INDEQS: Informed Neural controlled Differential EQuationS
cs.LG, stat.ML
Submitted: 2026-06-17
Updated: 2026-10-01
Code: https://github.com/Mitchi1/indeqs
License: http://creativecommons.org/licenses/by/4.0/
The gist: Neural Controlled Differential Equations (NCDE) provide a powerful continuous-time framework for forecasting time series, but standard graph-based extensions typically learn spatial structure purely
Terminology
Abstract
Neural Controlled Differential Equations (NCDE) provide a powerful continuous-time framework for forecasting time series, but standard graph-based extensions typically learn spatial structure purely from data, even in settings where a directed graph structure is known a priori. We introduce Informed Neural controlled Differential EQuationS (INDEQS), a graph-based NCDE forecasting method that incorporates prior knowledge of a directed graph at distinct architectural positions. INDEQS separates inner mixing of hidden states across graph nodes from outer mixing between vector field and control, and offers both a lightweight graph-constrained variant and a more expressive variant, learning additional graph connections from data via adaptive graph convolutions. To systematically study when graph informedness is beneficial in forecasting, we devise a continuous advection simulation on directed graphs, yielding synthetic spatio-temporal datasets with known ground-truth flow structure. We then evaluate INDEQS on two real-world tasks: river discharge forecasting on a hydrological network and traffic flow prediction on PeMS08. Across these synthetic and real-world benchmarks, outer informedness consistently improves mean absolute error over an uninformed NCDE with comparable parameter count, particularly on larger graphs, while inner informedness offers a more parameter-efficient alternative when strict adherence to a known adjacency is desired. A comparison of discrete convolutional and continuous-time decoders further shows that continuous decoders yield better accuracy and greater temporal flexibility on real-world tasks. An implementation of INDEQS and the advection simulation is available at https://github.com/Mitchi1/indeqs.
Sources
- Synthetic Datasets for Machine Learning on Spatio-Temporal Graphs using PDEs
- An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling
- Continuous Graph Flow
- Modeling Advection on Directed Graphs using Mat'ern Gaussian Processes for Traffic Flow
- Neural Controlled Differential Equations for Online Prediction Tasks
- Graph Neural Ordinary Differential Equations
- Temporal Graph Networks for Deep Learning on Dynamic Graphs
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