生成时间: 2026年08月14日 05:51:38
重点关注论文数量: 0
了解领域论文数量: 3
不相关论文数量: 8
了解领域论文（摘要翻译）
1. 带电流涨落的随机电阻性 Hall–MHD：通过收敛的保结构有限元方法获得鞅弱解
Stochastic resistive Hall–MHD with current fluctuations: martingale weak solutions via a convergent structure-preserving finite element method
作者: Agus L. Soenjaya
类别: math.NA, math.AP, math.PR
发布日期: 2026-08-13
arXiv ID: 2608.13407v1
链接: http://arxiv.org/abs/2608.13407v1
摘要翻译
中文标题: 带电流涨落的随机电阻性 Hall–MHD：通过收敛的保结构有限元方法获得鞅弱解
摘要翻译: 我们在有界凸多面体区域上研究受非线性理想导电边界条件约束的随机电阻性 Hall–磁流体动力学 (Hall–MHD) 系统。该系统由动量方程中的乘性高斯强迫以及感应方程中的结构化 curl 型噪声驱动，后者来源于广义欧姆定律中电阻项和 Hall 项的随机扰动。我们基于相容的离散 de Rham 复形，构造了一种全离散、线性隐式、保结构的混合有限元方法，该方法精确保持磁场高斯定律。我们证明了离散解到有限能量弱鞅解的子序列收敛，从而得到了带 curl 型噪声的随机 Hall–MHD 系统的构造性存在性结果。数值实验说明了该方法的随机动力学、精确散度保持性和收敛性。
2. 基于自编码器的降阶神经常微分方程中误差传播分析
Analysis of Error Propagation in Autoencoder-Based Reduced-Order Neural Ordinary Differential Equations
作者: Jingyi Zhang, Gwanghyun Jo
类别: math.NA
发布日期: 2026-08-13
arXiv ID: 2608.13132v1
链接: http://arxiv.org/abs/2608.13132v1
摘要翻译
中文标题: 基于自编码器的降阶神经常微分方程中误差传播分析
摘要翻译: 神经常微分方程降阶模型通常能达到可比的局部预测精度，但其长期外推行为可能差异显著。为分析这种差异，我们建立了一个路径积分恒等式，将局部偏差注入与学习到的隐空间动力学中的放大过程分开。相应的多步 Jacobian 范数量化了传输敏感性，并区分不同的传播模式。在 Burgers 和 Gray–Scott 系统上的实验展现出两种不同的误差演化模式。在 Burgers 系统中，预测误差保持有界，并且主要与持续的局部偏差相关。相比之下，Gray–Scott 系统在外推过程中表现出明显的放大现象，其中 Jacobian 范数作为敏感性诊断指标，而非物理预测精度的直接指示。
3. BTF-PINN：在无边界训练下施加 Dirichlet 边界条件
BTF-PINN: Enforcing Dirichlet Boundary Conditions Without Boundary Training
作者: Wenyu Dong, Shuo Zhang
类别: math.NA
发布日期: 2026-08-13
arXiv ID: 2608.12823v1
链接: http://arxiv.org/abs/2608.12823v1
摘要翻译
中文标题: BTF-PINN：在无边界训练下施加 Dirichlet 边界条件
摘要翻译: 齐次 Dirichlet 边值问题集中体现了用非插值型方法求解 Dirichlet 问题的核心困难。我们提出 BTF-PINN（Boundary-Training-Free Physics-Informed Neural Network），一种用于求解齐次 Dirichlet 边值问题的仅内部策略，它不需要边界训练、边界惩罚或贴合边界的参数化。其关键思想是将本质边界条件嵌入到新设计的无边界损失函数中。我们证明了所提出的仅内部变分形式与原始边值问题之间的等价性，建立了残差权重的尖锐阈值条件，并基于该泛函的强制性建立了收敛性分析。在高维问题、不规则几何区域和各向异性椭圆方程上的数值实验表明了 BTF-PINN 的有效性。与标准边界惩罚 PINN 的对比进一步表明，BTF-PINN 取得了更优的边界迹精度。
不相关论文（基本信息）
1. 自适应求积法与双指数公式在任意精度浮点运算下的性能评估
Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic
作者: Tomonori Kouya
类别: math.NA, cs.PF
发布日期: 2026-08-13
arXiv ID: 2608.13187v1
链接: http://arxiv.org/abs/2608.13187v1
摘要: Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D—that is, Ninomiya’s adaptive 9-point Newton–Cotes rule extended with a sequence of higher-order rules—and Takahasi and Moris’ double exponential (DE) formula. We evaluate them for Kahaner’s 21 test problems. For both absolute tolerances $10^{-50}$ and $10^{-100}$, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as $1/\sqrt{x}$, it requires about $5.4\times10^{7}$ function evaluations at $10^{-100}$, roughly $7\times10^{4}$ times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.
