laplacians provide energy-decreasing interpretations, while extensive experiments across physics simulations, graph classification (MUTAG), and higher-order combinatorial complexes demonstrate consistent performance gains over strong baselines. Overall, CTNNs offer a principled, scalable path to multi-scale, structure-aware learning on Euclidean and non-Euclidean domains with broad applicability.

Abstract

We introduce copresheaf topological neural networks (CTNNs), a powerful unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data, including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by formulating model design in the language of copresheaves, a concept from algebraic topology that generalizes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning, such as long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results establish CTNNs as a principled multi-scale foundation for the next generation of deep learning architectures.

Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework

TL;DR

Copresheaf Topological Neural Networks (CTNNs) introduce a unifying framework for deep learning on structured data by equipping each local region (cell) of a combinatorial complex with its own latent space and learning directional, cell-to-cell transport maps . This copresheaf-based message passing generalizes graph neural networks, sheaf neural networks, and topological neural networks, enabling multiscale, anisotropic, and task-specific information flow across diverse domains. The paper develops CNMs, CAMs, CIMs, and a general CMPNN framework, and instantiates architectures such as Copresheaf Transformers, Copresheaf GNNs, and CopresheafConv layers. Theoretical connections to diffusion and quiver Laplacians provide energy-decreasing interpretations, while extensive experiments across physics simulations, graph classification (MUTAG), and higher-order combinatorial complexes demonstrate consistent performance gains over strong baselines. Overall, CTNNs offer a principled, scalable path to multi-scale, structure-aware learning on Euclidean and non-Euclidean domains with broad applicability.

Abstract

We introduce copresheaf topological neural networks (CTNNs), a powerful unifying framework that encapsulates a wide spectrum of deep learning architectures, designed to operate on structured data, including images, point clouds, graphs, meshes, and topological manifolds. While deep learning has profoundly impacted domains ranging from digital assistants to autonomous systems, the principled design of neural architectures tailored to specific tasks and data types remains one of the field's most persistent open challenges. CTNNs address this gap by formulating model design in the language of copresheaves, a concept from algebraic topology that generalizes most practical deep learning models in use today. This abstract yet constructive formulation yields a rich design space from which theoretically sound and practically effective solutions can be derived to tackle core challenges in representation learning, such as long-range dependencies, oversmoothing, heterophily, and non-Euclidean domains. Our empirical results on structured data benchmarks demonstrate that CTNNs consistently outperform conventional baselines, particularly in tasks requiring hierarchical or localized sensitivity. These results establish CTNNs as a principled multi-scale foundation for the next generation of deep learning architectures.

Paper Structure

This paper contains 107 sections, 6 theorems, 59 equations, 11 figures, 19 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Combinatorial Complexes and Neighborhood Structures
  5. Sheaves and Copresheaves on Directed Graphs
  6. Copresheaf Topological Neural Networks
  7. Copresheaf neighborhood matrices
  8. Copresheaf-based message passing
  9. Architectures Derived from the Copresheaf Framework
  10. Copresheaf transformers
  11. Experimental Evaluation
  12. Evaluations on Physics Datasets
  13. Model and training
  14. Results
  15. Graph Classification
  16. ...and 92 more sections

Key Result

Theorem 1

(SNNs are CMPNNs) Let $G = (V, E)$ be an undirected graph equipped with a cellular sheaf $\mathcal{F}$ assigning vector spaces to vertices and edges, and linear maps $\mathcal{F}_{x \trianglelefteq e}$ for each vertex $x \in e$. Then for each edge $e = \{x, y\} \in E$, the SNN message passing from $

Figures (11)

