生成时间: 2026年09月28日 18:02:19

重点关注论文数量: 0

了解领域论文数量: 7

不相关论文数量: 12

了解领域论文（摘要翻译）

1. NEXT：物理信息神经谱指数时间差分架构

NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures

作者: Márcio Marques, Leonardo Mendonça, Leonardo M. Moreira, Christian Júnior de Oliveira, Vitor Balestro, Tiago Novello, Daniel Yukimura, Pavel Petrov, Lucas Nissenbaum

类别: cs.LG, math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31539v1

链接: http://arxiv.org/abs/2609.31539v1

摘要翻译

中文标题: NEXT：物理信息神经谱指数时间差分架构

摘要翻译: 物理信息神经网络（PINNs）构建含时偏微分方程解的神经表示，自然地融合物理知识与观测数据，这使得它们非常适合正向与逆向 PDE 问题。然而，PINNs 已知会遭受谱偏差和缺乏因果性的问题。神经谱架构（NeuSA）是最近提出的 PINNs 替代方案，缓解了这两个问题，但它们的数值积分对于许多相关物理问题中出现的刚性微分方程会变得不稳定。本研究提出神经谱指数时间差分架构（NEXT），它将 NeuSA 中 PDE 解的谱表示与高阶指数积分器相结合。在该方法中，由 PDE 诱导的向量场的线性刚性部分通过矩阵指数被精确积分，而可能非线性的剩余部分则由神经网络建模。NEXT 的有效性通过在一组刚性 PDE 上的基准实验得到验证，在这些实验中，NEXT 稳定且准确，而 NeuSA 在数值上发散。还表明 NEXT 可应用于反问题，其中模型必须从稀疏数据中学习未知参数或边界条件。本工作使用的所有代码公开于：https://github.com/marcioh2m/next.git 。

2. 用于可压缩 Euler 方程的热力学相容半隐式有限体积格式

A thermodynamically compatible semi-implicit finite volume scheme for the compressible Euler equations

作者: M. Dumbser, G. Puppo, S. Rinaldi, A. Thomann

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31065v1

链接: http://arxiv.org/abs/2609.31065v1

摘要翻译

中文标题: 用于可压缩 Euler 方程的热力学相容半隐式有限体积格式

摘要翻译: 本文针对气体动力学中的可压缩 Euler 方程，构造了一种新的半隐式 (SI)、双曲且热力学相容 (HTC) 格式。其中的挑战在于将熵不等式作为主要演化方程求解，而不是求解总能量守恒律；在我们的框架中，总能量守恒律是一个附加守恒律，需要由数值格式作为相容离散化的结果来满足。所提方法具有预测-校正特征，其中我们首先在不考虑熵产生的情况下求解等熵问题。在此过程中，熵以及动量通量中的非线性对流项被显式离散，而质量通量和压力项则被隐式处理。然后，将离散动量方程代入离散质量守恒方程，得到一个关于压力的弱非线性系统，可用嵌套 Newton 型算法高效求解。在此预测的基础上，加入熵中的非负产生项，从而保证数值格式满足单元熵不等式。为保证离散层面上的总能量全局守恒，引入了一种新的修正全局 Abgrall 型通量校正。所提出的数值方法允许时间步长仅受物质波速限制，而不受声速限制。这使得该格式特别适用于低 Mach 数区域的流动。给出一维和二维空间中的数值结果，以评估新格式的理论性质。

3. 无穷拉普拉斯问题与 $p$-Laplace 问题的深度学习求解器与代理模型

Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems

作者: Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See

类别: math.NA, cs.LG

发布日期: 2026-09-25

arXiv ID: 2609.30809v1

链接: http://arxiv.org/abs/2609.30809v1

摘要翻译

中文标题: 无穷拉普拉斯问题与 $p$-Laplace 问题的深度学习求解器与代理模型

摘要翻译: 我们研究了使用神经网络求解器处理无穷拉普拉斯问题与 $p$-Laplace 问题，这些问题在非线性分析中具有基础性地位，并且有实际应用。我们的方法采用物理信息神经网络（PINNs）和深度算子网络（DeepONets），来处理与较大 $p$ 值（从 $2$ 到 $1000$）相关的计算挑战，覆盖多种二维和三维区域。与传统基于物理的求解器相比，我们的方法具有优势，尤其是在三维情形下，此时基于网格的求解器处理这些问题会变得非常昂贵。我们还针对这两个问题的 PINN 近似建立了条件收敛结果，并针对参数化 $p$-Poisson 问题建立了 DeepONet 的通用逼近结果。我们通过数值实验证明了这些神经网络求解器的有效性，并将它们的性能与常规方法进行了比较。