2. Banach 空间中的自适应 Schauder 随机镜像下降
Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
作者: Jinhui Bai, Shuai Lu, Lei Shi
类别: math.OC, math.NA
发布日期: 2026-08-13
arXiv ID: 2608.13121v1
链接: http://arxiv.org/abs/2608.13121v1
摘要: In this paper, we extend stochastic mirror descent (SMD) to infinite-dimensional Banach spaces for solving a class of risk functional minimization problems, where stochastic gradient information is only available through sampling. We first choose the Bregman distance according to the uniform convexity properties of the Banach space. For the non-uniformly convex space $\mathcal{L}_{ρ_X}^1(Ω)$, we instead construct a Bregman distance induced by the entropy function. Based on a Schauder basis of the Banach space, we introduce a family of finite-dimensional subspaces that adapt to the sample size $n$. At each SMD iteration, we restrict the subproblem to the corresponding finite-dimensional subspace and project the stochastic gradient onto the associated finite-dimensional dual space, thereby introducing a new regularization strategy. This regularization strategy allows us to explicitly solve the SMD subproblem efficiently and to achieve a bias–variance trade-off. The algorithm requires $\mathcal{O}(n^{1+θ})$ time and $\mathcal{O}(n^θ)$ memory, where $θ>0$ can be chosen arbitrarily small when the minimizer has sufficient regularity. By developing a new analytical framework, we prove that the proposed algorithm achieves a convergence rate of $\mathcal{O}\left(n^{-1/p_1}\right)$ up to logarithmic factors, where $p_1\geq2$ is determined by the convexity properties of the underlying space. In the misspecified setting where the minimizer satisfies only weaker regularity conditions, we show that the proposed algorithm still converges to the minimum. We further extend the algorithm to inverse problems and validate its effectiveness in numerical experiments.
3. 度量图上的热方程的透明边界条件
Transparent Boundary Conditions for the Heat Equation on Metric Graphs
作者: Jasur Matrasulov, Jambul Yusupov, Matthias Ehrhardt
类别: math-ph, math.AP, math.NA
发布日期: 2026-08-13
arXiv ID: 2608.12937v1
链接: http://arxiv.org/abs/2608.12937v1
摘要: We study transparent boundary conditions (TBCs) for the time-dependent heat equation in a branching, quasi-one-dimensional domain modeled as a metric star graph. By combining the classical concept of TBCs with the theory of partial differential equations (PDEs) on networks, we derive an exact vertex condition that ensures unobstructed thermal flow across junctions. Specifically,we derive a sum rule for the diffusion coefficients that eliminates thermal backflow at the vertex.We demonstrate the validity of this analytical model numerically using a Crank-Nicolson finite-difference method, which confirms smooth, unobstructed heat propagation through the network.These results provide a practical mathematical framework for tunable control and optimization of thermal diffusion in low-dimensional structures. In particular, combining the well-known concept of the TBCs and theory for PDE on metric graphs, we propose a mathematical model providing a control tool for thermal diffusion in networks.
4. Strang投影分裂积分器鲁棒二阶收敛性的一个反例
A Counterexample to Robust Second-Order Convergence of the Strang Projector-Splitting Integrator
作者: Shiheng Zhang
类别: math.NA
发布日期: 2026-08-13
arXiv ID: 2608.12893v1
链接: http://arxiv.org/abs/2608.12893v1
摘要: The classical Strang projector-splitting integrator is widely observed to converge with order two, whereas the error analysis that remains uniform as the smallest singular value retained in the low-rank approximation tends to zero proves only order one. We show that this gap is intrinsic under the standard assumptions. We construct $3\times3$ matrix differential equations that are $C^2$ in time and smooth in the matrix variable, with rank-two initial data that satisfy uniform boundedness, Lipschitz, tangency-defect, and regularity bounds. Nevertheless, the exact-subflow Strang method has a nonzero $h^2$ term in the local error over one periodic forcing cycle consisting of four Strang steps. Repetition of that cycle rules out a global second-order bound whose constant and stepsize threshold are independent of the retained singular values. The mechanism is a rapid rotation of the factor directions associated with the small singular value: first-order consistency is preserved, but the changing projection spaces prevent the cancellation normally expected from a symmetric Strang composition. Hence the robust first-order result cannot, under these assumptions alone, be upgraded to robust second order for the classical projector-splitting method.
