生成时间: 2026年05月05日 05:45:56

重点关注论文数量: 3

了解领域论文数量: 3

不相关论文数量: 44

重点关注论文（完整分析）

1. 带有先验界的椭圆型低维拟周期解的数值构造

Numerical Construction of Elliptic Lower-Dimensional Quasi-Periodic Solutions with a Priori Bound

作者: Mingwei Fu, Bin Shi

类别: math.NA

发布日期: 2026-05-03

arXiv ID: 2605.01864v1

摘要翻译: 近期，Fu 和 Shi[2026] 针对近可积系统发展了构造全维拟周期解的数值框架。该方法基于交替格式，有效克服了角变量中的长期漂移——这是辛积分子的一项根本性局限。然而，在许多应用中，例如限制性三体问题，低维拟周期解更为重要。由于正规频率的存在导致复杂的共振现象，这类解的构造更具挑战性。除了子空间共振，还必须考虑第一和第二 Melnikov 条件以消除小除数。本研究将所提出的交替数值格式推广到椭圆型低维拟周期解的计算。我们针对 Hénon-Heiles 模型和 Fermi–Pasta–Ulam (FPU) 模型进行了数值实验，验证了方法的有效性。此外，我们强调扰动不仅仅是一个实系数多项式，而是一个实值函数。因此，相关的扰动算子表现出 Gevrey 衰减，且不具有 Hankel 结构。同时，我们利用预解恒等式进一步简化了多尺度分析，表明全局逆可以通过粘合过程用局部逆线性表示。这一表示揭示出与尺度相关的相互作用结构：短程以弱相互作用为主，而长程则出现强相互作用。这种平衡确保逆的 Gevrey 衰减保持均匀控制。而且，在这一线性表示中，反演条件为局域化性质提供了更清晰的刻画。

Comment: 41 pages, 6 figures

链接: http://arxiv.org/abs/2605.01864v1

1. 研究对象和背景

近可积哈密顿系统是解析动力系统的核心研究对象, 在行星轨道稳定性, 晶格振动等跨尺度问题中起关键作用. 在作用-角坐标下, 系统由如下哈密顿量描述:

$$ H(I,\theta)=H_0(I)+\varepsilon H_1(I,\theta), $$

其中 $I\in\mathbb{R}^n$, $\theta\in\mathbb{T}^n$, $\varepsilon$ 是小参数. 无扰动 ($\varepsilon=0$) 时, 运动是拟周期的:

$$ I(t)=I_0,\quad \theta(t)=\theta_0+\omega(I_0)t,\quad \omega(I_0)=\nabla H_0(I_0). $$

在扰动下, 考虑初始作用 $I_0$ 附近的平移 $J=I-I_0$, 并泰勒展开可得等效哈密顿量:

$$ H(J,\theta;I_0)=\langle\omega(I_0),J\rangle+\varepsilon H_1(J,\theta;I_0), $$

其中 $H_1$ 吸收了高阶项. 若频率映射 $\omega: I_0\mapsto\omega(I_0)$ 是微分同胚 (例如满足等能非退化条件 $\det\begin{pmatrix}\nabla^2 H_0 & \nabla H_0 \\ (\nabla H_0)^\top & 0\end{pmatrix}\neq0$), 则可重新参数化到频率空间:

$$ H(J,\theta;\omega)=\langle\omega,J\rangle+\varepsilon H_1(J,\theta;\omega). $$

在此框架中, 经典的 KAM 理论断言: 若频率满足 Diophantine 条件, 且不为共振关系 $\langle k,\omega\rangle =0$ (对 $k\in\mathbb{Z}^n\setminus\{0\}$), 则大多数不变环面在小扰动下得以保持. 然而, 在许多物理系统中, 某些作用分量可能为 0, 例如限制性三体问题中与主星偏心率相关的作用量为零. 此时, 频率分解为切向部分 $\omega_T\in\mathbb{R}^m$ 和法向部分 $\omega_N\in\mathbb{R}^{n-m}$, 哈密顿量可写为

$$ H(J,\theta,z,\bar z;\omega_T)=\langle\omega_T,J\rangle+\langle\omega_N z,\bar z\rangle+\varepsilon H_1(J,\theta;z,\bar z;\omega_T), $$

其中 $z_i=\sqrt{I_i}e^{i\theta_i}$ 是复变量. 对应的不变环面是 $m$ 维的 (低维环面), 嵌入在 $2n$ 维空间中. 当法向频率为实数 ($\omega_N\in\mathbb{R}^{n-m}$) 时称为椭圆型, 否则为双曲型. 椭圆低维环面的存在性涉及更复杂的共振条件: 除了标准切向共振 $\langle k,\omega_T\rangle =0$ ($k\in\mathbb{Z}^m\setminus\{0\}$) 外, 还需避免第一 Melnikov 条件

$$ \langle k,\omega_T\rangle -\omega_j =0,\quad j=m+1,\dots,n, $$

和第二 Melnikov 条件

$$ \langle k,\omega_T\rangle -\omega_{j_1}+\omega_{j_2}=0,\quad \langle k,\omega_T\rangle -\omega_{j_1}-\omega_{j_2}=0,\quad j_1\neq j_2. $$

Eliasson (1988) 和 Pöschel (1989) 的 KAM 定理证明了在适当非共振条件下大多数椭圆低维环面仍存在.

数值上构造拟周期解面临长期相位漂移的困难: 传统的辛积分器虽能保持几何结构, 但角度变量的误差线性累积会导致长期偏离. Fu 和 Shi (2026) 提出一种交替格式, 将常微分方程组转化为代数系统, 通过直接求解 Fourier 系数克服了漂移问题, 成功构造了全维拟周期解. 本文将该方法推广到需处理法向频率的椭圆低维情形, 核心困难在于小分母更密集, 且扰动是实值函数 (非多项式), 导致相应的扰动算子具有 Gevrey 衰减但无 Hankel 结构; 同时作者利用预解恒等式简化了多尺度分析, 将全局逆矩阵表示为局部逆的线性组合, 揭示了短程弱耦合, 长程强耦合的机制, 从而统一控制逆矩阵的 Gevrey 衰变. 本文的目标是建立一个带有先验误差界的收敛数值格式, 以任意精度构造椭圆低维拟周期解.

