生成时间: 2026年08月18日 04:58:23

重点关注论文数量: 1

了解领域论文数量: 7

不相关论文数量: 33

重点关注论文（完整分析）

1. 色散共振的渐近引导学习与符号回归

Asymptotics-guided learning and symbolic regression for dispersive resonances

作者: Konstantinos Alexopoulos, Josselin Garnier

类别: math.NA, cs.LG, math-ph, math.AP, physics.optics

发布日期: 2026-08-17

arXiv ID: 2608.16152v1

摘要翻译: 我们研究了色散介质中的共振预测问题，该问题被表述为体积积分算子的非线性谱问题。主要思想不仅是将渐近分析用作基线近似，还将其用作构建预测性修正模型的指导。我们利用亚波长展开所建议的特征（包括二维特有的对数尺度）来学习渐近共振与参考共振之间的残差。所得修正显著改善了对单共振器和二聚体的预测，并且符号回归为学习到的残差生成了紧凑的公式。结果表明，渐近分析不仅可以用于近似共振，还可以用于设计特征空间，在该空间中数据驱动的修正变得精确、低维且可解释。

Comment: 25 pages, 11 figures, 6 tables

链接: http://arxiv.org/abs/2608.16152v1

1. 研究对象和背景

本文研究色散介质中共振频率的预测问题。研究对象由二维 Helmholtz 方程描述：

$$ \nabla\cdot\left(\frac{1}{\mu(x)}\nabla u(x)\right) + \omega^2 \varepsilon(x,\omega) u(x) = 0, \quad x \in \mathbb{R}^2, $$

其中 $\omega \in \mathbb{C}$ 为角频率，$\varepsilon(x,\omega)$ 为可能色散且复值的介电常数，$\mu(x)$ 为磁导率。背景介质均匀，参数为 $(\varepsilon_0,\mu_0)$，谐振子占据有界区域 $D \subset \mathbb{R}^2$。谐振子内部的介电常数采用 Lorentz 型色散模型：

$$ \varepsilon_r(\omega) = \varepsilon_0 + \frac{\alpha}{\beta - \omega^2 - i\gamma\omega + \eta k_0^2}, \quad k_0 = \omega\sqrt{\varepsilon_0\mu_0}, $$

其中 $\alpha,\beta,\gamma,\eta \in \mathbb{R}$ 为材料参数，$\gamma \ge 0$ 表示阻尼。共振被定义为齐次问题存在非平凡外向解的频率，即如下非线性谱问题：

$$ \text{Id} + \omega^2 \xi(\omega) K_D^\omega \quad \text{不可逆}, $$

其中 $\xi(\omega) = \mu_0(\varepsilon_r(\omega)-\varepsilon_0)$ 为对比度，$K_D^\omega$ 是体积积分算子：

$$ K_D^\omega[u](x) = \int_D G(x,y;\omega) u(y) \, dy, $$

$G$ 是背景介质的自由空间 Green 函数。在二维中 Green 函数具有对数奇性：

$$ G(x,y;\omega) = -\frac{1}{2\pi}\log|x-y| + R(x,y;\omega), $$

这导致亚波长展开中出现对数项。该问题在光子学、超材料、钙钛矿谐振器等领域有重要应用，例如光学传感、光伏和发光器件。

从数学角度看，亚波长尺度的渐近分析提供了共振频率的显式近似。设谐振子为 $D = \delta B$（$0 < \delta \ll 1$，$B$ 为固定参考域），则体积积分算子的特征值具有展开形式，其中含 $\delta^2 k_0^2 \log(\delta k_0)$ 等项。然而，渐近展开在远离严格亚波长区域时精度下降，且高阶项（尤其是耦合和复杂几何情形）难以显式计算。另一方面，现有机器学习方法通常直接学习从参数到共振的完整输入-输出映射，依赖高容量模型，需要大量数据且难以解释。

本文试图填补的空白是：提出一种渐近引导的残差学习框架，将渐近模型作为基线，仅学习参考共振与渐近共振之间的残差。该残差使用由亚波长展开（包括二维特有的对数项）启发的特征进行建模，从而在保持低维和可解释性的同时提高预测精度。

2. 主要定理或主要结果

本文证明了以下主要结果：

  1. 残差学习框架：设 $\omega_{\mathrm{asymp}}$ 为渐近模型给出的共振，$\omega_{\mathrm{ref}}$ 为高保真参考共振，定义残差 $\Delta\omega = \omega_{\mathrm{ref}} - \omega_{\mathrm{asymp}}$。通过 Ridge 回归学习映射 $X \mapsto \Delta\omega$，其中 $X$ 包含物理几何参数（$\delta, \gamma$，以及 dimer 情形的 $\kappa$）、渐近模型输出（如 $\omega_{\mathrm{asymp}}$ 的实虚部、近似特征值）和物理信息特征。物理信息特征基于二维子波长展开中出现的尺度： $$ \delta^2 q_{\mathrm{asymp}}^2, \quad \delta^2 q_{\mathrm{asymp}}^2 \log\delta, \quad \delta^2 q_{\mathrm{asymp}}^2 \log q_{\mathrm{asymp}}, $$ 其中 $q_{\mathrm{asymp}} = |\omega_{\mathrm{asymp}}|$。对 dimer 的每个分支类似使用 $q_j = |\omega_{j,\mathrm{asymp}}|$。Ridge 回归形式为 $\Delta\omega \approx WX + b$，最小化正则化最小二乘泛函 $$ \min_{W,b} \sum_{n=1}^N \|W X_n + b - \Delta\omega_n\|^2 + \alpha_R \|W\|^2。 $$ 作者通过数值实验表明，该框架相比纯渐近模型显著提高了预测精度。
  2. 符号回归公式：使用 PySR 对残差的实部和虚部分别进行符号回归。对于单谐振子，一个紧凑的修正公式为： $$ \omega_{\mathrm{sym}} = \omega_{\mathrm{asymp}} + \Delta\omega_{\mathrm{sym}}, $$ 其中（用 $\omega_{\mathrm{asymp},r}$ 和 $\omega_{\mathrm{asymp},i}$ 表示渐近共振的实部和虚部） $$ \mathrm{Re}(\Delta\omega_{\mathrm{sym}}) = \delta^2 \omega_{\mathrm{asymp},r} \left(0.02398765\, \omega_{\mathrm{asymp},r}^2 + 0.00012753597\, \omega_{\mathrm{asymp},r}\omega_{\mathrm{asymp},i} + 0.0053167343\right), $$ $$ \mathrm{Im}(\Delta\omega_{\mathrm{sym}}) = 0.02947997\, \omega_{\mathrm{asymp},i} \left(\delta^2 q_{\mathrm{asymp}}^2 + \delta^2 + \delta^2 q_{\mathrm{asymp}}^2 \log q_{\mathrm{asymp}}\right)。 $$ 这些公式表明实部由 $\delta^2$ 调制的渐近共振主导，虚部包含与特征尺度一致的频率依赖结构。对于 dimer 情形，符号回归分别应用于两个共振分支（对称和反对称模式），并利用特征值劈裂 $\Lambda = |\lambda_{1,\mathrm{asymp}} - \lambda_{2,\mathrm{asymp}}|$ 作为输入变量。从提供内容中无法确认为 dimer 得到的具体符号表达式。
  3. 消融研究：作者通过消融实验表明，物理信息特征（尤其是 $\delta^2 q_{\mathrm{asymp}}^2 \log q_{\mathrm{asymp}}$ 等对数项）对准确修正是必要的；去除这些特征会显著降低精度。
  4. 辅助符号回归：作者提出用渐近下一阶修正的代理项辅助符号回归，将符号修正拆分为代理项和数据驱动项：$\Delta\omega_{\mathrm{SR}} = \Delta\omega_{\mathrm{proxy}} + \Delta\omega_{\mathrm{data}}$。实验表明这种辅助策略改善或稳定了学习到的公式，支持残差受预期渐近尺度支配的解释。
  5. 噪声鲁棒性：作者研究了训练共振含噪的影响，表明所提框架对中等噪声保持稳健，尤其是当训练样本数量增加时。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍是渐近展开在远离严格亚波长区域时失效，且高阶项难以显式计算；对于耦合系统，相互作用效应引入分支依赖的修正，进一步加剧了困难。作者的真正创新点在于：不放弃渐近模型，而是将其作为基线，用机器学习学习残差；并且不是任意选择学习特征，而是从二维子波长展开的已知结构中提取特征，使残差学习成为低维、可解释的问题。

