math.NA · 2026-07-22 · No. 61

Numerical Analysis, 2026-07-22.

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

01 — The papers

3 entries
  1. 01

    1-Lipschitz Neural Networks on Hadamard Manifolds

    Davide Murari, Marta Ghirardelli, Ben Adcock, Elena Celledoni, Brynjulf Owren, Carola-Bibiane Schönlieb

    math.NA · cs.LG

    Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann...

    arxiv.org/abs/2607.19335 · PDF

  2. 02

    Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(Ω)$ A Priori Error Bounds with Application to Mean Escape Time Computation

    Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou

    math.NA · cs.LG · math.ST · stat.ML

    Motivated by the numerical computation of the Mean Escape Time (MET) $τ:Ω\to\mathbb{R}$ of a stochastic process from a bounded domain $Ω\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $ρ$. Combining...

    arxiv.org/abs/2607.19167 · PDF

  3. 03

    A Second-Moment Theory for Floating-Point Reduction Trees

    Piyush Sao, Narasinga Miniskar, Pedro Valero-Lara, Keita Teranishi, Sudip Seal

    math.NA · cs.DC · cs.IT · cs.MS

    Summation error depends on partial-sum order, which standard worst-case bounds omit. To capture this dependence, we derive an exact mean-square error (MSE) recurrence for a binary reduction tree T under conditionally unbiased rounding. With unit roundoff u, the constant-nu model sets the local variance at pre-rounding value x to nu u^2 x^2. Its leading tree-dependent cost for the input vector p is p^T K_T p, where the common-ancestor kernel...

    arxiv.org/abs/2607.18758 · PDF

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