2. 主要定理或主要结果

主要结论集中于定理 1.1 (收敛性与先验界). 设 $\Omega\subseteq\mathbb{R}^m$ 为有界域, $\tau>m-1$ 固定. 存在临界阈值 $\varepsilon_0=\varepsilon_0(H_1,\Omega,a)>0$, 使得对任意 $0<\varepsilon\leq\varepsilon_0$, 以下成立: (1) 坏频率集合的测度: 存在常数 $\kappa=\kappa(n,m,\tau,\Omega)>0$ 和 $\delta=\delta(\varepsilon)>0$, 满足 $\lim_{\varepsilon\to0}\delta=0$, 且坏频率集 $\Omega_\star$ 满足 $\mathrm{mes}(\Omega_\star)\leq\kappa+\delta$. (2) 迭代序列: 对任意初值 $\omega_T^{(0)}\in\Omega\setminus\Omega_\star$ 和 $\hat{z}_p^{(0)}=0$, 交替格式 (1.18) 生成迭代序列 $(\omega_T^{(r)},\hat{z}_p^{(r)})$. (3) 收敛性: 存在 Gevrey 指数 $s=s(\varepsilon)$ 使得该序列保持在 $(\Omega\setminus\Omega_\star)\times\mathbf{K}(s)$ 中, 并收敛到精确解 $(\omega_T^\star,\hat{z}_p^\star)$, 其中 $\mathbf{K}(s)$ 为具有 Gevrey 衰减 $\sup_k\|\hat{z}(k)\| e^{|k|s}\leq1$ 的函数空间. (4) 先验界: 存在常数 $M=M(\varepsilon)$, 使得收敛速度由下述超指数界刻画:

$$ \|\hat{z}^{(r)}-\hat{z}^\star\|\leq \exp\!\left(-\frac{3}{2}(M s)^r\right),\quad |\omega_T^{(r)}-\omega_T^\star|\leq \exp\!\left(-\frac{3}{2}(M s)^r\right). $$

此外, 对于 $t\in[0,M^r]$, 近似解 $z^{(r)}(t)$ 满足误差界:

$$ \|z^{(r)}(t)-z^\star(t)\|\leq \exp(-(M s)^r). $$

这意味着对任意 $t\geq0$, 当 $r\to\infty$ 时 $z^{(r)}(t)$ 收敛到真实拟周期解 $z^\star(t)$.

辅助性结果包括: 在 Gevrey 衰减集 $\mathbf{K}(s)$ 中, 向量场 $\hat{X}$, 其 Jacobi 矩阵 $S$, 以及二阶导数均有界 (命题 2.1–2.3); 引入的算子 $B$ 也有界 (命题 2.4); 频率漂移 $\omega_T'-\omega_T$ 被 $\varepsilon$ 控制, 且频率映射是微分同胚 (命题 2.5). 在小尺度下, 若频率不属于近共振集 $\Omega_M^\tau$, 截断算子 $L_N$ 的逆的算子范数为 $O(\varepsilon_N^{-1})$, 且矩阵元素具有 Gevrey 衰减 $\exp(-|k-k'|s)$ (定理 3.3). 大尺度的相应结论在第 4 章通过多尺度分析得出, 并满足实现条件 3.1. 最终在第 5 章综合得到迭代引理并证明主定理.

3. 研究方法、关键技术和核心工具

克服长期漂移: 代数化与交替迭代传统辛积分直接离散化 ODE 会引起角度变量的长期误差积累. 本文沿用 Fu–Shi 方法将动力学转化为求解 Fourier 系数的代数方程 (1.12). 将解空间按共振模式分解为 $Q$-方程 (1.14) 确定切向频率, $P$-方程 (1.15) 确定非共振系数. 构造交替迭代格式 (1.18) 更新频率与 Fourier 系数, 本质是一种牛顿型方法: 频率由 $Q$-方程显式修正, 非共振系数用截断线性化算子 $L_N = T+\varepsilon B$ 的逆步进. 关键引入修正算子 $B$ (1.19), 它源于 $\hat{X}_q$ 对 $\hat{z}_p$ 的依赖, 用以保障迭代映射的收缩性.
小分母与截断逆的控制低维情形下, 分母 $\langle k,\omega_T'\rangle$ 及与法向频率的差可能导致极小的除数. 本文引入了依赖于尺度的阈值 $\varepsilon_N=\exp(-(\log N)^{15})$, 并设立实现条件 (3.2)–(3.3): 要求截断逆的范数有界且远距离阵元具有 Gevrey 衰减. 为满足这些条件, 需排除“坏”频率. 在小尺度 ($N\leq M$) 下, 通过测度估计 (引理 3.2) 排除满足近共振条件 (3.5)–(3.7) 的频率, 这些条件涵盖了标准共振, 第一和第二 Melnikov 共振. 在此补集中, 利用对角占优技术证明了 $L_N^{-1}$ 的有界性和 Gevrey 衰减 (定理 3.3).
多尺度分析与预解恒等式对于大尺度 ($N>M$), 单靠简单的对角占优不足. 本文的创新之一是利用预解恒等式 (resolvent identity) 将全局逆矩阵表示为局部逆的线性组合, 从而将问题分解为不同尺度盒子的耦合. 这种方法揭示出短程弱相互作用, 长程强相互作用的平衡机制, 保证了逆矩阵的 Gevrey 衰变在整个盒子上的统一控制, 同时使得逆存在的条件更清晰地归结为局部可逆性和耦合的衰减性. 具体的归纳构造建立在两种小尺度基情形之上.
Gevrey 衰减空间与算子的估计关键假设是 Fourier 系数属于 Gevrey 空间 $\mathbf{K}(s)$, 即 $\|\hat{z}(k)\|\leq e^{-|k|s}$. 在此空间内, 因为扰动 $H_1$ 是实值函数 (不限于多项式), 推导出向量场 $\hat{X}$ 及其各阶导数仍具有同类型的 Gevrey 衰减 (命题 2.1–2.3), 纵然没有 Hankel 结构. 这些估计保证了牛顿迭代中线性化算子的有效性和误差的快速缩减. 特别地, 命题 2.3 中二阶导数在截断区域 $\Lambda_N$ 上的界为 $O((2N+1)^{m/2})$, 这被容忍因为最终误差是超指数衰减.
迭代引理的归纳证明论文通过迭代引理 (Iterative Lemma) 将局部估计组装起来: 每一步迭代将误差范数从 $r$ 步提升到 $r+1$ 步, 并获得超指数收缩. 证明中交替使用了小尺度和大尺度的反演结果, 以及 Gevrey 衰减所保证的线性化算子的精细性质.

4. 与之前工作的比较

与 Fu 和 Shi (2026) 的比较: 前驱工作针对全维拟周期解, 该情形仅需避免切向共振 $\langle k,\omega\rangle=0$, 且作用量各分量非零. 本文将其交替数值格式推广到椭圆低维拟周期解, 此时部分作用为零, 法向频率出现, 导致必须额外规避第一, 第二 Melnikov 共振条件 (1.6)–(1.7). 技术上, 全维情形处理 $n$ 维频率 $\omega$ 的 Diophantine 条件, 而低维情形只需处理 $m$ 维切向频率 $\omega_T$ (因为分母只涉及 $\omega_T$), 这带来两个影响: 一方面使得小分母的出现频率可能更低 (多重特征值), 降低了问题的刚性; 另一方面因共振条件增多, 排除坏频率的测度估计更复杂 (需引入 $\Omega_{1,M}^\tau$ 和 $\Omega_{2,M}^{+,\tau}$). 此外, 修正算子 $B$ 的引入也是为适应低维扩展而设计.