证明和分析框架包括以下关键技术：

  1. 体积积分算子的子波长展开：通过变量替换 $x = \delta \tilde{x}$，将算子改写为 $$ K_D^\omega[u](\delta \tilde{x}) = \delta^2 \int_B \left[-\frac{1}{2\pi}\log\delta - \frac{1}{2\pi}\log|\tilde{x}-\tilde{y}| + R(\delta\tilde{x},\delta\tilde{y};\omega)\right] \tilde{u}(\tilde{y}) d\tilde{y}, $$ 从而分离出 $\delta$ 的显式依赖。算子层面的展开为 $$ K_B^\omega \simeq (\log(\delta k_0) + \gamma_E) K_B^{(-1)} + K_B^{(0)} + \delta^2 k_0^2 \log(\delta k_0) K_B^{(1)} + \cdots, $$ 其中 $\gamma_E$ 为 Euler-Mascheroni 常数。这一展开揭示了下一阶修正的尺度为 $\delta^2 k_0^2 \log(\delta k_0)$，以及分解 $\delta^2 q^2 \log(\delta q) = \delta^2 q^2 \log\delta + \delta^2 q^2 \log q$，这正是特征设计的物理依据。
  2. 隐函数定理推导残差尺度：定义标量共振函数 $\Phi(\omega,\delta) = 1 + \omega^2 \xi(\omega)\lambda(\delta,\omega)$，其中 $\lambda$ 为体积积分算子对应的特征值。设 $\omega_{\mathrm{asymp}}$ 满足 $\Phi_{\mathrm{asymp}}(\omega_{\mathrm{asymp}},\delta)=0$，且 $\partial_\omega \Phi_{\mathrm{asymp}}(\omega_{\mathrm{asymp}},\delta) \neq 0$，则一阶残差为 $$ \Delta\omega \simeq -\frac{\omega_{\mathrm{asymp}}^2 \xi(\omega_{\mathrm{asymp}}) \Delta\lambda(\delta,\omega_{\mathrm{asymp}})}{\partial_\omega \Phi_{\mathrm{asymp}}(\omega_{\mathrm{asymp}},\delta)}, $$ 其中 $\Delta\lambda = \lambda - \lambda_{\mathrm{asymp}}$。这从理论上说明频率残差继承特征值展开的尺度，从而证明将 $\delta^2 q^2$、$\delta^2 q^2 \log\delta$、$\delta^2 q^2 \log q$ 作为特征是合理的。
  3. Ridge 回归作为低维线性校正：使用线性模型 $\Delta\omega \approx WX + b$，联合学习实部和虚部。线性模型是刻意的选择，因为渐近近似已捕捉主导行为，且特征包含预期的高阶尺度，残差应为这些量的低维组合。特征标准化、训练-测试划分和正则化参数细节见附录 A（从提供内容中无法确认具体数值）。
  4. 符号回归（PySR）：使用 PySR 框架搜索显式解析表达式拟合残差，而不是完整共振。目标为 $\Delta\omega$ 的实部和虚部，输入变量为物理信息特征。对 dimer 还使用特征值劈裂 $\Lambda$。符号回归不是黑盒发现，而是渐近引导的工具，用于提取紧凑公式。
  5. 辅助符号回归：将符号修正分解为 $\Delta\omega_{\mathrm{proxy}}$（基于渐近尺度的代理）和 $\Delta\omega_{\mathrm{data}}$（数据驱动残差），先拟合代理项再用符号回归拟合数据项。这种方法利用了已知的渐近结构，稳定了学习过程，并支持残差确实由预期尺度支配的解释。
  6. 误差度量：采用相对误差 $E(\omega) = |\omega_{\mathrm{pred}} - \omega_{\mathrm{ref}}|/|\omega_{\mathrm{ref}}|$，直接比较渐近模型和修正模型的精度。

4. 与之前工作的比较

本文与已有工作的比较可从以下几个维度展开：

  1. 与纯渐近分析工作的比较：文中引用了体积积分算子谱性质的渐近展开工作 [2,3]。这些工作给出了亚波长区域共振的显式公式，例如单谐振子和耦合系统中的模式劈裂。但本文指出，这些渐近模型的精度在远离严格亚波长区域时恶化，且高阶修正难以显式计算。本文的推进在于：不试图推导更高阶展开，而是用数据驱动方式学习展开中缺失的项。残差学习框架将渐近模型作为基线，利用已知的展开结构设计特征，从而将渐近分析的适用范围扩展到更广的参数区间。代价是需要训练数据，且得到的是数值修正而非严格解析展开。
  2. 与机器学习直接学习完整映射的比较：文中与物理科学中的机器学习应用（[22] 算子学习与 PDE，[8,11,14,25,26] 光子学逆向设计）进行对比。这些方法通常直接学习从结构参数到光学响应的完整映射，依赖高容量模型，需要大量数据且难以解释。本文的差异在于：学习任务不是重新发现完整的共振图，而是修正渐近模型的残差。由于残差具有由渐近展开决定的低维结构，可以使用简单的线性 Ridge 回归或简洁的符号公式，从而显著降低数据需求和模型复杂度，并提高可解释性。消融实验表明物理信息特征至关重要，这进一步证实了结构引导的优势。
  3. 与符号回归在光子学中应用的比较：文中引用了符号回归在拓扑光子带 [9]、生物和仿生光学响应 [18,19] 以及物理发现 [12] 中的应用。这些工作通常将符号回归作为黑盒发现工具，直接搜索完整物理量的表达式。本文则将符号回归应用于渐近残差，并使用渐近引导的特征作为输入变量。这使得符号搜索聚焦于缺失的高阶项，而不是重新发现主导项。文中强调符号公式是数据驱动的拟合修正，不应视为严格渐近展开。此外，辅助符号回归策略（先拟合代理项）进一步利用了渐近结构，改善了公式的稳定性和可解释性。
  4. 与既有耦合系统渐近模型的比较：对于 dimer 系统，文中提到先前的渐近模型 [3] 描述了耦合导致的模式劈裂。但耦合系统的残差包含高阶自相互作用和耦合修正，比单谐振子更复杂。本文的分支-wise 学习和特征设计（包括特征值劈裂）旨在捕捉这些效应。从提供内容中无法确认为 dimer 的符号公式，但数值实验表明框架有效。
  5. 局限性和代价：本文的框架固定了材料参数（$\alpha,\beta,\eta$），仅变化 $\delta,\gamma,\kappa$，因此不是对任意材料族的学习问题。训练数据需要高保真数值模拟生成参考共振，这可能计算成本较高。噪声鲁棒性实验限于中等噪声。此外，符号公式是特定参数范围内的拟合，不具有渐近展开的普适性。作者未明确与其他数据驱动方法进行定量比较（例如精度、数据需求），这部分从提供内容中无法确认。

了解领域论文（摘要翻译）

1. 三维光滑曲面波散射的一种FFT加速边界积分方程方法

An FFT-Accelerated Boundary Integral Equation Method for Wave Scattering by Smooth Surfaces in Three Dimensions

作者: Jun Li, Huiyi Li, Wangtao Lu

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16208v1

链接: http://arxiv.org/abs/2608.16208v1

摘要翻译

中文标题: 三维光滑曲面波散射的一种FFT加速边界积分方程方法

摘要翻译: 对于轴对称曲面上的波散射，快速傅里叶变换（FFT）方法为加速标准边界积分方程（BIE）求解器提供了有效工具。由于积分算子具有卷积性质，曲面积分方程可以在生成曲线上解耦为一系列曲线积分方程。三维基本核的傅里叶系数可以通过基于Miller算法的三项递推关系快速计算。对于非轴对称曲面，这些成熟技术将失效。 本文提出了一种新的FFT加速边界积分方法，用于任意形状光滑曲面上的波散射。奇异核的傅里叶系数现在满足高阶递推关系。尽管它们可以通过标准Olver算法以最优线性复杂度求解，但事实证明，一种奇异性交换方法——将每个核重写为一个光滑函数与一个轴对称相关奇异因子的乘积——在实际中要快得多。因此，Miller算法与标准FFT卷积相结合，可以得到一种${\cal O}(M\log M)$的方法来计算核的${\cal O}(M)$个傅里叶模式，达到与轴对称曲面完全相同的复杂度阶数！借助这些基于FFT的高效程序，我们将曲面积分方程重写为${\cal O}(M)$个弱奇异曲线积分，用基于面板的广义高斯求积法对其进行离散，并获得高精度的线性系统来近似波场。大量数值实验验证了新方法的有效性。

