{
  "abstract": "Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder ---a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to ---and, in some bifurcating regimes, better than--- that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24332",
  "author_total": 4,
  "authors": [
    {
      "name": "Alessandro Della Pia"
    },
    {
      "name": "Lucia Russo"
    },
    {
      "name": "Ioannis Kevrekidis"
    },
    {
      "name": "Constantinos Siettos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/121r72hhd3mvw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24332",
  "fulltext_url": null,
  "id": "121r72hhd3mvw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24332",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7214042945",
  "pdf_url": "https://arxiv.org/pdf/2609.24332",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/121r72hhd3mvw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2609.24332",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24332"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "Autoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows",
  "title_text": "Autoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/121r72hhd3mvw",
  "venue": "arXiv",
  "versions": []
}