与经典 KAM 理论的比较 (如 Pöschel, Eliasson): 经典 KAM 通过无穷次辛变换消除非共振项, 本质是分析性的存在性结果, 并不提供直接数值构造方案. 本文则给出一个数值可实现的迭代格式, 且在每一步都有明确截断和误差控制, 最终给出超指数收敛的先验界. 同时, 本文去除了扰动 $H_1$ 为多项式的要求, 允许一般的实值函数, 从而覆盖更广的物理模型. 这一推广使得扰动算子失去 Hankel 结构, 但本文证明了 Gevrey 衰减仍足以维持所需估计.

与基于辛积分的传统数值方法比较: 传统辛积分器 (如 Feng 1995, Hairer et al. 2006) 在长时间积分时会因相位误差积累导致角度变量的长期漂移, 丧失真实的拟周期特征. 本文方法通过直接求解 Fourier 系数代数系统, 本质上避免了在时域上推进, 从而根除了时间步长带来的漂移问题. 文中强调, 所构造的 $z^{(r)}(t)$ 对所有 $t\geq0$ 收敛到精确解, 即提供了任意长时间区间上的精确近似, 这是经典时间步进方法无法保证的.

与 Craig–Wayne–Bourgain (CWB) 方案的对比: CWB 方案 (Bourgain 1997, 2005) 原本是在 PDE 环境中构造低维拟周期解的解析方法, 采用 Liouvillian 坐标和惩罚方法. 本文虽是数值格式, 但汲取了 CWB 直接处理哈密顿量而不预先分解的思想, 用于椭圆低维情形, 并结合多尺度分析和预解恒等式提供了新的数值刚性控制手段.

局限与代价: 与所有 KAM 型方法类似, 本文需要排除坏频率集, 其测度虽随 $\varepsilon$ 减小而可控, 但严格大于零, 故不能保证对所有频率成立. 此外, 先验界中的常数 $M$ 和 $s$ 依赖于 $\varepsilon$, 当 $\varepsilon$ 很小时 $M$ 会极大, 导致实际收敛速度在初期可能较慢, 但理论上实现超指数收敛.

2. Maxwell à la Helmholtz：基于 Helmholtz 算子的完美电导体三维散射直接边界积分方程

Maxwell à la Helmholtz: Direct boundary integral equations for 3D scattering by perfect electric conductors via Helmholtz operators

作者: Carlos Pérez-Arancibia, Catalin Turc

类别: math.NA, physics.comp-ph

发布日期: 2026-05-03

arXiv ID: 2605.01670v1

摘要翻译: 本文是 [Burbano-Gallegos, P’erez-Arancibia, and Turc, ESAIM: M2AN, 60(1):273–315, 2026] 的直接公式化对应篇，该文针对光滑完美电导体 (PEC) 障碍物的时谐电磁散射，发展了完全依赖 Helmholtz 边界积分算子的间接仅组合场边界积分方程 (BIEs)。本文利用 Maxwell PEC 散射问题与一对向量 Helmholtz 边值问题（一个关于电场，一个关于磁场）之间的同一等价性，来推导直接的 BIE 公式，其未知量为总场的 Dirichlet 和 Neumann 迹，并分解为其法向和切向表面分量。这些未知量具有直接的物理意义：特别地，磁场公式在其解中给出表面电流。这两个场迹分量的混合正则性要求引入一个特制的乘积 Hölder 空间，这是间接方法所没有的独特特征。我们证明，所得到的直接电场和磁场仅组合场积分方程（D-ECFOIE 和 D-MCFOIE）在所有频率下唯一可解，并引入 Calderón 型正则化（RD-ECFOIE 和 RD-MCFOIE）使它们成为第二类 Fredholm 方程。我们进一步考察了影响电场公式的低频崩溃，并引入一个修正方程，该方程施加物理电荷守恒约束，从而为任意接近零的频率恢复数值精度和良好条件的线性系统。基于密度插值方法的高阶 Nyström 求解器并在 Julia 包 Inti.jl 中实现的数值实验，验证了所提公式在一系列几何体和频率上的精度和稳健性。

Comment: 无

链接: http://arxiv.org/abs/2605.01670v1

1. 研究对象和背景

本文研究时谐电磁波在三维空间中被光滑完美电导体(PEC)障碍物散射的直接边界积分方程(BIE)方法，核心思想是将 Maxwell 散射问题完全转化为仅用亥姆霍兹(Helmholtz)边界积分算子表示的形式。考虑角频率 $\omega>0$ 的时谐场（时间因子 $e^{-i\omega t}$），设障碍物占据有界开区域 $\Omega\subset\mathbb{R}^3$，边界 $\Gamma=\partial\Omega$ 为 $C^{2,\alpha}$ ($\alpha\in(0,1)$) 光滑的闭曲面，并可能由多个互不相交的连通分支组成。外部区域 $\mathbb{R}^3\setminus\overline{\Omega}$ 充满各向同性均匀介质，介电常数 $\epsilon>0$，磁导率 $\mu>0$，波数 $k=\omega\sqrt{\epsilon\mu}>0$。入射电磁场 $(\bm{E}^i,\bm{H}^i)$ 在包含 $\Omega$ 的开集上满足 Maxwell 方程 $\nabla\times\bm{E}^i - i\omega\mu\bm{H}^i = \bm{0}$，$\nabla\times\bm{H}^i + i\omega\epsilon\bm{E}^i = \bm{0}$。散射场 $(\bm{E}^s,\bm{H}^s)$ 满足：在外域中满足 Maxwell 方程

$$ \nabla\times\bm{E}^s - i\omega\mu\bm{H}^s = \bm{0},\quad \nabla\times\bm{H}^s + i\omega\epsilon\bm{E}^s = \bm{0}, $$

边界上满足 PEC 条件：切向电场分量为零，即 $\mathsf{P}_t\gamma\bm{E}^s = -\mathsf{P}_t\gamma\bm{E}^i$，其中 $\gamma$ 为外 Dirichlet 迹，$\mathsf{P}_t$ 为到切平面的投影算子；并在无穷远处满足 Silver–Müller 辐射条件。该散射问题的适定性在 Hölder 空间框架下已由前人工作（如[3]）建立。

传统 Maxwell BIE（如 EFIE、MFIE、CFIE）以表面切向矢量场为未知量，核具有超强奇异性，面临低频崩溃、需要 Calderón 预处理等困难。近年来，研究者尝试通过引入标量势或利用等价性将 Maxwell 问题转化为 Helmholtz 方程问题。文献[3]提出了“Maxwell à la Helmholtz”框架：证明了 Maxwell PEC 散射问题等价于一对关于电场 $\bm{E}^s$ 和磁场 $\bm{H}^s$ 的向量 Helmholtz 边值问题(BVP)，其中边界条件仅依赖于各自场的 Dirichlet 和 Neumann 迹的切向/法向分量。具体地，$\bm{E}^s$ 满足

$$ \Delta\bm{E}^s + k^2\bm{E}^s = \bm{0} \ \text{在}\ \mathbb{R}^3\setminus\overline{\Omega}, $$$$ \mathsf{P}_t(\gamma\bm{E}^s) = -\mathsf{P}_t(\gamma\bm{E}^i), \quad \mathsf{P}_\nu(\partial_\nu\bm{E}^s + 2H\gamma\bm{E}^s) = -\mathsf{P}_\nu(\partial_\nu\bm{E}^i + 2H\gamma\bm{E}^i), $$

