生成时间: 2026年10月19日 16:52:27

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了解领域论文数量: 8

不相关论文数量: 13

了解领域论文（摘要翻译）

1. 混合椭圆—双曲 Monge–Ampère 问题：建模与最小二乘求解器

A Mixed Elliptic–Hyperbolic Monge–Ampère Problem: Modeling and a Least-Squares Solver

作者: G. V. Carvalho, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp, L. Kusch, W. L. IJzerman

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23544v1

链接: http://arxiv.org/abs/2610.23544v1

摘要翻译

中文标题: 混合椭圆—双曲 Monge–Ampère 问题：建模与最小二乘求解器

摘要翻译: 我们考虑一个由平行光到远场反射器的自由曲面光学设计所启发的输运问题。经典表述依赖于椭圆型或双曲型 Monge–Ampère 方程，分别导致全局凸/凹或鞍形反射器几何。然而，光滑反射器阵列的设计可能需要同时结合两类曲率的光学表面。为此，我们引入一个混合椭圆—双曲 Monge–Ampère 系统，其通过将源区域划分为一个椭圆子区域和一个双曲子区域得到，并为其数值逼近建立了一种最小二乘方法。数值例子展示了所提出框架计算孤立和周期混合型输运解的能力，从而得到光滑的反射器构型。

2. 无需稳定化或Piola变换的曲面双调和问题的有限元方法

A Finite Element Method for the Surface Biharmonic Problem without Stabilization or Piola Transforms

作者: Jiachen Gu, Xia Ji, Shuo Zhang

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23492v1

链接: http://arxiv.org/abs/2610.23492v1

摘要翻译

中文标题: 无需稳定化或Piola变换的曲面双调和问题的有限元方法

摘要翻译: 针对闭合光滑曲面上的双调和问题，我们构造了一种既不需要稳定化也不需要Piola变换的非协调有限元方法。该简约富化四次（REQ）单元仅依赖于顶点值和标量边矩。相同的局部构造适用于分片平面曲面的每个三角形。双线性形式由逐单元的Laplacian内积组成，并允许标准的有限元组装。通过将平面顶点残差与角亏损贡献分离，我们在平面网格和闭合分片平面曲面上建立了弱离散Miranda–Talenti型估计。这些估计保证了在充分细化的插值曲面网格上的均匀稳定性。我们证明了该方法的适定性、能量范数下的一阶收敛性，以及分裂$H^1$范数和$L^2$范数下的二阶收敛性。在三个曲面上的数值试验显示了预测的收敛阶，并考察了离散Miranda–Talenti常数和角亏损的缩放行为。

3. 面向带不确定性 Vlasov–Poisson 系统的保结构半拉格朗日双保真方法

A Structure-Preserving Semi-Lagrangian based bi-fidelity method for the Vlasov–Poisson system with uncertainties

作者: Xiaofeng Cai, Jiaojiao Guo, Liu Liu

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23484v1

链接: http://arxiv.org/abs/2610.23484v1

摘要翻译

中文标题: 面向带不确定性 Vlasov–Poisson 系统的保结构半拉格朗日双保真方法

摘要翻译: Vlasov–Poisson (VP) 系统的不确定性量化需要进行重复的动态模拟，从而带来巨大的计算成本。我们发展了一种保结构的半拉格朗日 (SL) 双保真近似，用于宏观可观测量。采用 MUSCL 作为无压 Euler–Poisson 系统的低保真求解器，从而能够更经济地探索随机参数空间。采用守恒 SL 格式与高阶间断 Galerkin 相结合，作为无碰撞 VP 系统的高保真求解器；其质量守恒、正性保持和低数值耗散为双保真重构提供了更可靠的高保真快照。我们进一步推导了一个条件误差界，该误差界将投影误差、离散误差和模型误差分离。针对确定性和不确定性 VP 问题的数值实验，包括多物种和高维情形，表明所提出的框架实现了准确且稳健的不确定性传播。

4. 弱非径向介质中高频强散射的加速 Born 级数

Accelerated Born series for high-frequency strong scattering by weakly non-radial media

