Extending Graph Neural CDEs to Directed and Heterogeneous Graphs

Abstract:

In recent years, Neural Differential Equations (NDEs) [1, 2] have emerged as a powerful bridge between dynamical systems and deep learning, finding applications in both supervised learning and generative modeling. In graph representation learning, controlled differential equations (CDEs) have been leveraged to simultaneously capture the evolution of node features and graph topology [3, 4], with promising applications such as traffic forecasting [4]. Building on our recent work where we successfully integrated permutation equivariance into Graph Neural CDEs for undirected, homogeneous graphs, this thesis aims to extend the framework to additional graph modalities—specifically, directed and heterogeneous graphs. Such an extension is expected to address real-world challenges in fields like clinical sciences or climate modeling, where interactions are inherently directional and multimodal.

Research Questions:

1. How can the existing permutation equivariant framework be adapted to directed and heterogeneous graphs?
2. What theoretical guarantees—such as preservation of equivariance under heterogeneous permutations and universal approximation properties—can be maintained or enhanced in these extended settings?
3. How does the performance of the extended model compare with existing approaches on synthetic benchmarks and real-world datasets?

Prerequisites:

• Proficiency in deep learning and at least one deep learning framework (preferably using JAX, PyTorch also sufficient)
• Basic command of differential equations and group theory

Contact:

• Torben Berndt

References:

[1] P Kidger. On neural differential equations. PhD thesis, University of Oxford, 2021.

[2] R. T. Q. Chen et al. Neural ordinary differential equations. Advances in Neural Information Processing Systems, 2018

[3] T. Qin and B. Walker et al. Learning dynamic graph embeddings with neural controlled differential equations. Preprint, 2023

[4] J. Choi et al. Graph Neural Controlled Differential Equations for Traffic Forecasting. AAAI, 2022