{
  "abstract": "We propose a semigroup-aware structured kernel neural operator for source-driven evolution PDEs. Unlike generic time-conditioned kernel models, the proposed formulation represents temporal dependence through a latent exponential propagator \\(e^{tA_θ}\\), where \\(A_θ\\) is a learned latent generator. This provides a structured transition from a non-ODE kernel representation to an ODE-inspired kernel operator whose latent temporal evolution satisfies the composition principle of evolution semigroups. For time-dependent forcing terms, a direct kernel-based evaluation requires repeated quadrature over delayed kernel evaluations. To address this computational bottleneck, we derive a latent-first reformulation of the source term: the forcing field is projected onto the kernel-induced latent space, where its accumulated effect is represented as the solution of a forced linear latent dynamical system. For a fixed learned kernel, this reformulation is exact. The resulting system can then be integrated using a \\(ϕ_1\\)-based exponential integrator, yielding a source-evaluation path that is structurally aligned with the learned latent propagator. The resulting framework combines geometry- and physics-conditioned low-rank kernel factorization with alternative source-evaluation paths based on either explicit Duhamel quadrature or latent exponential integration. Dissipativity constraints on the latent generator and identity-preserving corrections near the initial time are optional architectural components. Numerical experiments on advection-iffusion-reaction (ADR) problems compare the non-ODE and semigroup-aware formulations, evaluate the quadrature and \\(ϕ_1\\) source paths, and assess predictive accuracy and computational cost.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39845",
  "author_total": 5,
  "authors": [
    {
      "name": "Naseem Abbas"
    },
    {
      "name": "Vittorio Colao"
    },
    {
      "name": "Davide Macrì"
    },
    {
      "name": "Matthieu Barreau"
    },
    {
      "name": "William Spataro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1j26prrnjre8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39845",
  "fulltext_url": null,
  "id": "a1j26prrnjre8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39845",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39845",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1j26prrnjre8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39845",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39845"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs",
  "title_text": "A Semigroup-Aware Kernel Neural Operator with Latent Exponential Integration for Forced PDEs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/a1j26prrnjre8",
  "venue": "arXiv",
  "versions": []
}