4. 稀疏连接秩启发神经网络

Sparsely connected rank-inspired neural network

作者: Yunqing Huang, Wu Su, Nianyu Yi, Peimeng Yin

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.30750v1

链接: http://arxiv.org/abs/2609.30750v1

摘要翻译

中文标题: 稀疏连接秩启发神经网络

摘要翻译: 我们提出了一种用于函数逼近和偏微分方程（PDEs）数值求解的稀疏连接秩启发神经网络（SCRINN）。通过引入结构化稀疏连接架构以及正交性和归一化引导的损失函数，SCRINN 在构造近似标准正交神经基的同时，促进其求值矩阵的秩逐步增长，并减少可训练参数的数量。我们建立了一个两阶段框架，其中首先构造近似标准正交神经基，随后使用最小二乘方法确定输出权重。针对函数逼近以及多种稳态、时间依赖和非线性偏微分方程的数值实验表明，SCRINN 能够以显著更低的训练成本持续获得高逼近精度。代码已在 GitHub 上公开。

5. 具有 Radon 测度跟踪和逐点控制约束的 Darcy 最优控制整体式间断 Galerkin 框架

A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints

作者: SeongHee Jeong, Sanghyun Lee

类别: math.NA, math.OC

发布日期: 2026-09-25

arXiv ID: 2609.30707v1

链接: http://arxiv.org/abs/2609.30707v1

摘要翻译

中文标题: 具有 Radon 测度跟踪和逐点控制约束的 Darcy 最优控制整体式间断 Galerkin 框架

摘要翻译: 我们研究非均质多孔介质中由 Darcy 方程支配的椭圆型最优控制问题，其中控制带有逐点箱约束。目标泛函通过一个 Radon 测度构造，这使得期望压力状态可以在不同维数的观测集上跟踪，包括点、曲线和子区域，并在单一表述内完成。状态方程和伴随方程采用对称内罚间断 Galerkin 方法离散，得到对强渗透率间断稳健的局部质量守恒逼近，而控制则用分片常数逼近。状态、伴随和控制作为主要未知量保留在单一耦合最优性系统中，并通过原始-对偶主动集策略整体求解。我们建立离散的稳定性和适定性，并推导两个变量的先验 $L^2$ 误差估计。主要困难是由测度值跟踪数据引起的伴随状态正则性降低。该分析通过由连续最优状态驱动的中间伴随来控制由此产生的伴随-控制耦合。数值实验证实了点、曲线和子区域观测的预期收敛阶，展示了与网格无关的原始-对偶主动集迭代次数，并展示了局部质量守恒。

6. 一种用于各向异性椭圆问题的内罚间断 Galerkin 离散的非重叠谱加性 Schwarz 方法

A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems

作者: Maurice S. Fabien, Sijing Liu, Marcus Sarkis

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.30639v1

链接: http://arxiv.org/abs/2609.30639v1

摘要翻译

中文标题: 一种用于各向异性椭圆问题的内罚间断 Galerkin 离散的非重叠谱加性 Schwarz 方法

摘要翻译: 我们设计并分析了一种用于各向异性椭圆问题的内罚间断 Galerkin 离散的非重叠加性 Schwarz 预条件子。该预条件方法与 Krylov 子空间迭代相结合，被证明独立于高度间断（且各向异性）的跳跃系数以及子域尺寸。为了提高有效性，考虑了各种辅助空间以减小粗网格算子的规模。我们展示了如何修改加性 Schwarz 预条件子，使其适用于非对称 IPDG 格式。若干数值实验验证了理论并证实了预条件子的稳健性。

7. 冻结 Calderón：形状不确定性下 Maxwell 散射的鲁棒预条件

Freezing Calderón: Robust Preconditioning for Maxwell Scattering under Shape Uncertainty

