Continuous Simplicial Neural Networks

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 network (COSIMO), a novel SNN architecture derived from PDEs on simplicial complexes. We provide theoretical and experimental justifications of COSIMO's stability under simplicial perturbations. Furthermore, we investigate the over-smoothing phenomenon, a common issue in geometric deep learning, demonstrating that COSIMO offers better control over this effect than discrete SNNs. Our experiments on real-world datasets demonstrate that COSIMO achieves competitive performance compared to state-of-the-art SNNs in complex and noisy environments.

@article{einizade2025_2503.12919,
  title={ Continuous Simplicial Neural Networks },
  author={ Aref Einizade and Dorina Thanou and Fragkiskos D. Malliaros and Jhony H. Giraldo },
  journal={arXiv preprint arXiv:2503.12919},
  year={ 2025 }
}