INDEQS: Informed Neural controlled Differential EQuationS

2026-06-17

AI Summary

This work addresses a key limitation in existing neural graph differential equation approaches for time series forecasting, which often disregard known directed graph structures and instead infer spatial dependencies solely from data. To remedy this, we propose INDEQS—a neural controlled differential equation framework that explicitly incorporates prior directed graph structure at the architectural level by decoupling internal node state mixing from external control–vector field interactions. We develop both a lightweight constrained variant and an adaptive enhanced version, and introduce a continuous-time simulation benchmark based on directed-graph advection processes to systematically evaluate the efficacy of graph priors. Experiments on synthetic data as well as real-world river discharge and traffic flow tasks demonstrate that integrating graph priors significantly reduces mean absolute error, particularly in large-scale graphs, and that our continuous-time decoder achieves superior accuracy and greater temporal modeling flexibility compared to discrete convolutional alternatives.

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.

Problem

• Neural Controlled Differential Equations
• graph-based forecasting
• directed graph prior
• spatio-temporal prediction
• time series forecasting

Innovation

• Neural Controlled Differential Equations
• graph-informed modeling
• adaptive graph convolution
• continuous-time forecasting
• spatio-temporal dynamics