作者: Paul Escapil-Inchauspé, Carlos Jerez-Hanckes

类别: math.NA

发布日期: 2026-09-24

arXiv ID: 2609.30559v1

链接: http://arxiv.org/abs/2609.30559v1

摘要翻译

中文标题: 冻结 Calderón：形状不确定性下 Maxwell 散射的鲁棒预条件

摘要翻译: Calderón 预条件恢复了 Maxwell 电场积分方程边界元离散的与网格无关的条件性，但组装对偶重心算子可能主导计算成本。我们证明，该预条件子可以在一个标称几何上组装一次，并在保持连通性的形状扰动中重复使用。RWG-BC 自由度的精确组合对齐传输了对偶配对，仅使对偶网格算子保持冻结。在一致离散稳定性和几何算子连续性假设下，一项双参数算子预条件分析给出了一个谱条件数界，该界在网格尺寸上一致，并在张量化后对随机 Galerkin 维数也一致。在光滑和非光滑散射体上的数值实验表明，在显著的形状变化下，冻结预条件子紧密跟踪新组装的 Calderón 预条件子。在 NASA almond 上的一项 Monte Carlo 实验，其中在构造每个形状之前筛选其可容许性，以确保没有任何实现被拒绝，使两种策略之间的差异小于实现之间的变化，并且该处的网格加密使冻结迭代次数保持不变。在我们的稠密实现中，重用对每个新实现带来约 30 倍的实测加速；基于已发表的压缩组装计时的成本模型估计给出约 2 倍。

不相关论文（基本信息）

1. 皮质骨与松质骨电学性质的数值模拟：一个简化模型

Numerical Simulation of Electrical Properties in Cortical and Trabecular Bone: A Simplified Model

作者: María José Cervantes, Catalina A. Cely-Ortíz, C. Manuel Carlevaro, Ramiro M. Irastorza

类别: physics.med-ph, math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31584v1

链接: http://arxiv.org/abs/2609.31584v1

摘要: The electrical properties of biological tissues depend on their composition and microstructure and determine their response to applied electric fields. In bone tissue, these properties are closely related to its composition, porosity, and microstructural organization. However, simplified models that link bone microstructure (trabecular and cortical) to its effective electrical properties while accounting for the electrical anisotropy of cortical bone remain scarce. In this work, a multiscale approach was developed, combining finite element simulations with Bruggeman-based homogenization models. Trabecular bone was modeled using microCT derived geometries, and its effective conductivity was analyzed as a function of BV/TV and free water content. Cortical bone was represented by a simplified three-dimensional network of Haversian canals (axial = z) and Volkmann canals (transverse), incorporating their directional organization through an anisotropic Bruggeman formulation. The models captured the conductivity range reported experimentally for trabecular bone (approximately 25-200 mS/m) and reproduced the anisotropic behavior observed in the cortical FEM simulations, with higher conductivity in the axial direction. In the generated cortical geometries, 65.5% of the total porosity was associated with axial Haversian canals. At the macroscale, the tibia model with lower trabecular and cortical BV/TV and reduced cortical thickness showed an approximately 38-39% reduction in impedance relative to the reference model. These results indicate that microstructural variations in bone tissue are reflected in its effective electrical properties and in the macroscopic impedance of the model, providing a physical and computational basis for future investigations into the sensitivity of electrical measurements to changes in bone microstructure.

2. 边界元方法的边值修正

Boundary Value Correction for the Boundary Element Method

作者: Erik Burman, Peter Hansbo, Mats G. Larson

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31555v1

链接: http://arxiv.org/abs/2609.31555v1

摘要: We develop a boundary element method for the Laplace equation with Dirichlet boundary conditions in a bounded three-dimensional domain. The method uses flat panels of maximum diameter (h) on an interior surrogate boundary and corrects the imposed boundary values by a first-order Taylor expansion along the panel normals. The correction uses the normal flux already present in the Calderón formulation and introduces no additional unknown field. Under uniform assumptions on the surrogate geometry and boundary operators, the penalised form is coercive when (γδ_h\le c_0<4), where (γ) is the penalty parameter and (δ_h) is the maximum normal offset. The error estimate separates trace approximation from a geometry term of size (γ^{1/2}δ_h^2|u|{H^3(Ω)}). With (γ\simeq h^{-1}), the sufficient condition (δ_h\lesssim h^{(k+1)/2}) yields an error of order (h^{k+1/2}) for (P_k/P{k-1}) Dirichlet/flux spaces under the stated regularity and uniform trace bounds. Thus an offset of order (h^2) supports the estimate through cubic Dirichlet approximation. We analyse two penalty variants and present numerical experiments on a sphere and an ellipsoid to examine convergence and sensitivity to geometry, penalty scaling and quadrature.