2. 参数网格上曲面双调和方程的重构拉普拉斯方法

A Reconstructed-Laplacian Method for the Surface Biharmonic Equation on Parametric Meshes

作者: Shuo Yang

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16139v1

链接: http://arxiv.org/abs/2608.16139v1

摘要翻译

中文标题: 参数网格上曲面双调和方程的重构拉普拉斯方法

摘要翻译: 我们发展并分析了一种基于重构曲面拉普拉斯算子的连续/间断伽辽金（CDG）方法，用于光滑闭曲面上的双调和方程。在次数为 $r\ge1$ 的拟合参数网格上使用次数为 $k\ge2$ 的连续映射有限元，而次数为 $(k-2)$ 的间断提升修正了破碎的 Laplace-Beltrami 算子，以处理双侧余法向通量跳跃。所得的完全平方形式在均值为零的空间上对每个固定的 $β>0$ 都是强制的，无需足够大的罚参数。在标准几何假设下，我们证明了能量误差和重构拉普拉斯误差为 $\mathcal O(h^{k-1}+h^r)$，$L^2$ 误差为 $\mathcal O(h^{q_k}+h^{r+1})$，其中 $q_2=2$，且对 $k\ge3$ 有 $q_k=k+1$。基准计算支持这些收敛阶，而曲面 Swift-Hohenberg 实验说明了该方法向非线性拉普拉斯主导模型的推广。

3. T-网格上多项式样条空间的维数与维数稳定性的同调分解

A Homological Decomposition for the Dimension and Dimensional Stability of Polynomial Spline Spaces over T-Meshes

作者: Bingru Huang

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15684v1

链接: http://arxiv.org/abs/2608.15684v1

摘要翻译

中文标题: T-网格上多项式样条空间的维数与维数稳定性的同调分解

摘要翻译: 我们研究平面T-网格上具有给定光滑阶的双次数$(m,m')$多项式样条空间的维数与维数稳定性。同调维数公式将样条维数写为欧拉示性数项与修正项之和。对于固定的有序双次数，欧拉示性数项由网格结构和给定的光滑阶决定。修正项可以写成附着于最大内部线段（MIS）的系数空间之商。我们证明了一个加权删除定理：当可用的顶点关系生成某条MIS的系数空间时，该被加项可以从商中删除而不改变修正项。此操作既不改变T-网格，也不改变其链复形。重复删除后留下一个加权完全不可对角化分量（CNDC）。我们证明加权CNDC与合格MIS的删除顺序无关，并且在结构类中对于相应配对是相同的。因此，修正项可以仅使用加权CNDC中的MIS来表示，同时保留来自原始T-网格的所有关系。维数稳定性等价于剩余关系空间维数的恒定性。特别地，空的加权CNDC是稳定性的充分条件。我们还推导了剩余修正项的上界，并将其与Mourrain基于所有MIS的上界进行比较。

4. 超越场精度：逆物理信息神经网络参数误差的双轴诊断

Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Error

作者: Yifan Zhang, Qian Tao

类别: cs.LG, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15373v1

链接: http://arxiv.org/abs/2608.15373v1

摘要翻译

中文标题: 超越场精度：逆物理信息神经网络参数误差的双轴诊断

摘要翻译: 逆物理信息神经网络（PINNs）能够在准确重建场的同时返回错误的物理参数。我们引入了一种两轴训练后诊断方法，将指定观测与估计协议下的有限样本分辨率与最终学习场和残差度量所编码的有符号参数偏好分离开来。第一轴使用匹配的正向估计器对噪声观测进行重复拟合。在已知的合成真值下，第二轴冻结场和残差视图，并计算朝向附近残差剖面最小值的局部分数位移。端点一致性随后检验联合训练是否在同一最终视图下实现该偏好。在三个合成的一维标量参数偏微分方程中，匹配正向估计的平均绝对相对误差范围为2.34%至17.46%。位移在锁定种子、架构和新鲜噪声重训练中跟踪冻结剖面最小值（r从0.945到0.982），并在240次新鲜噪声RBA运行中跟踪传递的有符号对数误差（r = 0.994；237/240个正确方向）。耦合的两参数达西检验验证了完整矩阵计算。这两个轴是互补的诊断坐标，而非可加性误差分量或可部署的无预言机估计器。它们共同将后续工作引向观测、残差证据或端点传递。

5. 二维标量电磁问题的协调与非协调Trefftz逼近

Conforming and nonconforming Trefftz approximations for two-dimensional scalar electromagnetic problems

作者: Igor Tsukerman

类别: physics.comp-ph, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15352v1

链接: http://arxiv.org/abs/2608.15352v1

摘要翻译

中文标题: 二维标量电磁问题的协调与非协调Trefftz逼近

摘要翻译: Trefftz函数局部且精确地满足微分方程；拟Trefftz函数则以预设的高阶精度近似满足。本文考虑二维标量电磁问题的（拟）Trefftz逼近。已有的离散化方法包括柔性局部逼近法（FLAME）、Trefftz单元（$T$-单元）以及Trefftz间断伽辽金（Trefftz-DG）方法。新的发展包括梯度增强FLAME（GEFLAME）、协调Trefftz–FLAME有限元（TFF）以及通过Schur–DtN约化得到的全场Bloch波矢-频率解。应用涵盖静电学、散射、奇异场以及周期结构中的Bloch波。 这些方法在协调性与灵活性之间做出了不同的折衷。FLAME和GEFLAME将Trefftz函数直接纳入局部差分格式；TFF使用逐单元FLAME格式将多项式迹提升到单元内部；$T$-单元将逐单元Trefftz空间与这些迹弱匹配；而Trefftz-DG通过通量和罚项耦合分片Trefftz空间。 在所报告的波散射算例中，GEFLAME在相当的代数代价下，其场和梯度误差比二次有限元离散低数个数量级。局部奇异函数增强消除了主要的重入角误差。协调TFF和拟协调$T$-单元允许标准的有限元组装，但暴露出多项式边迹作为精度瓶颈。对于周期介质，双线性Trefftz-DG格式给出了解析的多项式波数-频率特征问题，而无需将Bloch乘子限制在单位圆上。Schur–DtN约化产生紧凑的边界响应问题。

6. MiNO：用于偏微分方程的余切丛传播子学习

MiNO: Cotangent-bundle propagator learning for PDEs

作者: Gnankan Landry Regis N’guessan, Bum Jun Kim

类别: cs.LG, cs.CE, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15187v1

链接: http://arxiv.org/abs/2608.15187v1

摘要翻译

中文标题: MiNO：用于偏微分方程的余切丛传播子学习

摘要翻译: 用于偏微分方程的科学机器学习通常针对解场（如物理信息神经网络）或解映射（如神经算子）。我们研究第三个目标：传播子本身，即相空间中的一个相位和振幅。其动机在于正则性上的差距。一个被输运的间断在空间和时间上是不光滑的，但移动它的规则可以是一个携带单位振幅的多项式相位，因此产生演化的对象可以比它所产生的场光滑得多。微局部神经算子（MiNO）学习该对象，利用程函方程描述相位、输运方程描述振幅，并通过振荡积分恢复解。尖锐前沿和焦散因此属于传播几何，而非逐点拟合的场。小的残差所证实的不仅仅是被重建的场。它们将学习到的典范关系（承载奇异性的几何）置于接近精确典范关系的位置，并将可训练误差与频率截断尾部区分开来。在一个匹配预算的不连续平流基准测试中，MiNO 在10,000步内达到其有限重建窗口的精度极限（该极限以闭式形式预测）后停止改进，而具有神经正切核损失平衡的物理信息神经网络则停留在其初始误差附近。在光滑平流中，MiNO 的平均误差为 $3.84\times10^{-3}$，而有监督的傅里叶神经算子为 $3.12\times10^{-2}$。单分支 MiNO 是所比较模型中规模最小的，并且一个训练好的生成器可以在不重新训练的情况下服务五种未见过的初始条件。

7. 用于开孔多孔材料均质化的混合专家代理模型

Mixture of experts surrogate model for the homogenization of open-porous materials