5. 六阶椭圆奇异摄动问题的一种稳健且最优方法
A robust and optimal method for a sixth-order elliptic singular perturbation problem
作者: Mingqing Chen, Jianguo Huang
类别: math.NA
发布日期: 2026-08-13
arXiv ID: 2608.12886v1
链接: http://arxiv.org/abs/2608.12886v1
摘要: In this paper we propose a robust nonconforming finite method for a gradient-elastic Kirchhoff plate (GEKP) model with a small parameter $0<ι<1$. While the method employs an $H^3$ cubic finite element for the discrete space, it employs an interpolation mapping into an $H^2$ nonconforming finite element space in both the lower-order terms of the bilinear form and the right-hand side. Notably, the numerical method is robust with respect to the parameter $ι$ and uniformly convergent to order $h$ without any stabilized terms. Finally, some numerical experiments are performed to verify the theoretical findings.
6. 线性化 Cahn-Hilliard-Cook 方程的有限元逼近
Finite element approximation of the linearized Cahn-Hilliard-Cook equation
作者: Ali Mesforush, Stig Larsson
类别: math.NA
发布日期: 2026-08-13
arXiv ID: 2608.12833v1
链接: http://arxiv.org/abs/2608.12833v1
摘要: The linearized Cahn-Hilliard-Cook equation is discretized in the spatial variables by a standard finite element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. The backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn-Hilliard equation. The results should be interpreted as results on approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn-Hilliard-Cook equation.
7. 具有拟均匀二维投影的点序列的代数构造
Algebraic constructions of point sequences with quasi-uniform two-dimensional projections
作者: Takashi Goda
类别: math.NT, math.NA, stat.ME
发布日期: 2026-08-13
arXiv ID: 2608.12690v1
链接: http://arxiv.org/abs/2608.12690v1
摘要: Motivated by sequential space-filling designs for computer experiments, we study algebraic constructions of extensible point sets in the $d$-dimensional unit cube whose two-dimensional coordinate projections are all quasi-uniform. Our two constructions share a common parametrization in terms of finite configurations of distinct rational directions on the projective line $\mathbb{P}^1(\mathbb{Q})$. First, using a cubic number field, we construct explicit Kronecker sequences for which the mesh ratios of all two-dimensional coordinate projections remain uniformly bounded over every initial segment of length $N\ge 2$. Second, using a real quadratic field, a split prime, and a compatible $p$-adic embedding, we construct nested rank-1 lattice designs with the same uniform projection property at every nesting level. The proofs combine algebraic norm estimates with transference principles between simultaneous and dual Diophantine approximation, yielding lower bounds for the separation radii and upper bounds for the covering radii, both of optimal order in the number of points, uniformly over all coordinate pairs. We also investigate how the choice of rational projective coefficients affects the resulting mesh ratios. This leads to a minimax problem for finite configurations on $\mathbb{P}^1(\mathbb{Q})$, in which one seeks to minimize the maximum mesh ratio over all two-dimensional coordinate projections. These constructions provide extensible point sets with uniformly controlled bivariate geometry.
8. 使用采样局部 WeakIdent 识别变化的偏微分方程
Identifying changing partial differential equations using Sampled Local WeakIdent
作者: Wenbo Hao, Mengyi Tang, Sung Ha Kang
类别: math.NA
发布日期: 2026-08-12
arXiv ID: 2608.12479v1
链接: http://arxiv.org/abs/2608.12479v1
摘要: We propose Sampled Local WeakIdent (SLW-Ident), a framework for identifying changing governing equations from a single set of given data. Different from a typical approach of using finite element based approximation to represent varying coefficients, this paper explores a local approach in identification of differential equations. First, we present the power of Local WeakIdent which gives good local identification and is also computationally efficient with a small patch size, yet it can be sensitive to local perturbations. We propose SLW-Ident which stabilizes the identification process and also incorporates global information: we first sample patches in the whole given domain, identify equations for each sampled patch, then use residual error of these equations to find the transitions between different equations. We refer to a region where the support of the identified equation does not change to be a region of one equation. Within each region of one equation, we pick the most frequently identified equation as the identified equation, and find constant as well as varying coefficient PDEs within each region of one equation. This is justified by an uncertainty quantification theory that gives the relation between statistical error of dominant support selection and the number of patches. We provide various numerical experiments showing that SLW-Ident accurately recovers the regions of one equation and the governing equations for changing PDEs with varying coefficients even with noisy given data.