3. 面向复杂三维几何的谱/hp 元方法的低阶精化预处理

Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries

作者: Parv Khurana, Henrik Wüstenberg, Spencer J. Sherwin, David Moxey, Julien Hoessler, Athanasios Chatzopoulos

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31443v1

链接: http://arxiv.org/abs/2609.31443v1

摘要: Low-order refined (LOR) preconditioning replaces a high-order operator with a low-order discretisation on a refined nodal mesh. For tensor-product elements, the two operators are spectrally equivalent with bounds independent of the polynomial order $P$, but the construction does not extend directly to simplex and mixed-element discretisations. This work makes two contributions: it extends LOR preconditioning to simplex and mixed-element discretisations, including triangular, tetrahedral, and prismatic elements, and establishes a generalised Vandermonde transformation linking the modal and nodal LOR formulations, showing that the resulting preconditioned spectra and Krylov convergence are independent of the high-order basis. Numerical experiments show controlled iteration growth on triangular meshes despite increasing condition number, and controlled iteration counts up to $P=6$ on tetrahedral, prismatic, and mixed-element meshes. A single algebraic multigrid V-cycle per outer iteration gives the best balance of iteration count and cost. The method is applied to a production incompressible Navier-Stokes simulation of a race-car front-wing and wheel configuration, discretised on a mesh of $2.87\times10^6$ mixed prismatic and tetrahedral elements giving $32.2\times10^6$ pressure degrees of freedom at $P=3$. LOR reduces the mean pressure conjugate gradient (CG) iteration count from $235.3$ to $5.5$, and the pressure-solve time over 1000 timesteps by 16.1%, relative to the default production static-condensation diagonal preconditioner in Nektar++. The one-time cost of constructing the LOR preconditioner is amortised over the production simulation.

4. 基于免纤维束成像的弥散张量成像的解剖连接及其在EEG脑成像中的应用

Anatomical Connectivity from Tractography-free Diffusion Tensor Imaging and Its Application to EEG Brain Imagin

作者: Joonas Lahtinen

类别: math.NA, math.DS, math.PR

发布日期: 2026-09-25

arXiv ID: 2609.31410v1

链接: http://arxiv.org/abs/2609.31410v1

摘要: This proof-of-concept paper demonstrates how the Kalman filter, as a brain imaging method, can be enhanced by incorporating raw diffusion tensor imaging (DTI) data without tractography, in addition to EEG recordings. An efficient algorithm is designed to streamline the otherwise tedious process of inferring connectivity between brain regions from DTI data. The DTI-based evolution model is applied to the Kalman filter and the Standardized Kalman filter, and these are compared with their conventional counterparts using the random walk evolution model. The numerical experiments use synthetic somatosensory and auditory evoked potentials. The results show that the DTI-modelled Kalman filter can obtain deep activity in the brainstem and thalamus. Moreover, the connection model limits the estimated spreads to the correct or nearest anatomical brain region.

5. 带 Kraichnan 输运噪声的随机二维 Euler 方程的全离散逼近的收敛性

Convergence of a fully discrete approximation for the stochastic 2D Euler equations with Kraichnan transport noise

作者: Abhishek Chaudhary, Ujjwal Koley, Andreas Prohl

类别: math.PR, math.AP, math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31369v1

链接: http://arxiv.org/abs/2609.31369v1

摘要: We study a fully discrete approximation theory for the two-dimensional incompressible Euler equations on $\mathbb{T}^{2}$, in vorticity form, driven by Kraichnan-type transport noise and with $\mathbb{H}^{-1}(\mathbb{T}^2)$-valued initial vorticity. The main new ingredient is a finite-dimensional coercivity estimate for the torus Kraichnan model, which captures at the discrete level the regularizing mechanism induced by the rough transport noise. Together with a new compactness strategy based on continuous equation-based interpolants and localized discrete estimates, this coercivity estimate gives tightness of the approximations and shows that subsequential limit is a weak martingale solution of the Euler equations.

