{
  "abstract": "Memory-dependent evolution equations replace the time derivative of the state by a convolution with a memory kernel K(t). They are used in anomalous diffusion, viscoelasticity and tumour-growth modelling, usually with K fixed in advance, most often as the power-law kernel of a Caputo derivative. We study the joint recovery of K and the state u with a physics-informed neural network (PINN) trained by a block-coordinate Variable Projection (VarPro) scheme, for four kernel parametrizations: a non-negative Prony series, a fractional (Caputo) power law, a stretched exponential and a shape-constrained spline. In the best-performing runs on a synthetic benchmark, the state is recovered with relative errors below 0.05% for all four families, while kernel recovery depends strongly on the family. The fractional kernel is recovered almost exactly, the stretched exponential to 5.7%, and the Prony series with a kernel-function reconstruction error of 2–4% in its best runs. A local sensitivity analysis at the exact solution, which requires no training, shows that the individual Prony parameters are severely ill conditioned: the column-normalized condition number of the sensitivity matrix is more than two orders of magnitude larger when the true decay rates are a factor of four apart than when they are well separated. Fitted against the exact state, without a network, the kernel and the three active Prony modes are recovered to a rounding error, while the redundant zero-weight modes of the over-parametrized representation remain non-identifiable. A perturbation of only 0.1% raises the active-mode parameter error to tens of percent while the kernel error stays near 0.1%, and this error tracks the condition number. A Gauss–Newton/Laplace approximation and an eight-replicate training ensemble agree that the rates are far less well determined than K(t). A spline that assumes only positivity, monotonicity and convexity reaches a kernel error of about 6%. The results show that an accurate state does not guarantee accurate recovery of the kernel or of its parameters, and that state accuracy, kernel-function accuracy and parameter identifiability should be assessed separately.",
  "abstract_shown": true,
  "affiliations": [
    "CIMA and Department of Mathematics, Instituto Superior de Engenharia de Lisboa–ISEL, Polytechnic University of Lisbon, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisbon, Portugal"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "José Alberto Rodrigues",
      "orcid": "0000-0001-5630-7149"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/235bspzgn9t56.bib",
  "cited_by": 0,
  "doi": "10.3390/math14203689",
  "fulltext_url": null,
  "id": "235bspzgn9t56",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/20/3689",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7212498277",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/235bspzgn9t56.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:29Z",
      "native_id": "10.3390/math14203689",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/20/3689"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "openalex:3109|Model Reduction and Neural Networks",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Learning Memory Operators with Physics-Informed Neural Networks: State Recovery, Kernel Reconstruction, and Parameter Identifiability",
  "title_text": "Learning Memory Operators with Physics-Informed Neural Networks: State Recovery, Kernel Reconstruction, and Parameter Identifiability",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:29Z",
  "url": "https://research.draconitic.com/paper/235bspzgn9t56",
  "venue": "Mathematics",
  "versions": []
}