math.NA · 2026-06-02 · No. 13

Numerical Analysis, 2026-06-02.

2 new papers in math.NA. Titles, authors, abstracts. Links to arXiv. Want this in your inbox every morning? Subscribe →

01 — The papers

2 entries
  1. 01

    Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers

    Henry Kasumba, Ronald Katende

    math.NA · cs.CE · cs.LG

    Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural network (PINN) is used not as the final solver, but...

    arxiv.org/abs/2606.02475 · PDF

  2. 02

    Spectral Audit of In-Context Operator Networks

    Zhiwei Gao, Liu Yang, George Em Karniadakis

    math.NA · cs.LG

    Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed...

    arxiv.org/abs/2606.02427 · PDF

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