6. 非均匀平板输运的扩散综合加速的一致收敛性

Uniform convergence of diffusion synthetic acceleration for heterogeneous slab transport

作者: Matthias Schlottbom

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31346v1

链接: http://arxiv.org/abs/2609.31346v1

摘要: The diffusion synthetic accelerated (DSA) source iteration is a standard solver for the radiative transfer equation. Using Fourier analysis, a convergence rate (ρ_\infty(c)\le0.2247,c) with (c) the maximum ratio of scattering to total cross section has been established for an infinite homogeneous medium. For slab geometry with inflow boundary conditions and arbitrary bounded cross sections we prove that the spectral radius of the DSA iteration is at most (ρ_\infty(c)). We show that this convergence rate carries over to a variational discretization of the DSA iteration on every conforming tensor-product Galerkin space whose angular factor contains the constants. The analysis rests on an exact min–max characterization of the spectral radius. From it we derive a checkable sufficient condition for a given rate. We verify this condition using a suitable energy-stable projection, a weighted angular average whose weight is chosen so that the condition holds with the rate (ρ_\infty(c)). Since the projection maps discrete spaces into discrete spaces, the argument applies verbatim to the discrete iteration. As a consequence, the condition number of the preconditioned system is uniformly bounded by (1+0.29,c), which implies rapid convergence of the preconditioned conjugate gradients method.

7. 一种新的用于耦合偏微分方程问题的算子分裂方法，及其在孔隙力学中的应用

A new operator splitting method for coupled PDE problems, with an application to poromechanics

作者: Francesca Marcon, Stefano Scialò

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31320v1

链接: http://arxiv.org/abs/2609.31320v1

摘要: The present work proposes a new operator splitting technique. The method leverages ideas used in PDE-constrained domain decomposition, adapting them to the objective of decoupling equations defined on the same geometric domain. The methodology is applied here to the resolution of coupled flow and deformation in the linear quasi-static Biot problem. An overview of its main characteristics is provided, also in comparison with the well established fixed-stress split for poroelasticity.

8. 能量极小化区域分解

Energy-minimizing domain decomposition

作者: Muhammad Hassan, Benjamin Stamm, Lambert Theisen

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31182v1

链接: http://arxiv.org/abs/2609.31182v1

摘要: We propose a novel domain decomposition framework based on energy minimization, applicable to a wide class of linear and nonlinear partial differential equations. This energy-minimizing domain decomposition (EMDD) method addresses problems whose solutions are minimizers of an energy functional, either unconstrained or constrained to a manifold, and thus covers source and eigenvalue problems in a single formulation. Besides introducing the general framework, we derive the algorithmic realization of the method for four classes of model problems, namely linear and nonlinear source and eigenvalue problems, and derive the computational complexity of the method. Numerical experiments demonstrate the robustness and efficiency of the method across all four model problems and show that it compares favourably against existing model-specific domain decomposition methods.

9. 时滞微分系统的算子-ADI平衡截断

Operator-ADI balanced truncation of delay differential systems

作者: Timo Reis

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.31044v1

链接: http://arxiv.org/abs/2609.31044v1

摘要: We develop a balanced truncation method for linear delay differential systems based on a finite-rank alternating-direction implicit (ADI) iteration for the associated operator Lyapunov equations. The method acts directly on the infinite-dimensional history-state realization and does not discretize the delay interval during Gramian approximation or balancing. The finite-rank reachability and observability factors are represented exactly in Takenaka–Malmquist coordinates, reducing the required computations to finite-dimensional linear algebra. These factors yield a finite-dimensional, delay-free reduced model. For real system data, the implementation remains in real arithmetic. Under suitable conditions on the ADI shifts and a gap in the Hankel singular values, we prove convergence in $\mathcal H_\infty$ to the reduced transfer function obtained by exact balanced truncation of the infinite-dimensional system. The construction extends to finitely many discrete delays. Numerical experiments validate the method against independent reference computations. The complete MATLAB implementation used in the experiments is provided as supplementary material.

10. 求解标量守恒律的一种消失粘性极小化运动 Fourier 谱方法

A vanishing-viscosity minimizing movement Fourier spectral method for scalar conservation laws