作者: Mirza Karamehmedović, Faouzi Triki

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23428v1

链接: http://arxiv.org/abs/2610.23428v1

摘要翻译

中文标题: 弱非径向介质中高频强散射的加速 Born 级数

摘要翻译: 我们提出一种实用方法，用于加速一维、二维和三维中高频 Helmholtz 散射的修正 Born 级数。我们考虑有界、非负、紧支撑的介质，并在二维和三维中允许对径向对称性的小偏离。该方法利用散射算子谱的近似来选择单个复松弛参数。从常数参考介质的预计算特征值出发，我们估计这些特征值向给定径向剖面的演化。首阶 Wentzel-Kramers-Brillouin (WKB) 近似将该谱传播约化为标量非线性方程，随后对介质的非径向部分进行一阶修正。估计出的特征值连同对省略谱的界，定义了一个用于选择松弛参数的凸极小极大问题。这种谱准备与源无关，因此可复用于同一介质中的多个散射问题。数值算例展示了在散射算子谱半径大于 1 的强散射情形下的收敛性。与直接积分方程和有限差分解算器的比较评估了精度、残差收敛和计算时间，并在所选算例中表明在可比精度下可节省计算量。

5. 参数化 Riemann 流形上抛物型方程的基于拟蒙特卡罗的无网格方法

Quasi-Monte Carlo based meshfree methods for parabolic equations on parameterized Riemannian manifolds

作者: Zhijian He, Zhiyuan Hui, Changtao Sheng, Bihao Su, Tao Zhou

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23341v1

链接: http://arxiv.org/abs/2610.23341v1

摘要翻译

中文标题: 参数化 Riemann 流形上抛物型方程的基于拟蒙特卡罗的无网格方法

摘要翻译: 我们针对参数化 Riemann 流形上线性抛物型方程的逐点逼近，建立并分析了一个随机路径采样框架，同时处理闭流形和带 Dirichlet 边界的流形。基于 Feynman-Kac 表示，我们针对由一致椭圆算子生成的流形扩散构造了一种基于参数坐标回缩的离散化，称为 p-Ret 格式，并用其构造 Monte Carlo (MC) 与 randomized quasi-Monte Carlo (RQMC) 方法。我们推导了 p-Ret 转移的局部弱相容性以及闭流形上时间离散的一阶弱收敛性。对固定 $h$，我们证明 RQMC 采样产生比标准 MC 渐近更快的方差衰减。在闭流形上，方差为 $o(n^{-1})$，并且在额外正则性下，对任意 $\varepsilon>0$ 满足更尖锐的界 $O(n^{-2+\varepsilon})$。对 Dirichlet 问题，尽管由数值首次退出引起不连续性，嵌套均匀 scrambled Sobol’ 采样在适当正则性假设下对任意 $\varepsilon>0$ 达到 $O(n^{-1-1/s+\varepsilon})$ 的方差，其中 $s$ 表示名义路径空间维数。更值得注意的是，我们进一步提出一种 Array-RQMC 策略，该策略基于逐级低维随机点集以及将高维粒子状态映射为一维排序的 Hilbert 空间填充曲线，并在适当假设下建立 $o(n^{-1})$ 的方差率。数值结果表明，两种 RQMC 方法相对于标准 MC 均降低方差和均方根误差 (RMSE)，并且对于内在维数高达 $50$ 的流形仍然有效。