及辐射条件；类似地，$\bm{H}^s$ 满足 $\Delta\bm{H}^s + k^2\bm{H}^s = \bm{0}$，边界条件

$$ \mathsf{P}_\nu(\gamma\bm{H}^s) = -\mathsf{P}_\nu(\gamma\bm{H}^i), \quad \mathsf{P}_t(\partial_\nu\bm{H}^s + \bm{R}\gamma\bm{H}^s) = -\mathsf{P}_t(\partial_\nu\bm{H}^i + \bm{R}\gamma\bm{H}^i), $$

其中 $\partial_\nu$ 为外法向导数（Neumann 迹），$\bm{R}$ 为形算子（Weingarten 映射），$H$ 为平均曲率。基于此等价性，[3]发展了间接组合场积分方程(ECFOIE 和 MCFOIE)，其未知量为辅助面密度，无直接物理意义。

本文作为[3]的直接法对应，利用相同的等价性，但通过格林表示定理直接导出以总场迹为未知量的边界积分方程。这些未知量（总电场的法向分量 $\mathsf{P}_\nu\gamma\bm{E}$ 与 Neumann 迹的切向分量 $\mathsf{P}_t\partial_\nu\bm{E}$，及对磁场类似量）具有清晰的物理意义：例如，磁公式的解直接给出表面电流 $\bm{J} = \nu\times\bm{H}$。由于迹的分量具有不同的光滑性（Dirichlet 迹比 Neumann 迹光滑一个阶），需要引入混合 Hölder 空间 $C^{p,q}_{\nu,t}(\Gamma)$ 来容纳解，这是直接法与间接法的一个显著区别。本文旨在填补直接公式在适定性分析、正则化和低频行为方面的理论空白，提供一个与间接法对应的完整数学框架，并给出高效数值实现。

2. 主要定理或主要结果

本文的主要结果包括以下几个方面。

直接电场和磁场组合场积分方程(D-ECFOIE 和 D-MCFOIE)的推导。 对于电场散射问题，利用格林表示定理写出总场 $\bm{E}=\bm{E}^s+\bm{E}^i$ 在外域的积分表示
$$ \bm{E}^s(\bm{x}) = \mathcal{D}[\mathsf{P}_\nu\gamma\bm{E}](\bm{x}) - \mathcal{S}[\mathsf{P}_t\partial_\nu\bm{E} - 2\bm{H}\mathsf{P}_\nu\gamma\bm{E}](\bm{x}), \quad \bm{x}\in\mathbb{R}^3\setminus\overline{\Omega}, $$
其中 $\mathcal{D}$ 和 $\mathcal{S}$ 为向量亥姆霍兹双、单层位势。取外 Dirichlet 和 Neumann 迹并组合得到 D-ECFOIE：
$$ -\frac{1}{2}\{\bm{\varphi}_t + (i\eta\bm{I} - 2\bm{H})\bm{\varphi}_\nu\} + \{\bm{T} + i\eta\bm{K} + 2(\bm{K}' + i\eta\bm{S})\bm{H}\}\bm{\varphi}_\nu - (\bm{K}' + i\eta\bm{S})\bm{\varphi}_t = \bm{f}, $$
其中未知量 $\bm{\varphi} = \bm{\varphi}_\nu + \bm{\varphi}_t$，$\bm{\varphi}_\nu = \mathsf{P}_\nu\gamma\bm{E}\in C^{1,\alpha}$，$\bm{\varphi}_t = \mathsf{P}_t\partial_\nu\bm{E}\in C^{0,\alpha}$；右端 $\bm{f} = -\partial_\nu\bm{E}^i - i\eta\gamma\bm{E}^i$；$\eta\in\mathbb{R}\setminus\{0\}$ 为耦合参数。类似地，针对磁场问题导出 D-MCFOIE，未知量为 $\mathsf{P}_\nu\partial_\nu\bm{H}$ 和 $\mathsf{P}_t\gamma\bm{H}$，其在 $C^{0,1}_{\nu,t}(\Gamma)$ 中。
唯一性定理。 定理 4.2 证明：对任意 $k>0$ 及 $\eta\neq0$，D-ECFOIE 在 $C^{1,0}_{\nu,t}(\Gamma)$ 中至多有一解；定理 5.1 给出 D-MCFOIE 在 $C^{0,1}_{\nu,t}(\Gamma)$ 中的唯一性。证明思路是将齐次方程的解对应到某个位势 $U$，该位势在障碍内部和外部都满足相应的边值问题，由 Helmholtz 内 Robin 问题的唯一性和外 PEC 问题的唯一性推知密度为零。
Calderón 型正则化与第二类 Fredholm 方程。 通过引入单层算子的组合构造出紧扰动，将 D-ECFOIE 转化为正则化方程 RD-ECFOIE：
$$ (-\tfrac{1}{2}\bm{I} + \bm{\mathcal{K}}_e)\bm{\varphi} = \tilde{\bm{f}}, $$
其中 $\bm{\mathcal{K}}_e$ 在 $C^{0,\alpha}(\Gamma,\mathbb{C}^3)$ 上为紧算子，从而 RD-ECFOIE 属于 Fredholm 第二类。结合唯一性即得解的存在性。类似地，对磁场形式有 RD-MCFOIE。
低频崩溃的修正。 当 $k\to0$ 时，D-ECFOIE 可能失效（因为静电场情形下电荷守恒条件未被适当反映）。本文提出修正方程
$$ \mathsf{L}_e\bm{\varphi} + \xi\bm{S}_0\mathsf{P}_\nu\mathsf{P}_t\bm{\varphi} = \bm{f}, $$
其中 $\mathsf{L}_e$ 为原 D-ECFOIE 算子，$\xi\in\mathbb{C}\setminus\{0\}$ 为稳定参数，$\bm{S}_0$ 为 Laplace 单层算子。证明对所有 $k>0$ 该修正方程唯一可解（在温和条件下 $\xi\neq -2$ 等）。数值实验显示，该修正对接近零的频率恢复了精度和良态迭代次数。
数值验证。 利用基于广义密度插值法(GP-DIM)的高阶 Nyström 方法，在 Julia 包 Inti.jl 中实现，针对不同拓扑和几何的光滑表面（包括多连通情形）及广泛频率范围的平面波入射，验证了所提公式的准确性和稳健性。