  • Figure 1: A copresheaf topological neural network (CTNN) operates on combinatorial complexes (CCs), which generalize Euclidean grids, graphs, meshes, and hypergraphs. A CTNN is characterized by a set of locally indexed copresheaf maps $\rho_{x_i \to x_j}$, defined between cells $x_i$ and $x_j$ in the CC, and directed from $x_i$ to $x_j$. The figure illustrates how a CTNN updates a local representation $\mathbf{h}_x$ of a cell $x$ using neighborhood representations $\mathbf{h}_y$ and $\mathbf{h}_z$, which are sent to $x$ via the learnable local copresheaf maps $\rho_{x \to x}$, $\rho_{y \to x}$ and $\rho_{z \to x}$.
  • Figure 2: A combinatorial complex of dimension 2.
  • Figure 3: (a) A combinatorial complex $\mathcal{X}=(\mathcal{S},\mathcal{X},\mathop{\mathrm{rk}}\nolimits)$ with $\mathcal{S}=\{a,b,c,d\}$, edges $\mathcal{X}^1=\{\{a,b\},\{b,c\},\{c,a\},\{d,b\},\{c,d\}\}$, and faces $\mathcal{X}^2=\{\{a,b,c\},\{d,b,c\}\}$. (b) Induced edge–adjacency digraph: nodes represent the edges of $\mathcal{X}$ and the edges represent the face adjacencies. (c) Copresheaf neighborhood matrix $\mathbf{G}^{\mathcal{N}_\triangle}$.
  • Figure 4: (a) A combinatorial complex $\mathcal{X} = (\mathcal{S}, \mathcal{X}, \mathop{\mathrm{rk}}\nolimits)$ with $\mathcal{S} = \{a, b, c\}$, cells $\mathcal{X} = \{\{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}\}$. The figure also indicates the induced directed graph $G_{\mathcal{N}_{\text{inc}}}=(\mathcal{V}_{\mathcal{N}},E_{\mathcal{N}})$ from the incidence neighborhood structure on the combinatorial complex $\mathcal{X}$. Each arrow $z \to y$ represents a directed edge from a 1-cell to a 0-cell where $y \subset z$, and is associated with a linear map $\rho_{z \to y}$ as part of the copresheaf. (b) The induced directed graph $G_{\mathcal{N}_{\text{inc}}}=(\mathcal{V}_{\mathcal{N}},E_{\mathcal{N}})$ from the incidence neighborhood structure $\mathcal{N}_{\text{inc}}$. (c) The copresheaf neighborhood matrix (CNM) $\mathbf{G}^{\mathcal{N}_{\text{inc}}}$, where rows are indexed by 0-cells $\{a\}, \{b\}, \{c\}$ and columns by 1-cells $\{a,b\}, \{b,c\}$. The matrix entries are linear maps $\rho_{z \to y}$ when $z \in \mathcal{N}_{\text{inc}}(y)$, and 0 otherwise. This matrix supports directional feature propagation from 1-cells to 0-cells.
  • Figure 5: Illustration of copresheaf higher-order message passing. Left-hand side: Shows a central cell $x$ (as a circle) with arrows pointing to boxes labeled $\mathcal{N}_1(x), \mathcal{N}_2(x), \ldots, \mathcal{N}_k(x)$, representing the collection of neighborhood functions $\mathfrak{N} = \{\mathcal{N}_k\}_{k=1}^n$. Right-hand side: Depicts the message-passing process for the same cell $x$. Each neighborhood $\mathcal{N}_k(x)$ produces an aggregated message using $\bigoplus_{y \in \mathcal{N}_k(x)} \alpha_{\mathcal{N}_k}(\rho_{y \to x}(h_y^{(\ell)}))$. These messages are then combined using the inter-neighborhood function $\beta$, shown as a box, with an arrow updating $x$.
  • ...and 6 more figures

Theorems & Definitions (33)

  • Definition 1: Combinatorial complex hajij2023tdl
  • Definition 2: Neighborhood function
  • Definition 3: Neighborhood matrix
  • Definition 4: Copresheaf on directed graphs
  • Definition 5: Cellular sheaf
  • Definition 6: Neighborhood-dependent copresheaf
  • Definition 7: Copresheaf neighborhood matrices
  • Definition 8: Copresheaf adjacency/incidence matrices
  • Definition 9: Copresheaf message-passing neural network
  • Theorem 1
  • ...and 23 more