6. 可压缩多相流热力学相容有限体积格式的构型数值粘性

A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows

作者: Davide Ferrari, Ilya Peshkov, Michael Dumbser

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23247v1

链接: http://arxiv.org/abs/2610.23247v1

摘要翻译

中文标题: 可压缩多相流热力学相容有限体积格式的构型数值粘性

摘要翻译: 我们考虑由变分方法导出的可压缩多相流双曲热力学相容（HTC）模型的流体子系统。其总能量在自然状态变量下通常不是凸的，因此作用于这些变量的标量数值粘性未必产生非负熵。以体积分数重新表述方程，可将各相数量的变化与其内禀状态的变化分离。这指导了在相变量中构造构型数值粘性；在相变量中能量是凸的，同时格式保持 HTC 模型的结构。在半离散层面，新格式满足：I）相质量、混合物动量和总能量的局部守恒；II）单元熵不等式，其产生项逐面、逐相非负，且不对演化变量作凸性假设且无需裁剪；III）在显式时间步条件下分密度和体积分数的正性；IV）接触平衡的精确保持；V）与密度加权体积分数格式的等价性。关键要素为：i）将每个相跨面的跳跃精确分解为以物质速度耗散的构型部分和以声速耗散的内禀部分；ii）这种耗散在相邻单元之间具有非负权重的交换形式；iii）通过每个单元的链式法则将耗散逐侧传递到演化变量，从而将能量缺陷转化为单流体能量的 Bregman 散度；iv）在相变量中构建的可逆部分的热力学修正。性质 I）–V）得到证明并数值验证，包括一个 Rusanov 粘性产生负熵的激波管以及激波–水柱相互作用。

7. PINN-LPCDD：用于可压缩多介质流动的具有可学习且物理一致区域分解的物理信息神经网络

PINN-LPCDD: Physics-Informed Neural Network with Learnable and Physically Consistent Domain Decomposition for Compressible Multi-Medium Flows

作者: Weiheng Zeng, Hao Wang, Jiaxin Zhu, Kun Wang, Tiegang Liu, Chengliang Feng

类别: physics.comp-ph, math.NA, physics.flu-dyn

发布日期: 2026-10-16

arXiv ID: 2610.22903v1

链接: http://arxiv.org/abs/2610.22903v1

摘要翻译

中文标题: PINN-LPCDD：用于可压缩多介质流动的具有可学习且物理一致区域分解的物理信息神经网络

摘要翻译: 物理信息神经网络（PINN）在求解可压缩多介质流动时面临显著挑战，包括无法定位界面位置、对间断的逼近效果较差，以及难以处理大刚度差异介质之间的高压比问题。为解决这些问题，我们提出了具有可学习且物理一致区域分解的 PINN（PINN-LPCDD），该框架将锐利界面方法（SIM）融入 PINN 以求解可压缩多介质流动。首先，我们为每种介质设计具有可学习参数的独立子网络，以自动学习界面位置，从而使区域分解在物理上一致，并使界面处的接触间断自然保持锐利。其次，求解带人工粘性的 Euler 方程，以增强处理涉及激波问题时的鲁棒性。最后，利用 Euler 方程的尺度不变性来便于求解高压比问题。针对空气-空气、空气-氦气、空气-水以及空气-R22 问题的数值实验表明，所提方法在模拟多介质流动时具有可接受的精度。

8. de Rham 复形的非贴体离散化的几何多重网格预条件子

Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex

作者: Santiago Badia, Jordi Manyer

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.22865v1

链接: http://arxiv.org/abs/2610.22865v1

摘要翻译

中文标题: de Rham 复形的非贴体离散化的几何多重网格预条件子

摘要翻译: 我们提出两族多重网格预条件子，用于 de Rham 复形的 Riesz 映射在 H(grad)、H(curl) 和 H(div) 中的非贴体有限元离散化，这些离散化通过单元聚合进行稳定化。第一种是直接施加于非贴体算子的几何多重网格方法，其定义在一族共享相同边界表示的背景网格的层级上。其光滑子是在聚合网格上构造的星形片，从而每个片都能捕捉稳定化的核。第二种是多层辅助空间预条件子，它将在切割单元附近局部化的片光滑子与活动网格上的贴体问题相结合，而后者又由非嵌套几何多重网格方法求解。两者都被用作混合 Hodge Laplacian 的块对角预条件子的构建模块。在二维和三维、具有受控切割比例的几何以及一般几何上的数值实验表明，迭代次数对切割构型、网格尺寸和层数具有鲁棒性。我们还报告了高阶离散化、弱施加的 Dirichlet 边界条件以及基于面的稳定化的结果。

不相关论文（基本信息）

1. 线性和半线性次扩散方程在分级网格上向后 L2 方法的误差分析

Error analysis of a backward L2 method on graded meshes for linear and semilinear subdiffusion equations