作者: Axel Klawonn, Martin Lanser, Lucas Mager, Ameya Rege, Janine Weber-Hamacher

类别: math.NA

发布日期: 2026-08-14

arXiv ID: 2608.14809v1

链接: http://arxiv.org/abs/2608.14809v1

摘要翻译

中文标题: 用于开孔多孔材料均质化的混合专家代理模型

摘要翻译: 对于开孔多孔材料，将其微观结构特性纳入力学模拟中，对于准确捕获弹性变形构成了重大挑战。为应对这一困难，多尺度方法是耦合所考虑材料微观结构特征与宏观材料行为的常用工具。然而，当需要高精度时，由于每一步计算中需要求解大量微观问题，这些多尺度计算在计算上可能非常昂贵。在此，学习底层本构模型力学响应的代理模型可以显著降低多尺度方法的计算成本。在部分作者先前的工作中，梁框架模型已被用于模拟开孔多孔材料的微观结构，并与基于神经网络的代理模型相结合，以近似给定RVE（代表性体积单元）的材料行为。在本工作中，我们扩展了先前的研究，训练了一个更复杂的神经网络模型来预测多个RVE的力学行为，这些RVE在最大孔径和孔径分布上有所不同。具体而言，我们关注混合专家（MoE）模型，并比较不同的MoE架构及其在不同RVE上的性能。这种新方法降低了模拟多个RVE的计算成本，因为当考虑新的RVE时，MoE模型不需要额外训练。

不相关论文（基本信息）

1. AutoSR：通过搜索研究状态的自动符号回归

AutoSR: Automatic Symbolic Regression by Searching Research States

作者: Kejia Zhang, Youran Sun, Xinyu Ren, Chugang Yi, Haizhao Yang

类别: cs.SC, cs.AI, cs.LG, math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16876v1

链接: http://arxiv.org/abs/2608.16876v1

摘要: We introduce Automatic Symbolic Regression (AutoSR), a fully automated system that instantiates Research-Space Symbolic Regression by searching persistent scientific investigations rather than isolated equations. Finite, noisy data often yield numerically competitive expressions that imply very different behavior outside the observed regime, making numerical fit and syntactic complexity insufficient measures of scientific credibility. Existing approaches largely focus on improving expressions, yet the search typically retains little beyond the resulting formula and score, losing the scientific record, such as motivations and probes, that inform what to try next. AutoSR preserves this record in a \textbf{Research State}, coupling each candidate equation with the reasoning, computational evidence, and independent review developed along its branch. Proposer–reviewer agents develop these states under progressive-widening Monte Carlo tree search (PW-MCTS), which allocates computation across competing investigations, while the accumulated research record is ultimately synthesized into a final report that explains the leading relation and the basis for its selection. Across nine selected challenges from two benchmark suites, AutoSR recovers algebraically equivalent relations in every case, including three cp3-bench problems that no published system recovers and six structurally diverse LSR-Transform problems. Overall, AutoSR extends symbolic regression from equation-level search toward automated scientific investigation, allowing scientific knowledge and accumulated evidence to shape both what is explored and how the resulting equation is justified.

2. 加倍维度为平方应力带来良性景观

Doubling the dimension yields a benign landscape for the squared-stress

作者: Christopher Criscitiello

类别: math.OC, math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16799v1

链接: http://arxiv.org/abs/2608.16799v1

摘要: We consider the Euclidean distance geometry problem (EDG): given a subset of the pairwise distances of an unknown cloud of $n$ points in $\mathbb{R}^\ell$, recover the point cloud up to rigid motions. When $n$ is large, a popular practical approach is to minimize a nonconvex quartic, known as the squared-stress or s-stress, over point clouds in $\mathbb{R}^k$, with $k$ potentially larger than $\ell$. It is a long-standing open problem to understand the optimization landscape of the s-stress when all pairwise distances are known (Malone and Trosset, 2000; Parhizkar, 2013). It was recently shown that the landscape is not benign when $k=\ell$, and it was conjectured that the landscape becomes benign as soon as $k\ge \ell+1$ (Song et al., 2025; Criscitiello et al., 2026). Here, we show that the complete-graph s-stress has a benign landscape whenever $k\ge 2(\ell+1)$, establishing the conjecture up to a factor of two. A key idea is to view second-order criticality as a containment of two ellipsoids; finding a descent direction then corresponds to finding a separating hyperplane that violates this containment. This dual perspective yields the stated landscape result, and also applies to any measurement operator whose inverse satisfies a simple frame condition.

3. Cahn-Hilliard-Cook 方程的有限元逼近

Finite Element Approximation of the Cahn-Hilliard-Cook equation

作者: Ali Mesforush, Stig Larsson, Mihály Kovács

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16680v1

链接: http://arxiv.org/abs/2608.16680v1

摘要: We study the nonlinear stochastic Cahn-Hilliard equation per- turbed by additive colored noise. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to 1. We also prove strong convergence without known rate.

4. Vilenkin群上阶梯缩放函数的构造

Construction of step scaling functions in the Vilenkin group

作者: Sergei Lukomskii

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16516v1

链接: http://arxiv.org/abs/2608.16516v1

摘要: In Vilenkin’s group we present an algorithm for constructing a step scaling function with a given support and a constant on given cosets.

5. 图上带个体噪声的平均场博弈交错格式的收敛性与变分结构

Convergence and variational structure of a staggered scheme for mean field games with individual noise on graphs

作者: Jianbo Cui, Tonghe Dang

类别: math.NA, math.OC

发布日期: 2026-08-17

arXiv ID: 2608.16474v1

链接: http://arxiv.org/abs/2608.16474v1

摘要: We propose and analyze a time-staggered numerical scheme for mean field game (MFG) systems with individual noise on finite graphs. Numerically solving such coupled forward–backward systems is delicate because the density evolves in the open probability simplex and the coefficients may degenerate at its boundary. The scheme preserves mass and satisfies a discrete fundamental identity compatible with the Lasry–Lions monotonicity argument, leading to uniqueness of the numerical solution. By establishing a timestep-uniform positive lower bound for the density and uniform bounds for the value variable, we prove first-order convergence for every interior discrete solution. For potential MFGs, we establish a variational characterization by identifying the scheme with the KKT system of a convex discrete action, yielding existence of the discrete solution and an optimization-based realization. The resulting optimization problem is solved by a feasible primal–dual Newton method in mass-preserving coordinates. Numerical experiments confirm the predicted convergence rate and illustrate topology-dependent transport and congestion-driven route choice.

6. 无参考的对数能量-神谕恢复用于对称强制变分问题的神经近似：协调Riesz重构与存档级选择

Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection

作者: Karim Bounja, Lahcen Laayouni, Boujemaa Achchab, Abdeljalil Sakat

类别: cs.LG, math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16473v1

链接: http://arxiv.org/abs/2608.16473v1

摘要: Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual-energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower-upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle-non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, training-independent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost.

7. 裂隙介质中具有弹簧型与摩擦接触变形规律的弹性波传播

Elastic wave propagation in fractured media with spring-type and frictional contact deformation laws

作者: Ingrid Kristine Jacobsen, Jan Martin Nordbotten, Ivar Stefansson, Barbara Wohlmuth, Inga Berre

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16460v1

链接: http://arxiv.org/abs/2608.16460v1

摘要: Elastic wave propagation in fractured media is relevant to applications such as analysis of seismic waves and non-destructive characterization of materials. Understanding attenuation and scattering behavior arising from wave-fracture interaction is important for interpreting observations at both field and laboratory scales. This work presents a computational framework for elastic wave propagation in fractured media based on a mixed-dimensional discrete fracture-matrix representation. Fracture deformation is governed by four models of increasing complexity, ranging from widely used spring-based formulations to fracture contact mechanics models with friction, all incorporated within a unified computational framework. Many previous studies are often restricted to simplified wave fields, single fractures or subsets of the relevant fracture deformation mechanisms. In contrast, the proposed framework enables fully coupled simulation of elastic wave propagation with fracture deformation models that account for elastic normal deformation, frictional contact and fracture opening and closure. The elastic wave equation is discretized in space using the cell-centered finite volume method Multi-Point Stress Approximation with weak symmetry and in time using the Newmark method. The spatial discretization is locally conservative and applicable to general polyhedral grids, making it well suited for media containing fractures, material heterogeneities and anisotropy. The proposed framework is verified through numerical convergence analyses and is subsequently applied to wave propagation and fracture deformation in two- and three-dimensional media containing multiple intersecting fractures.

