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Continuous Simplicial Neural Networks

Aref Einizade, Dorina Thanou, Fragkiskos D. Malliaros, Jhony H. Giraldo

TL;DR

COSIMO introduces a PDE-based, continuous SNN operating on simplicial complexes to model higher-order interactions. By decoupling diffusion on the lower and upper Hodge Laplacians with time parameters $t_d$ and $t_u$, the model achieves dynamic receptive fields and explicit control over over-smoothing, supported by stability guarantees under simplicial perturbations. The paper provides both theoretical analyses and extensive experiments showing competitive performance on trajectory prediction, mesh regression, and node/graph classification tasks, while offering practical considerations for computation via exponential filtering and eigen-decomposition. Overall, COSIMO advances robust, continuous learning on higher-order structures with improved stability and interpretability, and the authors provide open-source code at https://github.com/ArefEinizade2/COSIMO.

Abstract

Simplicial complexes provide a powerful framework for modeling higher-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. The implementation codes are available in https://github.com/ArefEinizade2/COSIMO.

Continuous Simplicial Neural Networks

TL;DR

COSIMO introduces a PDE-based, continuous SNN operating on simplicial complexes to model higher-order interactions. By decoupling diffusion on the lower and upper Hodge Laplacians with time parameters and , the model achieves dynamic receptive fields and explicit control over over-smoothing, supported by stability guarantees under simplicial perturbations. The paper provides both theoretical analyses and extensive experiments showing competitive performance on trajectory prediction, mesh regression, and node/graph classification tasks, while offering practical considerations for computation via exponential filtering and eigen-decomposition. Overall, COSIMO advances robust, continuous learning on higher-order structures with improved stability and interpretability, and the authors provide open-source code at https://github.com/ArefEinizade2/COSIMO.

Abstract

Simplicial complexes provide a powerful framework for modeling higher-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. The implementation codes are available in https://github.com/ArefEinizade2/COSIMO.

Paper Structure

This paper contains 35 sections, 15 theorems, 44 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notation and Simplicial Complexes
  5. Discrete Simplicial Neural Network
  6. Continuous Simplicial Neural Network
  7. PDEs in Simplicial Complexes
  8. COSIMO as a Solution to the Simplicial PDEs
  9. Computational Complexity of COSIMO
  10. Stability Analysis
  11. Understanding Over-smoothing in SNNs
  12. Over-smoothing in Discrete SNNs
  13. Over-smoothing in COSIMO
  14. Experiments and Results
  15. Real-world Applications
  16. ...and 20 more sections

Key Result

Proposition 4.2

The solution to the descriptive sets of PDEs in Section sec:PDEs is given by: where $\mathbf{x}_{k,d}(0)$, $\mathbf{x}_{k,u}(0)$, and $\mathbf{x}_{k}(0, 0)$ are the initial conditions for the PDEs.

Figures (7)

  • Figure 1: Example of an abstract simplicial complex.
  • Figure 2: The PDE-based signal evolution on a simplicial complex, governed by independent diffusion processes on the lower and upper Hodge Laplacians and a coupled process integrating both spaces. The colors in the simplicial complexes represent the values of the underlying simplicial signals.
  • Figure 3: Over-smoothing results of discrete SNNs and COSIMO across different layer depths.
  • Figure 4: Stability analysis under varying SNRs.
  • Figure 5: Comparison of the performance error and GPU memory usage (in GB) across runtime (in seconds) (in both color and circle size) on both the Small and Full versions of the Shrec-16 dataset. The proposed COSIMO method has a good trade-off between runtime and memory usage while performing considerably better.
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 3.1: from yang2025hodgeaware
  • Remark 4.1
  • Proposition 4.2
  • Remark 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Theorem 5.1
  • Corollary 5.2
  • Theorem 5.3
  • Corollary 5.4
  • ...and 18 more