作者: Natalia Kopteva, Lu Wang

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23780v1

链接: http://arxiv.org/abs/2610.23780v1

摘要: Time-fractional linear and semilinear parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are considered. Their solutions exhibit an initial singularity of type $t^σ$ for any fixed $σ\in (0,1) \cup (1,2)$. A backward L2-type discrete fractional-derivative operator of order $3-α$ is considered on nonuniform temporal meshes. Sufficient conditions for inverse monotonicity of this operator areestablished. Hence, sharp pointwise-in-time error bounds are obtained on graded temporal meshes with arbitrary degree of grading. The theoretical findings are illustrated by numerical experiments.

2. 面向随机微分方程的谱延迟校正方法

Toward Spectral Deferred Correction Methods for Stochastic Differential Equations

作者: Lisa Binkowski, Matthew R. Christian, Sebastian Götschel, Michael L. Minion

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23693v1

链接: http://arxiv.org/abs/2610.23693v1

摘要: The numerical solution of stochastic differential equations is computationally very demanding. Often it is of interest to evaluate certain moments, e.g., the expectation, by Monte Carlo sampling, thus requiring the computation of numerous realizations. The discretization error of the numerical scheme induces bias in the computed Monte Carlo estimator. As popular numerical schemes, like Euler-Maruyama or Milstein, are low-order, even reasonably coarse error bounds for the simulation are costly to achieve. While the evaluation of moments requires only weak convergence, in other applications like parameter estimation, strong convergence is important, but higher-order is typically harder to obtain in the strong than in the weak sense. In this paper we develop an efficient method with the potential of achieving higher strong and weak convergence order. It is based on the smooth approximation of Brownian bridges combined with spectral deferred correction methods well known for the numerical solution of ordinary differential equations. We discuss convergence, and present numerical results.

3. 基于结构化草图的张量列格式快速矩阵-向量乘积

Fast matrix-vector products in tensor-train format via structured sketching

作者: Alberto Bucci, Stefan Güttel, John W. Pearson

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23451v1

链接: http://arxiv.org/abs/2610.23451v1

摘要: We propose a randomized, streamable algorithm that computes a tensor-train approximation of a matrix-vector product without ever forming the full intermediate tensor. We develop three contraction strategies for streaming tensor-train approximation (STTA): sequential, fused, and pre-sketched. We focus on the regime of large operator mode sizes and show that pre-sketching can provide significant speed-ups when a lower-rank approximation of the input is sought, by pre-computing all sketch-related contractions with the operator. The pre-sketched strategy can also accelerate successive randomized compression (SRC), although to a lesser extent since its second sweep depends on the input vector and cannot be pre-computed. We compare the three strategies in terms of computational cost and memory, and demonstrate their performance on synthetic and application-driven examples where rank reduction is relevant.

4. 矩阵函数的 Cayley 访问模型：可达类与代价

Cayley access models for matrix functions: reachable class and cost

作者: Gianna M. Del Corso

类别: quant-ph, math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23429v1

链接: http://arxiv.org/abs/2610.23429v1

摘要: We study how to compute $f(A)b$ on a quantum computer, where $A$ is Hermitian. When $f$ is nonsmooth or has localized difficulty in the spectrum, classical numerical linear algebra approximates it with rational functions rather than polynomials. We propose the Cayley transform, parameterized by $σ\in \mathbb{R}$ and $γ> 0$, as an alternative to the standard quantum access model. The two parameters let one concentrate accuracy where $f$ is difficult, which standard models can achieve only by raising the polynomial degree globally. This transformation converts Hermitian matrices to unitaries, allowing generalized quantum signal processing to apply Laurent polynomials bounded on the unit circle. We compare three access models (block encoding, matrix exponential, and Cayley encoding) in terms of query complexity and of the cost of their circuit realization. When the difficulty of $f$ is localized in the spectrum, the Cayley encoding reduces the required degree by a factor of three to four, and it represents the shifted resolvent exactly at degree one. For spectral projectors, the standard models cannot reach the target accuracy at fixed degree, because their approximants cannot concentrate resolution near the discontinuity. Realizing the Cayley encoding from a block encoding requires solving a shifted linear system, whose cost offsets its query-complexity gain over the standard models. For structured matrices arising in practice, however, low-depth circuits for the encoding can be constructed directly, and the gain is preserved at the circuit level.

