INDEQS: Informed Neural controlled Differential EQuationS

A new graph-based forecasting method called INDEQS incorporates prior knowledge of directed graph structures into Neural Controlled Differential Equations, a continuous-time modeling framework. The approach, submitted to arXiv on 17 June 2026, separates how information mixes across graph nodes from how it flows between the model's vector field and control signals. [1][2] Neural Controlled Differential Equations (NCDEs) offer a continuous-time framework for time-series forecasting, but standard graph-based versions typically learn spatial relationships entirely from data, ignoring known directed structures. [1][2] The INDEQS method, short for Informed Neural controlled Differential EQuationS, addresses this by injecting a priori graph knowledge at two distinct architectural positions: inner mixing, which governs how hidden states blend across nodes, and outer mixing, which controls interaction between the vector field and the control path. [2] The researchers provide two variants. A lightweight, graph-constrained version adheres strictly to the supplied adjacency matrix, while a more expressive variant learns additional connections through adaptive graph convolutions. [2] To isolate when informedness helps, the team built a continuous advection simulation on directed graphs, generating synthetic spatio-temporal datasets with known ground-truth flow. [2] On two real-world benchmarks — river discharge forecasting on a hydrological network and traffic flow prediction on the PeMS08 dataset — outer informedness consistently reduced mean absolute error compared to an uninformed NCDE with a similar parameter count, especially on larger graphs. [2] Inner informedness proved more parameter-efficient when strict adherence to a known adjacency was required. [2] The study also compared discrete convolutional decoders against continuous-time decoders, finding that continuous decoders delivered better accuracy and greater temporal flexibility on real-world tasks. [2] Code for INDEQS and the advection simulation is publicly available on GitHub. [2] The paper appeared on arXiv, the open-access e-print repository that hosts preprints across mathematics, physics, computer science, and related fields. [6] As of November 2024, arXiv was receiving roughly 24,000 submissions per month. [6] The work was posted under the machine learning category (cs.LG) and includes links to bibliographic tools and code-finding services through arXivLabs, a framework that lets community collaborators build experimental features on the platform. [4][5]