3. 研究方法、关键技术和核心工具

本文的分析与构造涉及数个关键技术点，下面逐一分述。

1. 乘积函数空间 $C^{p,q}_{\nu,t}(\Gamma)$ 的引入。 由于待求未知量由迹的切向和法向分量组成，它们在边界上的光滑性不同：Dirichlet 迹的分量比 Neumann 迹的分量高一阶导数。传统的 Hölder 空间无法同时容纳两者，故定义

$$ C^{p,q}_{\nu,t}(\Gamma) := \{ \bm{\varphi}:\Gamma\to\mathbb{C}^3 \mid \mathsf{P}_\nu\bm{\varphi}\in C^{p,\alpha}(\Gamma,\mathbb{C}^3), \ \mathsf{P}_t\bm{\varphi}\in C^{q,\alpha}(\Gamma,\mathbb{C}^3) \}, $$

配备范数 $\|\bm{\varphi}\|_{C^{p,q}_{\nu,t}} = \|\mathsf{P}_\nu\bm{\varphi}\|_{C^{p,\alpha}} + \|\mathsf{P}_t\bm{\varphi}\|_{C^{q,\alpha}}$，并证明它是 Banach 空间（引理 4.1）。这为解的适定性提供了恰当的分析框架。间接方法[3]中因未知量为辅助面密度且具有一致光滑性，无需此空间。

2. 利用等价性定理构建唯一性论证。 唯一性证明的核心是构造一个由未知密度生成的亥姆霍兹位势 $\bm{U}$，其内、外 Robin 迹满足齐次方程对应的边界条件。通过内区域 Helmholtz 方程 Robin 问题的唯一性（利用 $\eta\neq0$ 和 $k>0$ 的虚部符号）得内部迹为零，跳跃关系给出外部迹的表达式，进而验证它们满足等价性定理中的 PEC 边界条件。因为外部 PEC 问题解唯一，故外部迹也为零，从而密度为零。这一证明链条简洁且充分利用了[3]中建立的等价关系，避免了直接处理 Maxwell 方程的复杂性。

3. Calderón 正则化与 Fredholm 性质。 原始 D-ECFOIE 的算子包含双重、单层及超奇异算子，直接看不是第二类 Fredholm。本文构造正则化算子 $\bm{\mathcal{R}}_e$，其核心是利用 Laplace 单层算子 $\bm{S}_0$ 构造紧扰动：令

$$ \bm{\mathcal{R}}_e\bm{\varphi} := \bm{S}_0\left( \bm{P}_\nu\bm{\varphi} + \bm{P}_t\bm{\varphi} \right) + \text{紧项}, $$

使得复合后的主部成为恒等算子加减一个紧算子。具体地，通过计算跳跃关系，发现形如 $\bm{C}_\nu$ 和 $\bm{C}_t$ 的算子（引理 4.3）是紧的，从而确保正则化方程的整体算子 $\bm{A}_e = \bm{\mathcal{R}}_e\mathsf{L}_e$ 具有形式 $-\frac{1}{2}\bm{I} + \bm{\mathcal{K}}$，其中 $\bm{\mathcal{K}}$ 紧。紧性的证明利用了 Laplace 单层核的弱奇异性以及几何差量 $\nu_i(\bm{x})-\nu_i(\bm{y})$ 带来的可积性。正则化后方程的解即为原方程的解，故唯一性传递，从而由 Fredholm 择一性得解存在。这种“单层正则化”策略在声学散射中常用，但本文将其成功地移植到混合正则性的向量方程中，是一个技术亮点。

4. 低频机制的修正与电荷守恒约束的数学实现。 当 $k\to0$ 时，D-ECFOIE 的连续问题可能非唯一（因为静电极限下电场边界值问题引入多余自由度），导致数值矩阵条件数增长。本文通过分析发现，问题根源在于齐次解中包含不影响物理场的“无旋”分量。为此，在算子中加入约束项 $\xi\bm{S}_0\mathsf{P}_\nu\mathsf{P}_t$，强制电荷面散度条件。该修正本质上是将原方程限制到满足 $\mathsf{P}_\nu\mathsf{P}_t\bm{\varphi}=0$ 的子空间上，而这一条件正是静电情形下电荷守恒的等价表述。证明修正方程唯一可解的关键在于：任意频率下修正算子的零空间与物理散射问题解的唯一性相容；低频极限下，结合 Laplace 单层算子的正性与参数条件，确保了解的好条件数。该修正仅对电场公式需要，磁公式在单连通表面上自动免疫低频崩溃。

4. 与之前工作的比较

本文与多个已有工作形成对照，以下从几个维度展开比较。

1. 与间接法框架的比较（主要与[3]）。 本文是文献[3]中提出的“Maxwell à la Helmholtz”框架的直接法配套工作。[3]通过间接的场-唯一积分方程（ECFOIE 和 MCFOIE）求解，其未知量为辅助密度，出现在组合层位势的表述中，并无直接物理意义。而本文的直接公式未知量本身就是物理迹（总场的法向分量和 Neumann 迹的切向分量），因此磁公式的解可直接给出表面电流，而不需要额外后处理。此外，由于迹分量的混合正则性，直接法需要引入乘积 Hölder 空间 $C^{p,q}_{\nu,t}$，这在间接法中是不需要的。在适定性分析上，间接法通过位势理论论证解的存在唯一性，而直接法则通过构造位势并利用等价性定理导出唯一性，再用 Calderón 正则化转为 Fredholm 第二类得到存在性，两者路径不同但互相补充。低频行为方面，间接法的电场公式也存在类似崩溃，[3]中采用修改组合位势的方式予以克服；本文的直接法则通过添加约束项来修正，两者均是对电荷守恒条件的恢复，但实现方式各异。

2. 与直接场-唯一方法的先前尝试比较（主要与[30, 9]）。 文献[30]利用相同的非标准边界条件构造了直接 BIE，但其公式存在内共振（即某些内部特征频率导致唯一性丢失），且不是第二类 Fredholm，也没有分析低频行为。本文通过引入耦合参数 $\eta\neq0$ 并采用组合迹方程，完全消除了伪共振问题，证明了在所有波数下的唯一性（定理 4.2 和 5.1）。[9]针对球面情形部分改善了[30]的伪共振，但仅局限于球对称，且未处理磁公式或提供一般适定性理论。本文首次对任意光滑曲面提供了完备的理论，包括一般存在唯一性证明、正则化、低频修正。此外，本文的磁公式在[30, 9]中均未出现。

3. 与传统 Maxwell BIE 的比较。 传统的 Maxwell BIE（如 EFIE、MFIE、CFIE 及其电流-电荷重构形式[1,7,8,31]）通常涉及矢量面元 curl 或 div 算子，导致超强奇异核和低频谱困难。尽管已有高精度 Nyström 方法[2,11,12,17]和 Calderón 预处理技术[31]，但这些方法仍需处理与 Maxwell 系统直接相关的复杂算子。本文将问题完全转化为 Helmholtz 双、单层位势及其迹算子（$\bm{S}$、$\bm{K}$、$\bm{K}'$、$\bm{T}$），这些算子在声学散射中已经得到深入研究，具有完备的映射性质和数值实现经验。因此，本文的公式可以自然地继承 Helmholtz 声学 BIE 的成熟数值技术，如高精度 Nyström 方法、快速算法等，降低了实现难度和计算成本。尤其值得注意的是，文中磁公式的解直接是表面电流 $\bm{J}$ 的负值，无需像传统 CFIE 那样求解电流再求积，简化了后处理。

