COSMOS: Continuous Simplicial Neural Networks

2025-03-17

AI Summary

Existing simplicial neural networks (SNNs) rely on discrete filtering, limiting their ability to model continuous dynamics inherent in high-dimensional structured data. Method: We propose Continuous Simplicial Neural Networks (CSNNs), the first framework to incorporate partial differential equations (PDEs) defined on simplicial complexes into neural network design, establishing a continuous-domain neural dynamical architecture. CSNNs jointly leverage topological representations of simplicial complexes and PDE-based modeling; we theoretically prove their stability under simplicial perturbations and demonstrate that their continuous evolution mechanism effectively mitigates over-smoothing. Contribution/Results: On ocean trajectory prediction and deformable mesh regression—both under strong noise—CSNNs significantly outperform state-of-the-art discrete SNNs, achieving new SOTA performance. This validates the efficacy and robustness of continuous dynamical modeling for learning high-order structural data representations.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Simplicial complexes provide a powerful framework for modeling high-order interactions in structured data, making them particularly suitable for applications such as trajectory prediction and mesh processing. However, existing simplicial neural networks (SNNs), whether convolutional or attention-based, rely primarily on discrete filtering techniques, which can be restrictive. In contrast, partial differential equations (PDEs) on simplicial complexes offer a principled approach to capture continuous dynamics in such structures. In this work, we introduce COntinuous SiMplicial neural netwOrkS (COSMOS), a novel SNN architecture derived from PDEs on simplicial complexes. We provide theoretical and experimental justifications of COSMOS's stability under simplicial perturbations. Furthermore, we investigate the over-smoothing phenomenon, a common issue in geometric deep learning, demonstrating that COSMOS offers better control over this effect than discrete SNNs. Our experiments on real-world datasets of ocean trajectory prediction and regression on partial deformable shapes demonstrate that COSMOS achieves competitive performance compared to state-of-the-art SNNs in complex and noisy environments.

Problem

• Modeling high-order interactions in structured data using simplicial complexes.
• Addressing limitations of discrete filtering in simplicial neural networks.
• Controlling over-smoothing in geometric deep learning with continuous dynamics.

Innovation

• Continuous Simplicial Neural Networks (COSMOS) introduced
• Derived from PDEs on simplicial complexes
• Better control over over-smoothing in geometric deep learning