8. 用于实验数据的环面计算机断层成像

Torus computed tomography for experimental data

作者: Ella Salo, Alexander Meaney, Olli Koskela, Jesse Railo

类别: math.NA, math.FA

发布日期: 2026-08-17

arXiv ID: 2608.16453v1

链接: http://arxiv.org/abs/2608.16453v1

摘要: We implement the torus-based X-ray tomography method introduced by Ilmavirta, Koskela, and Railo in “Torus computed tomography”, SIAM J. Appl. Math., 80(4):1947–1976, 2020, for experimental X-ray tomographic data. The numerical implementation is extended to accommodate fan-beam measurements by converting the data to a parallel-beam format and mapping the projection angles to the closed-geodesic directions on the torus. In addition, we consider two extensions of the original framework: the Star TCT method which extends the frequency coverage of the reconstruction, and a numerical implementation of torus backprojection developed by Railo in “Fourier analysis of periodic Radon transforms”, J. Fourier Anal. Appl., 26(4):64, 2020, for which we also derive a corresponding regularized formulation. We demonstrate the methods on experimental X-ray data of a walnut and compare them with filtered backprojection. We also introduce a pointwise positivity constraint as a post-processing step, which substantially improves the reconstruction accuracy. The simulated data experiments are revisited using an updated implementation. The results indicate that the proposed extensions improve reconstruction quality and support the applicability of torus-based reconstruction methods to experimental data.

9. 再生核巴拿赫空间上统计反问题的收敛性分析

Convergence Analysis of Statistical Inverse Problems on Reproducing Kernel Banach Spaces

作者: Darrel K Joseph, M P Rajan

类别: math.NA, math.FA, stat.ML

发布日期: 2026-08-17

arXiv ID: 2608.16404v1

链接: http://arxiv.org/abs/2608.16404v1

摘要: Statistical inverse problems have garnered significant attention in recent years due to the growing importance of statistical learning theory and functional analytic approaches in the fields of machine learning and artificial intelligence. In this paper, we investigate the stable approximation of the element $u^{\dagger}$ that satisfies the equation $Au = g$, where $A$ is a linear operator that maps a Banach space into an appropriate function space. The function $g$ is observed only through independently and identically distributed data points that are corrupted by noise and assumed to follow an unknown distribution $ρ$. We employ the Tikhonov regularization scheme, leveraging statistical learning techniques and the framework of reproducing kernel Banach spaces to estimate the solution. We establish convergence and derive the convergence rate of the estimated solution with respect to the true solution as the number of data points increases, with the rate expressed in probabilistic terms. The theoretical findings are further supported by numerical experiments that demonstrate the effectiveness of the proposed approach.

10. Itô随机延迟微分方程的一种弱二阶Runge-Kutta方法

A weak order 2 Runge-Kutta method for Itô stochastic delay differential equations

作者: Alessia andò, Dimitri Breda, Faraz William

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16396v1

链接: http://arxiv.org/abs/2608.16396v1

摘要: We present a Runge-Kutta method of weak order 2 for the numerical time integration of stochastic delay differential equations. This scheme extends the class of second order Runge-Kutta methods introduced by A. Rößler in [SIAM J. Numer. Anal., 47(3):1713-1738, 2009] for stochastic ordinary differential equations. The proposed integrator is applicable to equations with discrete commensurable delays and is particularly efficient for problems involving multiple noise terms. Experimental confirmation of the weak order 2 is provided and MATLAB codes are freely available.

11. 广义修正分裂方法的收敛性分析：使用多指标级数

Convergence analysis of generalized modified splitting methods using multi-index series

作者: Eugen Bronasco, Mechthild Thalhammer

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16356v1

链接: http://arxiv.org/abs/2608.16356v1

摘要: We consider splitting methods for partial differential equations involving unbounded operators. For non-time-reversible dynamics, such as dissipative systems, negative splitting coefficients are generally not admissible because they require stepping backward in time, leading to an order barrier when all coefficients are required to be positive. We introduce generalized modified splitting methods to overcome this barrier. To analyze their convergence, we develop the corresponding multi-index series formalism, which provides a systematic framework for deriving order conditions. Using this formalism, we derive the order conditions and construct a generalized modified splitting method of order $6$. We also provide Python scripts that automate the generation and verification of order conditions, as well as the construction of new generalized modified splitting methods. Finally, we establish connections between the introduced multi-index series and related series formalisms from the literature, in particular, word series, Lie-Butcher series, and multi-index Butcher series.

12. 带局部自适应网格细化的分层稀疏网格粒子网格法

Hierarchical sparse-grid particle-in-cell method with locally adaptive mesh refinement

作者: F. Deluzet, C. Guillet, J. Narski, P. Pace

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16294v1

链接: http://arxiv.org/abs/2608.16294v1

摘要: In this paper, we introduce new approximation spaces and a locally adaptive refinement strategy for the hierarchical sparse-grid PIC (HSG-PIC) method to improve the bias while preserving the noise-reduction properties of sparse-grid methods. We first propose an energy-based approximation space, which optimizes the relation between the $\mathrm{H}^1$-norm error and the number of degrees of freedom, together with a family of generalized sparse-grid spaces that continuously connects classical sparse-grid and full-grid approximations. We then develop a locally adaptive approximation strategy based on hierarchical surpluses, combined with an efficient incremental refinement algorithm that avoids solving the Galerkin problem on the complete generalized space. Numerical experiments demonstrate that the proposed adaptive HSG-PIC method substantially improves the approximation of solutions with localized structures while maintaining the statistical advantages of sparse-grid discretizations. Compared with a standard full-grid PIC method, the adaptive approach achieves comparable or higher accuracy with a significantly reduced number of mesh nodes and particles. These results demonstrate the potential of adaptive sparse-grid PIC methods for efficient simulations of kinetic plasmas with complex solution structures.

13. 一种用于具有条件稳定性的抛物型反源问题的无伴随积分反馈方法

An adjoint-free integral feedback method for a parabolic inverse source problem with conditional stability

作者: Sedar Ngoma

类别: math.NA, math.AP

发布日期: 2026-08-17

arXiv ID: 2608.16133v1

链接: http://arxiv.org/abs/2608.16133v1

摘要: We study an inverse source problem for a linear non-autonomous parabolic equation with additive source term $f(t)+ζ(t,x)$, where the unknown component depends only on time and is recovered from an integral observation of the solution. After reducing the problem to an equivalent linear inverse problem, we establish existence and uniqueness of the Tikhonov-regularized solution and derive a first-order optimality condition. We show that the forward operator admits a Volterra representation in time, yielding a weak-norm stability estimate in $H^{-1}(0,T)$ and, under an a priori $H^r(0,T)$ bound on the source, a conditional Hölder stability estimate in $L^2(0,T)$. Motivated by this Volterra structure, we introduce an adjoint-free integral feedback method that reconstructs the source using only forward solves. We analyze the feedback iteration by establishing its well-definedness and fixed-point properties, convergence for exact data, and finite-iteration stability with respect to noisy data. We further show that, with an appropriate noise-dependent stopping rule, the method constitutes an iterative regularization scheme. Numerical experiments for smooth and piecewise constant sources, supplemented by temporal regularization and automatic parameter selection, demonstrate accurate and stable reconstructions in the presence of noise.

14. 具有异质局部化与局部边界条件的非局部问题的有限元逼近

Finite Element Approximation of Nonlocal Problems with Heterogeneous Localization and Local Boundary Conditions

作者: Yuyan Chang, Hui Liang, Zhonghua Qiao

类别: math.NA

发布日期: 2026-08-17

arXiv ID: 2608.16079v1

链接: http://arxiv.org/abs/2608.16079v1

摘要: This paper studies the finite element approximation of a one-dimensional nonlocal Poisson problem with heterogeneous localization and homogeneous local Dirichlet boundary conditions. These local boundary conditions induce localization kernels with spatially varying interaction neighborhoods, which lead to substantial numerical challenges for the assembly of the singular nonlocal stiffness matrix. An asymptotically compatible conforming finite element method is developed for the variational formulation, together with an exact geometric decomposition for the singular stiffness matrix assembly. Under additional smoothness assumptions on the localization profile, second-order operator consistency is established and error estimates are derived with the corresponding convergence orders. Numerical experiments confirm the theoretical convergence behavior and demonstrate the improved boundary behavior of the heterogeneous localization model.