5. Łojasiewicz 型条件与多项式受控噪声下随机梯度下降的收敛速率

Convergence rate of stochastic gradient descent under Łojasiewicz type condition and polynomially controlled noise

作者: Paweł Węgrzyn

类别: math.OC, math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23386v1

链接: http://arxiv.org/abs/2610.23386v1

摘要: Stochastic gradient methods are widely used for solving optimization problems in many scientific fields. Recently, their popularity has risen due to the increase in machine learning’s prominence. Their convergence is often studied under the convexity assumption, which is too restrictive in modern applications. In this paper, we investigate the almost sure convergence rate and expectation under a Łojasiewicz-type gradient domination condition. This framework includes both bounded and decaying noise, as well as noise with controlled polynomial growth. The obtained results extend existing convergence by allowing a broader class of noise sequences and explicitly capturing the effect of their asymptotic behavior on the convergence rate and applicability.

6. 近似离散Radon变换的显式斜逆

Explicit Oblique Inverses of the Approximate Discrete Radon Transform

作者: Donsub Rim

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23363v1

链接: http://arxiv.org/abs/2610.23363v1

摘要: The approximate discrete Radon transform (ADRT) is a fast, hierarchical approximation of the Radon transform whose levels have explicit spectral decompositions, but its Moore-Penrose inverse, needed for data outside the range, has no closed form. We construct explicit left inverses of the ADRT, oblique projections onto its range, that are exact for every choice of their parameters and amplify noise by a constant factor relative to the Moore-Penrose inverse. Realized with polynomial filters on the butterfly structure of the levels, they are fixed linear maps of cost $O(N^{5/2}\log^{3/2}N/σ_{\min})$ for an $N\times N$ image, where $σ_{\min}$ is the smallest singular value of the ADRT; with the measured growth $σ_{\min}\sim N^{1/4}$ this is $O(N^{2.25}\log^{3/2}N)$. A hybrid that keeps the explicit inverse of the cross-quadrant level is cheaper and closer to the Moore-Penrose inverse. In compiled implementations both outperform a polynomial solver of the normal equations from $N=256$ on, and their conditioning is traced to the aliasing of integer-sampled lines.

7. 多层高斯过程及其在反问题中的应用

Multilevel Gaussian Processes with Applications to Inverse Problems

作者: Stjepan Salatovic, Aretha L. Teckentrup, Sebastian Krumscheid

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23202v1

链接: http://arxiv.org/abs/2610.23202v1

摘要: We present a multilevel Gaussian process (GP) regression framework for approximating functions that are accessible only through a hierarchy of approximations of increasing accuracy and computational cost. One example is the response surface of a parametrized partial differential equation, where the hierarchy arises from the resolution of the numerical solver. Modeling level increments independently with Matérn-kernel GPs and allocating samples flexibly across discretization levels yields substantial computational savings over single-level GP regression. We derive $L^2$ convergence rates for the posterior mean and variance, accounting for maximum likelihood estimation of the marginal variance and for kernel misspecification. When increment regularity or solver cost is unknown a priori, an adaptive sequential-design algorithm with cost-adapted active learning criteria allocates samples across levels. We apply the surrogate to Bayesian inverse problems, for which Hellinger distance bounds between the true and approximate posteriors are known. Numerical experiments illustrate the theoretical rates and the resulting computational savings.

