INDEQS: Informed Neural controlled Differential EQuationS
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 modification to graph-based NCDE forecasting methods 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 the synthetic and the river-discharge tasks, 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 .
Related
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BibTeX
@misc{detzel2026indeqs,
title = {{INDEQS: Informed Neural controlled Differential EQuationS}},
author = {Michael Detzel and Gabriel Nobis and Kristiyan Blagov and Juri F. Schubert and Jackie Ma and Wojciech Samek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.19138},
eprint = {2606.19138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19138},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - INDEQS: Informed Neural controlled Differential EQuationS
AU - Michael Detzel
AU - Gabriel Nobis
AU - Kristiyan Blagov
AU - Juri F. Schubert
AU - Jackie Ma
AU - Wojciech Samek
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2606.19138
UR - https://arxiv.org/abs/2606.19138
AB - 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 modification to graph-based NCDE forecasting methods 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 the synthetic and the river-discharge tasks, 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 .
ER -