15. zenDot：集成大语言模型的量子TCAD平台，用于半导体量子器件设计与优化自动化

zenDot: An LLM-integrated quantum TCAD platform for semiconductor quantum-device design and optimization automation

作者: Zeheng Wang, Yan Liu, Yue Hao, Genquan Han

类别: quant-ph, math.NA, physics.app-ph

发布日期: 2026-08-17

arXiv ID: 2608.16048v1

链接: http://arxiv.org/abs/2608.16048v1

摘要: Semiconductor quantum-device design still lacks an integrated Technology Computer-Aided Design (TCAD)-like environment that connects material geometry, quantum many-body simulation, and automated design. Here we introduce zenDot, a large-language model (LLM)-integrated quantum TCAD platform that links a material-labelled device state to a unified condensed-matter physics toolbox. The device and calculation components are integrated into a desktop workbench, Python API, and an embedded LLM agent, allowing electrostatics, charge and transport characterization, correlated-state calculations, and qubit modelling to be executed within one reproducible environment. We demonstrate zenDot on a Si/SiO2 double quantum dot, where a single device state reproduces the characterization workflow and supports hybrid, tunnel-charge, and singlet-triplet qubit analyses. A platform-level universal-control scan revises the singlet-triplet operating point and reduces the predicted worst-gate infidelity by nearly 30-fold. Beyond analysis, the LLM agent directly operates the same physics environment as human users, proposing design changes, executing registered simulations, and iterating on solver-returned metrics under physics-aware validation. Across three demonstration tasks it completes 18 validated design iterations, including geometry modification followed by a full re-solve from the material stack. zenDot establishes a machine-operable quantum TCAD workflow that connects device physics with LLM-driven design exploration.

16. 无导数随机函数与确定性等式约束的自适应采样信赖域优化

Adaptive Sampling Trust Region Optimization for Derivative-free Stochastic Functions and Deterministic Equality Constraints

作者: Nicole Felice, Sara Shashaani, Lindon Roberts

类别: math.OC, math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15894v1

链接: http://arxiv.org/abs/2608.15894v1

摘要: We study optimization problems with noisy zeroth-order objective observations and deterministic nonlinear equality constraints with available derivatives. We propose a constrained variant of the adaptive-sampling trust-region derivative-free optimization algorithm—ASTRO-DF. The method builds quadratic local models from estimated objective values at interpolation points within a moving trust region and promotes feasibility through a Byrd–Omojokun composite-step based on linearized constraints, following an SQP-like framework. We prove almost sure convergence using a new constrained criticality test and present numerical results on an equality-constrained stochastic activity network problem.

17. 逆星偏差的有界独立性

Bounded independence for the inverse star discrepancy

作者: Kosuke Suzuki

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15865v1

链接: http://arxiv.org/abs/2608.15865v1

摘要: We give a random-bit-efficient construction for the inverse star discrepancy. For every fixed $u\in(0,1)$, $k$-wise independent uniform points $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$ with $k=O(d(1+\log(1+N/d)))$ satisfy the Monte Carlo bound $D_N^*(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N) =O(\sqrt{d/N})$ with probability at least $u$. Consequently, $N=O(d\varepsilon^{-2})$ and $k=O(d(1+\log\varepsilon^{-1}))$ suffice to attain discrepancy at most $\varepsilon$. The proof isolates the finitely many moments required by a chaining argument and gives explicit constants. A random vector-valued polynomial over a finite field realizes the required bounded independence on a grid using $O(d^2(1+\log(1+N/d))\log N)$ random bits, rather than the $Θ(dN\log(dN))$ bits used by independent grid sampling.

18. 一种在所有声学马赫数和阿尔芬马赫数下、适用于具有一般状态方程的理想MHD的保结构和压力正性的半隐式IMEX有限体积格式

A Structure- and Pressure-Positivity-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal MHD at All Acoustic Mach and Alfvén Mach Numbers with Generic Equation of State

作者: Zefeng Chen, Riccardo Demattè, Walter Boscheri, Stephen Millmore

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15837v1

链接: http://arxiv.org/abs/2608.15837v1

摘要: We present a conservative, structure-preserving, finite-volume scheme for ideal MHD that ensures pressure positivity, remains applicable across all Mach and Alfven regimes, and handles general nonlinear equations of state. The scheme splits the MHD system into three sub-systems according to characteristic wave scales: an advective part for hydrodynamic transport, a magnetic part for velocity-field coupling, and a pressure part for pressure-velocity coupling. Nonlinear advective terms are explicit, while the other two sub-systems are implicit, yielding a mild, velocity-based CFL condition that is supported by extensive numerical evidence. This makes the scheme suitable for gas-pressure or magnetic-pressure dominated regimes and the incompressible limit. The implicit discretisation gives a pressure equation that reduces to an elliptic form in the low-Mach limit for ideal gases and general thermodynamics. Pressure positivity is ensured via a local conservation-preserving modification of the pressure-internal-energy relation, avoiding a posteriori clipping while preserving conservation. The divergence-free constraint is enforced exactly via constrained transport. Second-order accuracy is achieved with an IMEX Runge-Kutta time integration, TVD reconstruction for explicit fluxes, and central discretisation for implicit terms. The scheme is validated against numerous benchmarks, including high- and low-Mach regimes, strongly magnetised flows, and standard MHD shock problems in 1D and 2D, demonstrating accuracy, stability, and excellent shock-capturing.

19. 空间变化的输运-反应方程量子模拟的资源分析

Resource Analysis for Quantum Simulation of Spatially Varying Transport-Reaction Equations

作者: Benjamin Jasperson, Krishna Garikipati, Siddhartha Srivastava

类别: quant-ph, math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15827v1

链接: http://arxiv.org/abs/2608.15827v1

摘要: Spatially varying coefficients are the primary source of circuit complexity in quantum simulation of linear advection-diffusion-reaction equations. This work presents a resource analysis of single-step quantum propagators constructed using sparse FABLE block encodings, Quantum Singular Value Transformation, and linear-combination-of-unitaries. We derive theoretical estimates for qubit count, gate complexity, and circuit depth in terms of the spatial discretization, sparsity, and polynomial degree, and compare these predictions with synthesized quantum circuits for one- and two-dimensional variable-coefficient problems. The resulting analysis quantifies the cost of encoding realistic transport operators and provides practical resource estimates for near-term implementations.

20. 三维非线性稳态辐射输运方程的稳健块预处理

Robust Block Preconditioning for 3D nonlinear steady-state radiation transport equations

作者: Yunpan Ma, Lingxiao Li, Changhui Yao

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15826v1

链接: http://arxiv.org/abs/2608.15826v1

摘要: In this work, based on the discrete ordinate method, we propose a robust block preconditioning strategy for the 3D nonlinear steady-state radiation transport equation with heat diffusion term. The presence of the diffusive term of the temperature equation prevents its elimination into a single equation for the radiation intensity. To overcome this difficulty, all physical variables are assembled into a single monolithic linear system. The heat flux and temperature are treated as independent variables in a mixed $H(\mathrm{div})$-conforming finite element formulation. The equation for radiation intensity is discretised by a discontinuous Galerkin method with upwind flux, where a vectorial finite element space is used to couples the radiation intensity in different directions within each element. We then construct a Newton-Krylov iterative solver to solve the nonlinear equations, for which the core part is efficient preconditioning. To accelerate the convergence of Krylov’s method, three block preconditioners are constructed, corresponding to different levels of approximation of the coupling between the temperature and radiation intensity. $P_{\mathrm{Schur}}$ retains the full coupling. $P_{\mathrm{Split}}$ drops the conductive contribution to the radiation block. $P_{\mathrm{BJ}}$ neglects the radiation-to-temperature coupling, retaining only the temperature-to-radiation coupling. Numerical experiments demonstrate the mesh independence and robustness of the proposed preconditioners.