8. 烧伤愈合形态弹性模型中的无量纲化、变分不等式与不确定性量化

Nondimensionalization, Variational Inequalities and Uncertainty Quantification in a Morphoelastic Model for Burn Injury Healing

作者: Wilbert den Hertog, Fred Vermolen

类别: math.NA, q-bio.TO

发布日期: 2026-10-16

arXiv ID: 2610.23177v1

链接: http://arxiv.org/abs/2610.23177v1

摘要: Mathematical modelling of burn injury healing involves coupling biological processes with mechanical deformation, resulting in computationally stiff, nonlinear partial differential equations that severely limit the scalability of existing numerical solutions. We present a computational framework addressing these challenges. By nondimensionalizing the model and performing scaling analysis, we show that chemotaxis and inertia are negligible, permitting a quasi-static momentum approach. To enforce non-negativity constraints, we implement a Variational Inequality formulation using the Finite Element Method in FEniCSx, and verify it via numerical convergence studies and mass balances. We identify high-order parameter interactions, specifically equilibrium collagen density and signalling dynamics, as the primary drivers of solver stability. A sensitivity analysis of nearly four million simulations reveals that isolated parameter variations predominantly govern the maximum wound contraction. In contrast, higher-order interactions determine the timing of maximum contraction and the permanent contracture. However, no significant pairwise parameter interactions were found, suggesting that the model’s behaviour is governed by higher-order interactions, highlighting the inherent complexity of the wound-healing process.

9. 径向符号界下卷积求积的稳定性与误差估计

Stability and Error Estimates for Convolution Quadrature under Radial Symbol Bounds

作者: Jingchuang Ren, Haibin Su

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23163v1

链接: http://arxiv.org/abs/2610.23163v1

摘要: We develop stability and error estimates for multistep convolution quadrature (CQ) under general radial bounds on the Laplace symbol. Instead of the classical single-power condition $|F(z)|\leq M|z|^{-μ}$, we assume $|F(z)|\leq CΦ(|z|)$ and consider two classes of radial functions: monotone bounds satisfying a scaling condition and regularly varying bounds with locally bounded above and away from zero. For strongly $A(α)$-stable multistep methods, we derive approximation and stability estimates for the CQ weights, as well as error bounds for power-type inputs $g_β(t)=t^{β-1}/Γ(β)$ with $β>0$. These estimates explicitly retain the dependence on the memory law through the factor $Φ(t_n^{-1})$. The framework accommodates symbols whose behavior cannot be described by a global single-power. It also preserves slowly varying and multiscale factors that would be lost under a coarser power majorant. The theory is applied to fractional viscoelastic models, distributed-order operators, and logarithmic-power memory operators. Numerical experiments with backward Euler and second-order backward differentiation CQ confirm the expected temporal convergence orders for direct responses. Complementary evolution examples demonstrate the corresponding temporal behavior for smooth and nonsmooth initial data.

10. 面向不规则颗粒阻力系数预测的几何感知隐表示学习

Geometry-Aware Latent Representation Learning for Drag Coefficient Prediction of Irregular Particles

作者: Zihao Li, Shota Deguchi, Mitsuteru Asai, Shunhua Chen, Naoto Mitsume

类别: math.NA

发布日期: 2026-10-16

arXiv ID: 2610.23087v1

链接: http://arxiv.org/abs/2610.23087v1

摘要: Predicting the drag coefficient $C_\mathrm{d}$ of irregular particles requires representing their complex three-dimensional geometry in a form suitable for predictive models. Empirical drag correlations and many data-driven models represent particle geometry through a small set of hand-crafted shape descriptors. Geometric differences omitted by these descriptors are unavailable to the predictor. We propose a variational autoencoder-multilayer perceptron (VAE-MLP) framework that learns a latent representation directly from voxelized particle geometry. Geometry reconstruction and drag coefficient prediction jointly guide the learning of this representation. We compare VAE-MLP with a descriptor-based MLP (Descriptor-MLP) and established empirical correlations using CFD data for synthetic particle geometries at Reynolds numbers of 10–200. When the evaluated geometric variations are represented in the training data, the two learned models achieve comparable accuracy for regular ellipsoids, while VAE-MLP yields lower errors for irregular particles. When high-order surface perturbations are excluded from training, VAE-MLP achieves lower overall mean absolute relative error in all five training runs. At the strongest tested perturbation, the mean errors across these runs are 4.97% for VAE-MLP and 9.88% for Descriptor-MLP. These results support learning geometric representations directly from particle geometry to improve drag prediction for the irregular geometries and unseen surface perturbations examined here.

