A technical paper introducing copresheaf topological neural networks as a unified deep learning framework for structured data, including graphs, meshes, images, point clouds, and topological manifolds.
AI Architectures for Physics
Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework
May 27, 2025
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.
What You’ll Learn
- Why conventional architectures struggle with heterogeneous, directional, and multiscale data
- How copresheaf topological neural networks generalize message passing across graphs, meshes, images, point clouds, and combinatorial complexes
- How CTNNs unify ideas from CNNs, GNNs, transformers, sheaf neural networks, and topological neural networks
- Why learnable directional transport maps can improve representation learning on structured data
- How copresheaf-based architectures perform across physics simulations, graph classification, token-relation learning, and higher-order structure recognition tasks
Asset Details
Type: Technical paper
Title: Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework
Authors: Mustafa Hajij, Lennart Bastian, Sarah Osentoski, Hardik Kabaria, John L. Davenport, Sheik Dawood Beer Mohideen, Balaji Cherukuri, Joseph G. Kocheemoolayil, Nastaran Shahmansouri, Adrian Lew, Theodore Papamarkou, Tolga Birdal
Organizations: Vinci4D, University of San Francisco, Technical University of Munich, Stanford University, PolyShape, Imperial College London, MCML
Topic: Copresheaf topological neural networks, structured data, topological deep learning, graph neural networks, transformers, physics-aware learning
Format: PDF
Audience: Machine learning researchers, AI researchers, scientific computing teams, simulation researchers, graph learning practitioners, and technical leaders exploring foundation
This paper reflects collaborative research across Vinci4D, University of San Francisco, Technical University of Munich, Stanford University, PolyShape, Imperial College London, and MCML on structure-aware deep learning methods for complex Euclidean and non-Euclidean data domains.