作者: Lun Ji, Buyang Li, Fangyan Yao

类别: math.NA

发布日期: 2026-09-25

arXiv ID: 2609.30876v1

链接: http://arxiv.org/abs/2609.30876v1

摘要: We propose a vanishing-viscosity minimizing movement (VVMM) Fourier spectral method for scalar hyperbolic conservation laws under periodic boundary conditions. The method is a fully discrete space–time residual-minimizing scheme, Fourier spectral in space and polynomial in time, stabilized by vanishing viscosity and heat-kernel postprocessing. A basic analytical difficulty in Fourier discretizations of nonlinear conservation laws is the interaction between Fourier projection and the nonlinear flux; the resulting projection residual contains unresolved high-frequency components. VVMM addresses this issue by selecting, on each time slab, the numerical solution that minimizes {a penalized cut-off viscous conservation laws residual} and then applies heat-kernel postprocessing to damp unresolved modes. We prove a fully discrete $L^1$ error estimate, in any fixed space dimension, that separates the vanishing-viscosity error, the temporal interpolation error, and the spatial Fourier truncation error. {Under exact minimization of the penalized cut-off residual and suitable parameter coupling,} the estimate yields, for each fixed $γ>0$, an $L^1$ convergence rate $N^{-1/2+γ}$, with a constant depending on $γ$. Numerical experiments in one and two dimensions illustrate stable shock capturing without visible Gibbs oscillations and convergence behavior consistent with the theory.

11. 水平海床运动生成的全非线性波下方的质点轨迹

Particle Trajectories Beneath Fully Nonlinear Waves Generated by Horizontal Seabed Motion

作者: João Vitor P. Poletto, David Andrade, Marcelo V. Flamarion, Roberto Ribeiro-Jr

类别: physics.flu-dyn, math.NA

发布日期: 2026-09-25

arXiv ID: 2609.30687v1

链接: http://arxiv.org/abs/2609.30687v1

摘要: We investigate fully nonlinear water waves and fluid-particle dynamics generated by the horizontal motion of a seabed obstacle with prescribed time-dependent velocity. The governing equations are the full Euler equations, formulated in a time-dependent conformal domain that simultaneously maps the moving free surface and seabed onto fixed boundaries. The main contribution of this work is a Lagrangian formulation for computing particle trajectories in the resulting genuinely unsteady conformal domain. We derive a closed-form trajectory system in the canonical domain in which the time dependence of the conformal map is entirely represented by the real and imaginary parts of an analytic function that can be evaluated spectrally from the surface. We apply the formulation to waves generated by horizontal submarine landslide motion and characterize particle displacements throughout the fluid as functions of the initial particle position and Froude number. Our findings identify distinct regions in which particle motion is predominantly associated with the moving seabed, the generated wave, or the combined action of both. The results reveal a transition in the dominant mechanism driving particle motion: at low Froude numbers, particle displacements are primarily associated with the moving seabed, whereas at high Froude numbers the generated wave becomes increasingly dominant, particularly near the free surface and away from the obstacle path.The numerical predictions are benchmarked against laboratory data available in the literature, showing good agreement and providing a quantitative assessment of the model accuracy. Moreover, laboratory topographies beyond those considered here can be readily incorporated into the numerical framework through a Hermite interpolation procedure.

12. Muon 的收敛性保证：新的参数区域与推广

Convergence guarantees for Muon: New parameter regimes and generalizations

作者: Arthur C. B. de Oliveira, Dhruv D. Jatkar, Guilherme S. Vicinansa, Eduardo D. Sontag

类别: math.NA, cs.AI, math.OC

发布日期: 2026-09-24

arXiv ID: 2609.30546v1

链接: http://arxiv.org/abs/2609.30546v1

摘要: In this paper, we establish the first asymptotic convergence guarantees for the Muon algorithm through a more accurate proxy for the Newton-Schultz iteration than the typical matrix sign function. We prove that, for appropriate choices of hyperparameters, the iterates satisfy $\lim_{k\to\infty}\|\nabla f(x_k)\|=0$, and, under a global Polyak-Łojasiewicz condition, that the sequence of function values converges linearly. The key insight is that the regularization, implicit in Muon’s Newton-Schulz implementation, induces a bounded preconditioner, exposing Muon as a \emph{preconditioned Polyak heavy-ball} method and enabling a classical Lyapunov analysis. This observation naturally motivates applying the same preconditioning structure to the Nesterov gradient evaluation. We formalize this idea by introducing \emph{Muesterov}, a Nesterov-based variant of Muon, and prove that it enjoys the same convergence guarantees, extending the theoretical framework beyond the heavy-ball setting. Numerical experiments on a scalar cross-entropy problem corroborate the theory and illuminate the joint role of the learning rate and the Newton-Schulz regularizer in controlling convergence. Preliminary numerical simulations training the nanoGPT dataset provide intuition regarding the relevance of the observations in this paper to practical applications.