{
  "abstract": "Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each time leaves the temporal and dynamical structure of the solution to be resolved within a broad hypothesis space. We introduce Learning$^2$, a representation-level framework that structures this space by coupling a primary representation to a second representation through a known physical transformation. The resulting cross-representation constraint restricts the effective hypothesis space and provides an ante-hoc, physically interpretable criterion for excluding solutions that satisfy the primary representation alone. We instantiate Learning$^2$ through EuLaNet, an Eulerian--Lagrangian representation for fluid dynamics. Given the velocity state $u(\\mathbf{x},t)$, EuLaNet constructs its induced Lagrangian flow map $X(\\mathbf{a},t)$ through $\\dot{X}(\\mathbf{a},t)=u(X(\\mathbf{a},t),t)$, from which material transport and finite-time deformation are derived. The resulting representation couples the predicted state to the dynamical consequences it induces, providing a second consistency criterion beyond state-level agreement. We formalize this construction through an effective hypothesis space $\\mathcal{H}_{L^2}\\subseteq\\mathcal{H}$ and define the conditions under which a consequence representation provides discriminative constraints on candidate solutions. EuLaNet is implemented as a model-independent representation module, separating the physical constraint from the downstream learning architecture. This construction provides an ante-hoc mechanism for physically interpretable constraint in scientific learning and offers a basis for developing and evaluating broader classes of Learning$^2$ architectures. The implementation is open-sourced to support the development and extension of the architecture across scientific domains.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06546",
  "author_total": 2,
  "authors": [
    {
      "name": "Sai Siddharth"
    },
    {
      "name": "Maniarasu Ravi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e4zvge4ssvw7j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06546",
  "fulltext_url": null,
  "id": "e4zvge4ssvw7j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06546",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7212060072",
  "pdf_url": "https://arxiv.org/pdf/2609.06546",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e4zvge4ssvw7j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06546",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06546"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Introductory Notes on Learning$^2$",
  "title_text": "Introductory Notes on Learning^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/e4zvge4ssvw7j",
  "venue": "arXiv",
  "versions": []
}