4. 技术框架的对比。 本文的分析建立在对曲面几何量（形算子 $\bm{R}$ 和平均曲率 $H$）的精细控制之上。这些几何量在边界条件中自然出现（参见定理 3.2），并引入到积分算子中（如 $2\bm{S}\bm{H}$ 项）。先前的研究大多未显式处理这些几何乘子对积分方程算子的影响。文中通过建立乘法算子有界性及与层势的复合紧闭性，保证了整个方程在正确函数空间中的映射性质。此外，所引入的紧算子 $\bm{C}_\nu$ 和 $\bm{C}_t$ 的构造及其紧性证明（引理 4.3）利用了表面法向的变化，这也在以前的工作中少见。

3. 结构保持的Maxwell方程最优控制及其在源隐身中的应用

Structure-Preserving Optimal Control of Maxwell’s Equations with Applications to Source Cloaking

作者: Harbir Antil, Yaw Owusu-Agyemang, Rohit Khandelwal, Jimmie Adriazola, Denis Ridzal

类别: math.OC, math.AP, math.NA, physics.comp-ph

发布日期: 2026-04-30

arXiv ID: 2605.00212v1

摘要翻译: 我们为含时Maxwell方程的最优控制开发了一个结构保持的求解框架。基于正问题弱形式的适定性理论, 我们首先分析了一个将Nédélec和Raviart–Thomas有限元与Crank–Nicolson时间步进耦合的正问题求解器。该求解器保持了de Rham结构, 强制了离散Gauss定律, 精确满足每步能量平衡, 并在由最优控制设置决定的问题数据的低正则性假设下收敛到弱解。为了控制Maxwell系统, 我们将时空电流密度的旋度作为源添加到Ampére定律中。旋度形式在无需辅助约束的情况下保证了电荷守恒。我们证明了控制-状态映射的适定性和连续性, 推导了对偶系统以及跟踪型目标泛函的梯度表示, 并构建了一个继承了正求解器结构保持性的离散优化格式。我们的离散平稳性条件与其连续对应形式相一致, 并且随着网格和时间精细化, 离散最优控制收敛到连续最优解。我们通过数值求解一系列源隐身模型问题, 展示了我们的最优控制形式和理论发展的优点。

Comment: 无

链接: http://arxiv.org/abs/2605.00212v1

1. 研究对象和背景

本文研究时变麦克斯韦方程组的最优控制问题。设 $\Omega\subset\mathbb{R}^3$ 为有界 Lipschitz 区域，时间区间 $(0,T)$，电场 $\mathbf{E}$ 与磁感应 $\mathbf{B}$ 满足

$$ \begin{cases} \varepsilon\partial_t\mathbf{E} - \nabla\times(\mu^{-1}\mathbf{B}) + \sigma\mathbf{E} = \chi_{\Omega_{\mathrm{src}}}\mathbf{I}_{\mathrm{src}} + \chi_{\Omega_{\mathrm{ctrl}}}\nabla\times\mathbf{z} & \text{在 }\Omega\times(0,T), \\ \partial_t\mathbf{B} + \nabla\times\mathbf{E} = 0 & \text{在 }\Omega\times(0,T), \\ \mathbf{E}\times\nu = 0 & \text{在 }\partial\Omega\times(0,T), \\ \mathbf{E}(0)=\mathbf{E}_0,\; \mathbf{B}(0)=\mathbf{B}_0 & \text{在 }\Omega, \end{cases} $$

并自动蕴含电高斯定律 $\partial_t\operatorname{div}(\varepsilon\mathbf{E}) + \operatorname{div}(\sigma\mathbf{E} - \chi_{\Omega_{\mathrm{src}}}\mathbf{I}_{\mathrm{src}})=0$ 和磁高斯定律 $\operatorname{div}\mathbf{B}=0$。其中 $\varepsilon,\mu\in L^{\infty}(\Omega)^{3\times3}$ 对称且一致正定，$\sigma\in L^{\infty}(\Omega)^{3\times3}$ 对称半正定；源电流 $\mathbf{I}_{\mathrm{src}}\in L^2(0,T;L^2(\Omega_{\mathrm{src}})^3)$，控制变量 $\mathbf{z}\in Z:=L^2(0,T;H(\operatorname{curl};\Omega_{\mathrm{ctrl}}))$ 以旋度形式 $\nabla\times\mathbf{z}$ 加入安培定律右侧，从而自动保证 $\nabla\cdot(\nabla\times\mathbf{z})=0$，无需额外电荷守恒约束。初值 $(\mathbf{E}_0,\mathbf{B}_0)\in L^2(\Omega)^3\times L^2(\Omega)^3$ 且 $\operatorname{div}\mathbf{B}_0=0$。边界为理想电导体条件 $\mathbf{E}\times\nu=0$。

最优控制的目标是在全时域 $[0,T]$ 上引导电磁场趋向于给定的目标场 $(\mathbf{E}_d,\mathbf{B}_d)$，观测区域为 $\Omega_{\mathrm{obs}}\subseteq\Omega$，同时极小化控制能量。目标泛函定义为

$$ J(\mathbf{E},\mathbf{B},\mathbf{z}) = \frac12 \int_0^T \left( \|\varepsilon^{1/2}(\mathbf{E}-\mathbf{E}_d)\|^2_{L^2(\Omega_{\mathrm{obs}})^3} + \|\mu^{-1/2}(\mathbf{B}-\mathbf{B}_d)\|^2_{L^2(\Omega_{\mathrm{obs}})^3} \right)\,dt + \frac12\|\mathbf{z}\|^2_Z, $$

其中控制范数 $\|\mathbf{z}\|_Z^2 = \int_0^T \left(\alpha_1\|\mathbf{z}\|^2_{L^2(\Omega_{\mathrm{ctrl}})^3} + \alpha_2\|\nabla\times\mathbf{z}\|^2_{L^2(\Omega_{\mathrm{ctrl}})^3}\right)dt$，$\alpha_1,\alpha_2>0$。

此前，在仅有低正则假设（例如 Lipschitz 区域与 $L^2$ 数据）下，文献[1]已建立上述前向问题的弱解适定性，并在半离散层面给出了保持 de Rham 结构、离散高斯定律和能量恒等式的 Nédélec/RT 空间有限元格式。然而，全离散格式的收敛性、特别是结合 Crank–Nicolson 时间步进的分析尚未完成；进一步，在该低正则框架下建立结构保持的最优控制理论——包括控制到状态映射的适定性、伴随方程推导、离散梯度一致性以及离散最优解的收敛性——则是本工作试图填补的核心空白。该问题在电磁隐身、隐蔽等源屏蔽应用中具有直接意义，但需要数值方案在粗网格和长时间尺度下仍能保持物理结构，而现有文献鲜有同时满足结构保持、能量守恒和收敛性保证的最优控制框架。