21. 守恒律系统的最大熵耗散数值格式

Maximal entropy dissipation numerical scheme for conservation law systems

作者: Marko Nedeljkov

类别: math.NA, math.AP

发布日期: 2026-08-16

arXiv ID: 2608.15716v1

链接: http://arxiv.org/abs/2608.15716v1

摘要: This paper presents a numerical finite volume method for conservation law systems that are adapted to the principle of maximal dissipation. The general assumptions are the existence of a strictly convex entropy functional and the finite propagation speed property of a given system. The procedure is based on a numerical flux construction obtained by the minimization of the entropy functional in each time step. The scheme satisfies assumptions of the Lax-Wendroff theorem. A limiting solution obtained by this scheme is compared with the classical weak solutions obtained by the Glimm or the Wave Front Tracking algorithm for one-dimensional systems.

22. 带乘性噪声的随机Cahn–Hilliard方程全离散逼近的时间一致强收敛阶

Uniform-in-time strong convergence rates of fully discrete approximations for stochastic Cahn–Hilliard equations with multiplicative noise

作者: Jiaqin He, Nan Deng, Shuhan Zhang, Wanrong Cao

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15625v1

链接: http://arxiv.org/abs/2608.15625v1

摘要: This paper investigates the uniform-in-time strong convergence rates of a fully discrete approximation for the stochastic Cahn–Hilliard equation driven by multiplicative noise in spatial dimensions $d\in\{1,2,3\}$. The proposed scheme combines a spectral Galerkin method in space with a backward Euler scheme in time. The main analytical difficulties arise from the state-dependent stochastic perturbation, the absence of a global monotonicity structure for the nonlinear term, and the fourth-order nature of the Cahn–Hilliard operator. In particular, these features make the derivation of uniform $L^{\infty}$-moment estimates highly nontrivial in three dimensions. For the continuous equation, by utilizing the Itô formula to $\|u\|^p$ and introducing the energy functional $\mathcal{E}(u(t))$, we derive the uniform moment boundedness of the solution. At the fully discrete level, we develop discrete energy estimates and close the required high-order moment bounds through an induction argument. Based on these regularity estimates, we deduce uniform-in-time strong convergence rates for the fully discrete scheme. Moreover, we prove the existence and uniqueness of invariant measures for both the exact dynamics and the fully discrete numerical dynamics. Numerical experiments are provided to confirm the theoretical findings.

23. 高维中的缩放边界求积格式：多面体与曲区域上的积分

Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions

作者: Eric B. Chin, N. Sukumar

类别: math.NA

发布日期: 2026-08-16

arXiv ID: 2608.15563v1

链接: http://arxiv.org/abs/2608.15563v1

摘要: We extend the scaled boundary cubature (SBC) scheme from planar regions to higher-dimensional regions described by oriented boundary patches. The resulting parametrization transforms integrals over compact regions in $\Re^d$ into sums of integrals over the boundary-patch parameter domains and a radial coordinate. In three dimensions, this yields a direct volume-integration rule for solids bounded by affine faces, triangular or tensor-product surface patches, B-spline patches, NURBS patches, and combinations of curved and affine boundary representations. For affine polytopes, recursive application of the scaled boundary map yields nested tensor-product rules over simplex sectors; in three dimensions, these reduce to tetrahedral-sector rules that apply equally to convex and nonconvex oriented polyhedra. We also develop transformations for weakly singular integrands. Placing the scaling center at a point singularity exposes the radial power cancelled by the Jacobian, while generalized radial scalings and Gauss–Jacobi quadrature handle fractional powers. A transverse scaled-boundary map provides the analogous construction for affine singular sets, with straight-line examples in three dimensions. Numerical examples verify polynomial exactness on affine polyhedra and a four-dimensional tesseract, rapid convergence on curved B-spline and NURBS solids, and the expected convergence improvements for point and line singularities. Near-boundary singularity tests also identify when additional patch subdivision or patch-parameter transformations are required.

24. 基于广义对偶数的多方向任意阶方向导数的高效计算

Efficient Computation of Arbitrary-Order Directional Derivatives in Multiple Directions via Generalized Dual Numbers

作者: F. Peñuñuri, K. B. Cantún-Avila, O. Carvente, R. Peón-Escalante

类别: math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15345v1

链接: http://arxiv.org/abs/2608.15345v1

摘要: Arbitrary-order directional derivatives along multiple (possibly distinct) directions are computed through a generalized dual-number formulation for both scalar- and vector-valued functions. The proposed framework combines generalized dual evaluations with an inclusion–exclusion reconstruction of symmetric multilinear forms, allowing general multidirectional derivatives to be reconstructed from repeated-direction evaluations without explicitly constructing higher-order derivative tensors. Mixed directional and mixed partial derivatives arise naturally as particular cases of the formulation. The methodology further enables the systematic computation of arbitrary-order kinematic quantities and the construction of Taylor-series methods for systems of ordinary differential equations through automatically generated time derivatives. Numerical examples include the computation of high-order directional derivatives in high-dimensional functions, mixed partial derivatives, arbitrary-order kinematic quantities, and Taylor-series integration methods. The implementation is developed in modern Fortran within an open-source framework compatible with the Fortran Package Manager ecosystem.

25. 波动方程的时空伽辽金边界元方法

Space-Time Galerkin Boundary Element Method for the Wave Equation

作者: Johannes Tausch

类别: math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15292v1

链接: http://arxiv.org/abs/2608.15292v1

摘要: The space-time Galerkin discretization of retarded layer potentials leads to a linear system where the coefficients are expressed in terms of integrals over the ansatz and test elements. They require carefully designed quadrature schemes because the kernel is singular in the origin and is discontinuous across the hyperbolicity cone. This paper introduces a new integration approach that leads to a scheme that converges exponentially with the number of quadrature points. The key here is to consider the integration domain as a convex polytope and to devise a decomposition into the convex hulls of simpler polytopes which are parameterized such that the singularity and discontinuity occurs in a single variable. The method is implemented for piecewise constant elements and tested on a scattering problem with known analytic solution.

26. Q阶收敛的$p$步推广

A $p$-step generalization of the Q-order of convergence

作者: Gabriel Jarry-Bolduc

类别: math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15202v1

链接: http://arxiv.org/abs/2608.15202v1

摘要: The notion of Q-order convergence is arguably the most important tool for describing the asymptotic behavior of a convergent sequence. Loosely speaking, it captures the``speed’‘of convergence of an iterative method. The concept of Q-order convergence is not always well suited for sequences whose errors do not decrease monotonically at every step. In this paper, we introduce the notion of $p$-step Q-order convergence. It generalizes the classical notion of Q-order convergence by comparing errors that are $p$ iterations apart rather than errors of successive iterates. This definition recovers classical Q-order convergence as the special case $p=1$. We show that it extracts meaningful convergence information from certain non-monotonic sequences for which the classical Q-order either does not exist or assigns an overly pessimistic classification. We develop the basic theory of the new notion and locate it within the classical hierarchy by proving that $p$-step Q-order at least $α$ implies R-order at least $α$. Natural applications include iterative methods whose updates alternate or cycle over multiple steps.

27. 基于GLT序列理论的块Jacobi/Gauss-Seidel与加性/乘性Schwarz预处理分析及其在区域分解离散化中的应用

Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations

作者: Carlo Garoni, Abdessadek Rifqui, Stefano Serra-Capizzano

类别: math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15183v1

链接: http://arxiv.org/abs/2608.15183v1

摘要: When a linear differential problem is discretized by a linear numerical method characterized by a mesh fineness parameter $n$, the computation of the numerical solution reduces to solving a linear discrete problem identified by a matrix $A_n$ whose size grows with $n$. The sequence of discretization matrices $\{A_n\}_n$ often falls within the class of generalized locally Toeplitz (GLT) sequences, even when the numerical method belongs to the family of domain decomposition methods (DDMs), as illustrated herein through examples. Four widely used preconditioners for DDM discretization matrices are the block Jacobi (BJ), block Gauss–Seidel (BGS), additive Schwarz (AS), and multiplicative Schwarz (MS) preconditioners. In this paper, we provide formal definitions of the BJ/BGS/AS/MS preconditioners for arbitrary multilevel block matrices. These definitions and the associated notations are inspired by the theory of GLT sequences and are proposed as alternatives to those commonly used by the DDM community. We analyze the structure of the BJ/BGS/AS/MS preconditioners when applied to multilevel block matrices $A_n$ belonging to a GLT sequence $\{A_n\}_n$. Every GLT sequence $\{A_n\}_n$ is uniquely associated with a special function $κ$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $κ$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $κ$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $κ^{AS}\approxκ$, and $κ^{AS}=κ$ whenever the overlaps in the subdomains used for the construction of $P_n^{AS}(A_n)$ vanish as $n\to\infty$. A numerical validation of these results in the context of isogeometric DDMs is presented.

