math.NA · 2026-07-16 · No. 55

Numerical Analysis, 2026-07-16.

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

    Approximation of solutions of parameter-dependent problems by residual neural networks

    Ana Carpio

    math.NA · cs.LG · math.AP

    We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions...

    arxiv.org/abs/2607.13574 · PDF

  2. 02

    Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

    Tianchi Yu, Ivan Oseledets

    math.NA · cs.AI · cs.CE · cs.LG

    For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising...

    arxiv.org/abs/2607.13566 · PDF

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