2. 主要定理或主要结果

本文的主要结果包括以下六个方面：

全离散前向格式的构建与收敛性。建立了 Crank–Nicolson 时间步进与 Nédélec/RT 空间离散相结合的完全离散格式 (6)。证明了每一时间步能量恒等式：
$$\frac{E^{n+1} - E^n}{\Delta t} + \|\sqrt{\sigma}\mathbf{E}_h^{n+1/2}\|^2_{L^2(\Omega)} = (\mathbf{f}^{n+1/2}, \mathbf{E}_h^{n+1/2}),$$
其中 $E^n = \frac12(\|\sqrt{\varepsilon}\mathbf{E}_h^n\|^2_{L^2(\Omega)} + \|\sqrt{\mu^{-1}}\mathbf{B}_h^n\|^2_{L^2(\Omega)})$。在源项 $\mathbf{f}\equiv0$ 时，该格式对 $\sigma\ge0$ 是能量耗散的（当 $\sigma\equiv0$ 时严格保守），并满足稳定性估计 (10)：
$$E^m + \Delta t \sum_{n=0}^{m-1}\|\sqrt{\sigma}\mathbf{E}_h^{n+1/2}\|^2_{L^2(\Omega)} \le C_T\left(E^0 + \sum_{n=0}^{m-1}\Delta t\|\varepsilon^{-1/2}\mathbf{f}^{n+1/2}\|^2_{L^2(\Omega)}\right),$$
其中 $C_T$ 仅依赖于 $T$、$\varepsilon$ 和 $\mu$ 的上下界，与 $h,\Delta t$ 无关。此外，离散磁高斯定律自动保持：若 $\operatorname{div}\mathbf{B}_h^0 = 0$，则对所有 $n$ 有 $\operatorname{div}\mathbf{B}_h^n = 0$。在低正则假设（Lipschitz 区域，数据仅为 $L^2$）下，证明了全离散解收敛到唯一弱解。
控制到状态映射的适定性。对任意 $\mathbf{z}\in Z$，将 $\chi_{\Omega_{\mathrm{ctrl}}}\nabla\times\mathbf{z}$ 视为 $\mathbf{f}$ 的一部分，由定理 1 知麦克斯韦系统存在唯一的弱解 $(\mathbf{E},\mathbf{B})$。定义控制到状态映射 $\mathcal{S}:\mathbf{z}\mapsto(\mathbf{E},\mathbf{B})$，它是从 $Z$ 到状态空间的仿射且有界线性映射，从而保证了最优控制的存在性（在凸二次目标函数和闭凸容许集下）。
伴随系统与梯度表示。在对偶能量空间中导出了伴随麦克斯韦系统，并证明了约化目标泛函 $j(\mathbf{z}) = J(\mathcal{S}\mathbf{z},\mathbf{z})$ 是 Gâteaux 可微的，其梯度表示为
$$\nabla j(\mathbf{z}) = \alpha_1\mathbf{z} - \alpha_2\Delta\mathbf{z} + \chi_{\Omega_{\mathrm{ctrl}}}\int_0^T (\nabla\times\mathbf{p})\,dt,$$
其中 $(\mathbf{p},\mathbf{q})$ 是伴随状态变量，满足终值条件 $\mathbf{p}(T)=\mathbf{q}(T)=0$ 和相应的伴随方程。一阶最优性条件为 $\nabla j(\mathbf{z}) = 0$，该条件在凸性假设下也是充分的。
离散梯度的一致性。将 Crank–Nicolson 和 Nédélec/RT 离散格式同时应用于状态和伴随方程，利用交换投影 (commuting projections) 和 de Rham 复形，导出了约化目标泛函的离散梯度，它与连续梯度是相容的，即离散最优性系统恰好是 Galerkin 最优性系统。这使得离散优化问题天然保持结构。
离散最优控制的收敛性。设 $(\mathbf{E}_{h,\Delta t},\mathbf{B}_{h,\Delta t},\mathbf{z}_{h,\Delta t})$ 为离散最优三元组，则当 $h,\Delta t\to0$ 时，存在子列弱收敛于连续最优三元组 $(\mathbf{E},\mathbf{B},\mathbf{z})$。若关于 $\mathbf{z}$ 的目标函数是严格凸的（即 $\alpha_1,\alpha_2>0$ 且观测包含能量范数），则全序列强收敛。
源隐身数值实验。使用上述结构保持优化框架，在一维、二维和三维设置下求解了源隐身问题（$\mathbf{E}_d=0,\mathbf{B}_d=0$），验证了理论的有效性。特别地，图 1 展示了一维情形下非分离控制 $\mathbf{z}(x,t)$ 优于分离控制 $u(x)a(t)$ 的结果。

3. 研究方法、关键技术和核心工具

本文的核心技术障碍在于：在 Lipschitz 区域和仅有 $L^2$ 正则性的数据下，同时保持离散格式的 de Rham 结构、能量恒等式和高斯定律，并建立全离散收敛性；进一步将这一低正则结构保持格式无缝扩展到最优控制框架，并取得离散最优控制的一致性收敛。

作者的主要创新与证明框架可分解为以下几个关键技术点：

一、Crank–Nicolson/Nédélec/RT 全离散格式的构造与能量恒等式

格式基于半离散框架[1] 的 Nédélec 空间 $\mathcal{N}_h^0\subset H_0(\operatorname{curl};\Omega)$ 离散电场 $\mathbf{E}_h$，Raviart–Thomas 空间 $\mathcal{RT}_h\subset H(\operatorname{div};\Omega)$ 离散磁感应 $\mathbf{B}_h$，并利用 de Rham 复形性质 $\nabla\times\mathcal{N}_h^0\subset\mathcal{RT}_h$ 以及 $\operatorname{div}(\nabla\times\mathcal{N}_h^0)=0$，从而自动满足 $\operatorname{div}\mathbf{B}_h^n=0$ 若初值散度为零。时间离散采用 Crank–Nicolson 中点平均，得到递推格式 (6)。

证明适定性的核心在于建立能量恒等式 (9)。为此，对 (6a) 取检验函数 $\psi_h = \mathbf{E}_h^{n+1/2}$，对 (6b) 取 $\phi_h = P_h(\mu^{-1}\mathbf{B}_h^{n+1/2})$，其中 $P_h$ 是到 $\mathcal{RT}_h$ 的 $L^2$ 正交投影（见附录 A.1）。关键技巧在于利用投影性质 $(\nabla\times\mathbf{E}_h^{n+1/2}, P_h(\mu^{-1}\mathbf{B}_h^{n+1/2})) = (\nabla\times\mathbf{E}_h^{n+1/2}, \mu^{-1}\mathbf{B}_h^{n+1/2})$，从而消去旋度耦合项。结合代数恒等式

$$\left( M\frac{\mathbf{u}-\mathbf{v}}{\Delta t}, \frac{\mathbf{u}+\mathbf{v}}{2}\right) = \frac{1}{2\Delta t}(\|\sqrt{M}\mathbf{u}\|^2 - \|\sqrt{M}\mathbf{v}\|^2),$$

