# Discrete and Continuous Deep Residual Learning Over Graphs

arXiv: https://arxiv.org/abs/1911.09554 · HTML: https://mlsanity.com/paper/1911.09554/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Nov 2019.

## Built on by (most cited first, 13 of 13)

- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Gradient Gating for Deep Multi-Rate Learning on Graphs](https://mlsanity.com/paper/2210.00513/) (2022)
- [ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networks](https://mlsanity.com/paper/2206.05437/) (2022)
- [TIDE: Time Derivative Diffusion for Deep Learning on Graphs](https://mlsanity.com/paper/2212.02483/) (2022)
- [Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FROND](https://mlsanity.com/paper/2404.17099/) (2024)
- [AbODE: Ab Initio Antibody Design using Conjoined ODEs](https://mlsanity.com/paper/2306.01005/) (2023)
- [Generalized energy and gradient flow via graph framelets](https://mlsanity.com/paper/2210.04124/) (2022)
- [Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study](https://mlsanity.com/paper/2401.04331/) (2024)
- [IGNN-Solver: A Graph Neural Solver for Implicit Graph Neural Networks](https://mlsanity.com/paper/2410.08524/) (2024)
- [Adaptive Initial Residual Connections for GNNs with Theoretical Guarantees](https://mlsanity.com/paper/2511.06598/) (2025)
- [Graph Anisotropic Diffusion](https://mlsanity.com/paper/2205.00354/) (2022)

## Abstract (arXiv)

In this paper we propose the use of continuous residual modules for graph kernels in Graph Neural Networks. We show how both discrete and continuous residual layers allow for more robust training, being that continuous residual layers are those which are applied by integrating through an Ordinary Differential Equation (ODE) solver to produce their output. We experimentally show that these residuals achieve better results than the ones with non-residual modules when multiple layers are used, mitigating the low-pass filtering effect of GCN-based models. Finally, we apply and analyse the behaviour of these techniques and give pointers to how this technique can be useful in other domains by allowing more predictable behaviour under dynamic times of computation.