28. 块编码矩阵多项式的精确高效电路构造

Exact and Efficient Circuit Construction for Block Encoding Matrix Polynomials

作者: Taehee Ko

类别: quant-ph, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15161v1

链接: http://arxiv.org/abs/2608.15161v1

摘要: We propose a direct and stable circuit compiling algorithm that explicitly and exactly constructs block encodings of matrix polynomials for Hermitian matrices. In a unified framework, our algorithm is directly applicable to both standard input models: block encoding and Hamiltonian simulation. For a target polynomial of degree $d$, our classical algorithm achieves a near-optimal time complexity of $\mathcal{O}(d\log d)$. Numerical results confirm this asymptotic scaling for polynomial degrees up to $10^7$ in about a minute on a standard CPU. We achieve this by developing a general-purpose diagonal block encoding of function values, which bridges standard quantum-state preparation techniques with an interpolation-based QSP framework.

29. 三维临界Zakharov–Kuznetsov方程：爆破与孤立子动力学

The 3D critical Zakharov–Kuznetsov equation: blow-up and soliton dynamics

作者: Christian Klein, Svetlana Roudenko, Nikola Stoilov

类别: math.AP, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15086v1

链接: http://arxiv.org/abs/2608.15086v1

摘要: We study the full three-dimensional dynamics of the $L^2$-critical Zakharov-Kuznetsov equation with the fractional nonlinearity $|u|^{4/3}u$, equivalently $u^{7/3}$ for real-valued functions. This equation is a higher-dimensional extension of the generalized Korteweg-de Vries equation. In the critical setting solutions to this 3D ZK equation may blow up in finite time or exhibit global time dynamics. The novelties of this work is to treat a non-integer power and to study the dynamics of solutions in a higher dimension. We first review the finite time blow-up in 2D critical ZK, then do a formal analysis of the slightly mass-supercritical blow-up dynamics, deriving the corrections to the blow-up rate and profile for the critical ZK equation in any dimension. We then perform a computational study of solutions, utilizing full 3D numerical simulations. In particular, we use a Fourier pseudospectral discretization and an integrating factor fourth-order Runge-Kutta method on a full three-dimensional grid. A multi-GPU implementation makes it possible to follow blow-up solutions in a full 3D setting. We examine perturbations of the ground state, Gaussian data, and nonsymmetric two-bump configurations. The computations show dispersive and concentrating regimes, in both cases with radiation emitted in a conic-type region opposite to the direction of propagation and convergence of the concentrating core toward a rescaled ground-state profile. The two-bump experiments also demonstrate that total mass alone does not determine the blow-up dynamics. We discuss the numerical evidence for the predicted blow-up rate and identify the pre-asymptotic and resolution limitations that remain near the blow-up time.

30. 几乎处处复相位恢复的最小测量数

The Minimal Measurement Number for Almost-Everywhere Complex Phase Retrieval

作者: Zhiqiang Xu

类别: cs.IT, math.NA

发布日期: 2026-08-15

arXiv ID: 2608.15003v1

链接: http://arxiv.org/abs/2608.15003v1

摘要: Let $d\geq2$ and let $\bm{f}_1,\ldots,\bm{f}_m\in\mathbb C^d$. We prove that if (m\leq 2d-1), then the intensity measurement map fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimal number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$.

31. 扩散问题的随机blob方法

Random blob methods for diffusion

作者: Katy Craig, Claire Murphy

类别: math.NA, math.AP

发布日期: 2026-08-14

arXiv ID: 2608.14914v1

链接: http://arxiv.org/abs/2608.14914v1

摘要: Linear and nonlinear diffusion equations arise in a range of phenomena of mathematical interest, including slow and fast diffusion, sandpile dynamics, height-constrained transport, the two-dimensional Navier-Stokes equation, and dynamics for sampling probability measures. In recent years, blob methods have attracted significant interest as an approach for numerically simulating these types of PDEs. To address the $O(N^2)$ computational bottleneck of blob methods, we consider stochastic discretizations of space and time. We compare the random batch method with a new approach, which we call the random multirate method. Both of these methods build on classical stochastic methods in the optimization literature, with many similarities to stochastic gradient descent and random coordinate descent. We find that, for linear and nonlinear diffusion equations, in the tradeoff between computational complexity and accuracy, the random multirate method has the best performance. On one hand, we prove that the random multirate method converges to the underlying ODE system at a rate of $O(k Δt)$, matching forward Euler, and show by example that this rate is theoretically sharp. On the other hand, we observe even better rates of convergence for the random multirate method when applied to ODEs arising from blob methods for diffusive PDEs. Finally, due to its ability to simulate a wide range of nonlinear diffusion equations, including height-constrained transport and sandpile dynamics, our method succeeds in capturing key features of PDEs for which few numerical approaches exist.

32. 用于模拟绕圆柱无粘高速流动的带激波捕捉的物理信息SUPG稳定有限元框架

A physics-informed SUPG-stabilized finite element framework with shock-capturing for simulating inviscid high-speed flows around a cylinder

作者: Süleyman Cengizci, Ömür Uğur

类别: physics.flu-dyn, math.NA

发布日期: 2026-08-14

arXiv ID: 2608.14883v1

链接: http://arxiv.org/abs/2608.14883v1

摘要: This study presents a hybrid computational framework for simulating non-reacting inviscid high-speed flows of nitrogen gas (N$_2$) around a circular cylinder. Owing to the strongly convection-dominated nature of the compressible Euler equations, the compressible-flow streamline-upwind/Petrov–Galerkin (SUPG) formulation is combined with the YZ$β$ shock-capturing technique to stabilize the finite element discretization in the presence of strong discontinuities. Building upon the stabilized solution, a physics-informed neural network (PINN) is employed as a post-processing correction stage (\underline{P}INN-\underline{A}ugmented \underline{S}UPG with \underline{S}hock-\underline{C}apturing—PASSC). The network is anchored to the finite element solution through a shock-weighted data-consistency loss, while the governing equations are enforced in a conservative space–time control-volume form supplemented by macroscopic conservation windows, an entropy-admissibility penalty, and the boundary conditions of the underlying problem. Two-dimensional simulations are performed for free-stream Mach numbers ranging from $2.0$ to $12.0$, and the results are assessed against analytical normal-shock and stagnation relations, the semi-empirical Billig correlation, and reference solutions from the literature. The correction is designed to improve the numerical representation of shocks by reducing localized discretization-induced oscillations and mesh-scale serrations while preserving the agreement of the stabilized solution with the analytical and semi-empirical reference quantities.

33. 正则性引导的数据同化：双曲守恒律的集合卡尔曼滤波的一种分层贝叶斯方法

Regularity-informed data assimilation: A hierarchical Bayesian approach to ensemble Kalman filtering for hyperbolic conservation laws

作者: Jan Glaubitz, Daniel Sharp, Mathieu le Provost, Youssef M. Marzouk

类别: math.NA, stat.ME

发布日期: 2026-08-14

arXiv ID: 2608.14840v1

链接: http://arxiv.org/abs/2608.14840v1

摘要: We propose a novel regularity-informed filtering framework for data assimilation in the context of hyperbolic conservation laws and other time-dependent partial differential equations. We focus on systems whose states exhibit steep gradients and jump discontinuities. While filtering is widely used to improve numerical simulations by incorporating observational data, traditional filtering methods lack awareness of the spatial regularity of states produced in these systems. As a result, data assimilation often produces unphysical state estimates, introducing spurious oscillations in smooth regions and smearing sharp features. To address this limitation, we introduce a filtering framework incorporating edge-preserving regularization into the filter’s analysis step; this framework balances simulation forecasts, observation data, and structural prior knowledge. We formalize this approach using the ensemble Kalman filter (EnKF) and a class of hierarchical generalized sparse Bayesian learning (GSBL) priors, which adaptively infer spatially varying hyperparameters to promote non-oscillatory behavior in smooth regions while preserving discontinuities. We demonstrate the effectiveness of the resulting GSBL-EnKF method on challenging benchmark problems governed by hyperbolic conservation laws. Our results show that preserving regularity during the analysis step can improve the physical realism and accuracy of filtering for complicated time-dependent systems, especially when quantifying the uncertainty of transient states of the system.