{
  "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 .",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Transactions on Machine Learning Research (TMLR), ISSN: 2835-8856, 2026",
  "arxiv": "2606.19138",
  "author_total": 6,
  "authors": [
    {
      "name": "Michael Detzel"
    },
    {
      "name": "Gabriel Nobis"
    },
    {
      "name": "Kristiyan Blagov"
    },
    {
      "name": "Juri F. Schubert"
    },
    {
      "name": "Jackie Ma"
    },
    {
      "name": "Wojciech Samek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/422wnnvpey0gn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.19138",
  "fulltext_url": null,
  "id": "422wnnvpey0gn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.19138",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2215|Traffic Prediction and Management Techniques"
    ]
  ],
  "openalex": "W7165132683",
  "pdf_url": "https://arxiv.org/pdf/2606.19138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/422wnnvpey0gn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:14Z",
      "native_id": "2606.19138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.19138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "INDEQS: Informed Neural controlled Differential EQuationS",
  "title_text": "INDEQS: Informed Neural controlled Differential EQuationS",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:14Z",
  "url": "https://research.draconitic.com/paper/422wnnvpey0gn",
  "venue": "arXiv",
  "versions": []
}