11. ReLU神经网络对光滑函数的最优逼近

Optimal approximation of smooth functions by ReLU neural networks

作者: Thomas Nagler

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-10-15

arXiv ID: 2610.22693v1

链接: http://arxiv.org/abs/2610.22693v1

摘要: We study uniform approximation of the Hölder ball $\mathcal H_1^s([0,1]^D)$ by fully connected ReLU networks. Except in the one-dimensional Lipschitz case, existing constructions remain separated from the information-theoretic efficiency limit. This article closes the gap for every $s>0$ and $D\ge1$: networks with width $O(N)$, depth $O(L)$, and parameter magnitude $O(1)$ attain uniform approximation error of order $(N^2L^2\log N)^{-s/D}$. No relation between width and depth is required. The construction uses new encoding and decoding mechanisms that exploit higher-order finite-difference bounds for local Taylor coefficients of the target function. As a statistical consequence, empirical risk minimizers over the corresponding clipped network classes attain the minimax regression rate $n^{-2s/(2s+D)}$ without additional logarithmic factors. The paper also gives a detailed exposition of the key approximation techniques, covering both established and new mechanisms.

12. 刚性问题的优化线性多步方法

Optimized linear multistep methods for stiff problems

作者: Tyler Jensen, David I Ketcheson, Lajos Lóczi

类别: math.NA

发布日期: 2026-10-15

arXiv ID: 2610.22685v1

链接: http://arxiv.org/abs/2610.22685v1

摘要: We design and test several new optimized linear multistep methods for stiff problems. Each method is A-stable, implicit, and second-order accurate. We focus on achieving the smallest possible error constant while also ensuring that stiff modes are damped. We recover some known optimal L-stable three- and four-step methods, and provide new and improved five-step methods. We also develop new methods that are not L-stable but provide a prescribed rate of damping in the infinitely stiff limit. Variable-step-size formulae are also provided for some of the new methods. Numerical tests demonstrate that the improved error constants lead to improved accuracy.

13. 稀疏随机矩形矩阵的奇异向量：离域相与半局域相之间的转变

Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases

作者: Ioana Dumitriu, Hai-Xiao Wang

类别: math.PR, math.CO, math.NA, math.ST

发布日期: 2026-10-15

arXiv ID: 2610.22658v1

链接: http://arxiv.org/abs/2610.22658v1

摘要: Consider the bipartite Erdős–Rényi graph $\mathbb{G}(n,m,d/\sqrt{mn})$ at the critical sparsity $d = b\log(N)$ for some $b > 0$ with $N = n + m$, and constant aspect ratio $γ= n/m \geq 1$. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Marčenko–Pastur (MP) distribution, is split into two phases: in the bulk, singular vectors are completely delocalized, while near the edges, singular vectors are semilocalized, with their mass concentrating on small disjoint balls, centered at vertices with extreme degrees. As $b$ decreases, we identify sharp thresholds for the emergence of outlier singular values. For explicit $b_{\rm R}^{\star}$ and $b_{\rm L}^{\star}$ as functions of $γ$, we prove that when $b > b_{\rm R}^{\star}$, there is no outlier outside the bulk; when $b_{\rm L}^{\star}< b < b_{\rm R}^{\star}$, only right outliers emerge; and when $b < b_{\rm L}^{\star}$, outliers are present on both sides, all with high probability. This removes an extra technical condition from our previous work arxiv:2508.01456, providing a complete characterization of the parameter space. In sparser regimes, in terms of left outliers, we provide some nontrivial generating mechanisms, demonstrating that local graph structure plays a key role in understanding extreme singular values. Our proofs rely on two technical inputs: (1) an entrywise local law in the MP bulk, where the deterministic equivalent of the resolvent depends on both variance profiles and normalized vertex degrees, different from the Gaussian setting; (2) spectral radius upper bounds for nonbacktracking matrices constructed from regularized random graphs, recently developed in arXiv: 2609.26615. Our results reveal fruitful phenomena for sparse random rectangular matrices at criticality, which are beyond the universality class of their Gaussian counterparts.