得到严格的能量平衡 (9)。这一投影技巧同时避免了显式使用 $H(\operatorname{div})$ 范数和时间导数的对偶范数，使得稳定性和收敛性分析可在纯 $L^2$ 框架下进行。

二、低正则假设下的收敛性分析

收敛性证明的核心是将全离散解序列通过分片常数和分片仿射重构嵌入到 Bochner 空间中，然后利用紧性论证。具体地，对离散解 $(\mathbf{E}_h^n,\mathbf{B}_h^n)$，构造分片常数轨道 $\mathbf{E}_{h,\mathrm{pc}}(t)$、中点提升 $\mathbf{E}_{h,1/2}(t)$ 和分片仿射轨道 $\mathbf{E}_{h,\mathrm{pl}}(t)$，以及类似对 $\mathbf{B}_h$ 的构造。由能量稳定性 (10) 可推出这些重构序列在如下空间中的一致有界性：

$$\mathbf{E}_{h,\mathrm{pl}} \in H^1(0,T;\mathcal{N}_h^0) \hookrightarrow C([0,T];L^2(\Omega)),\quad \mathbf{B}_{h,\mathrm{pl}} \in H^1(0,T;\mathcal{RT}_h) \hookrightarrow C([0,T];L^2(\Omega)),$$

以及 $\mathbf{E}_{h,\mathrm{pc}}$ 在 $L^\infty(0,T;L^2(\Omega))$ 和 $\mathbf{E}_{h,1/2}$ 在 $L^2(0,T;L^2(\Omega))$ 中的有界性。利用弱紧性可抽取子列弱收敛于某个极限 $(\mathbf{E},\mathbf{B})$，再由格式的相容性和弱解的定义验证该极限满足 (4)。唯一性由定理 1 保证，从而整个序列收敛。这里低正则性（尤其是仅 $L^2$ 数据）带来的挑战是如何处理 curl 项在弱形式中的对偶配对，通过将对偶配对重新解释为 $L^2$ 内积适当迁移到检验空间，结合离散序列的强收敛性质，成功避免了 $H(\operatorname{curl})$ 范数的紧性缺失。

三、控制问题的伴随推导与梯度表示

从控制到状态映射 $\mathcal{S}$ 的线性性出发，利用抽象优化理论可得最优控制的存在性。为获得一阶最优条件，需要计算约化目标 $j(\mathbf{z})$ 的梯度。作者采用 Lagrangian 方法：定义 Lagrangian

$$\mathcal{L} = J(\mathbf{E},\mathbf{B},\mathbf{z}) + \int_0^T \big[\langle\varepsilon\partial_t\mathbf{E} - \nabla\times(\mu^{-1}\mathbf{B}) + \sigma\mathbf{E} - \mathbf{f}, \mathbf{p}\rangle + \langle\partial_t\mathbf{B} + \nabla\times\mathbf{E}, \mathbf{q}\rangle\big]\,dt,$$

其中 $(\mathbf{p},\mathbf{q})$ 为伴随变量。通过对状态变量 $(\mathbf{E},\mathbf{B})$ 取 Gâteaux 导数并分部积分，并令状态方向导数前的系数为零，可导出伴随方程：给定终值条件 $\mathbf{p}(T)=0,\mathbf{q}(T)=0$，有

$$\begin{aligned} -\varepsilon\partial_t\mathbf{p} + \nabla\times(\mu^{-1}\mathbf{q}) + \sigma\mathbf{p} &= \chi_{\Omega_{\mathrm{obs}}}\varepsilon(\mathbf{E}-\mathbf{E}_d),\\ -\partial_t\mathbf{q} - \nabla\times\mathbf{p} &= \chi_{\Omega_{\mathrm{obs}}}\mu^{-1}(\mathbf{B}-\mathbf{B}_d), \end{aligned}$$

并配有相应的边界条件。然后，$j$ 的梯度表示为

$$\nabla j(\mathbf{z}) = \alpha_1\mathbf{z} - \alpha_2\Delta\mathbf{z} + \chi_{\Omega_{\mathrm{ctrl}}}\int_0^T (\nabla\times\mathbf{p})\,dt,$$

这里利用了 $\nabla\times\mathbf{z}$ 的伴随算子是 $\nabla\times$ 本身。该梯度公式结构清晰，可直接用于优化算法。

四、离散“优化—然后—离散”的一致性

为保证数值优化方案继承连续问题的结构，作者对最优性系统采用与正向问题相同的 CN-Nédélec/RT 离散：即先对连续最优性条件（状态、伴随、梯度方程）进行离散，得到离散最优性系统。关键的一步是证明这样得到的离散梯度与先离散目标泛函再对离散控制求导得到的结果一致，从而离散最优解恰好是离散 Galerkin 问题的解。这一致性依赖于交换投影：存在投影算子 $\Pi_h^{\mathrm{curl}}$ 和 $\Pi_h^{\mathrm{div}}$ 使得图表交换，进而离散伴随方程与连续伴随方程的离散形式相符，离散梯度无额外误差。这使得离散优化问题可直接采用标准算法求解，且结构保持性质得以继承。

五、离散最优控制的收敛性

最后，为证明离散最优解收敛到连续最优解，作者使用了如下策略：由离散最优性条件可导出关于 $\mathbf{z}_{h,\Delta t}$ 的一致估计，从而抽取弱收敛子列。利用状态和伴随方程离散解的强收敛性（来自第一部分），可将最优性条件中的项取极限，证明极限点 $(\mathbf{E},\mathbf{B},\mathbf{z})$ 满足连续最优性条件，因而是一个连续最优解。若控制代价是严格凸的，则最优控制唯一，于是整个序列收敛。

4. 与之前工作的比较

本文的贡献是在结构保持有限元、低正则分析、最优控制理论的交叉点上做出的，与现有工作的主要差异和推进体现在以下几个方面：

一、与已有结构保持有限元及 Maxwell 方程求解器的比较

文献[16, 12, 4, 19] 等奠定了 Nédélec/RT 空间和 de Rham 复形在电磁计算中的基础，但大多针对光滑材料、强解或至少 $H(\operatorname{curl})$ 的正则性。本文与其对话的文献[1] 首次在 Lipschitz 区域且仅 $L^2$ 数据（即 $\mathbf{f}\in L^2(0,T;L^2(\Omega)^3)$，初值 $\mathbf{E}_0,\mathbf{B}_0\in L^2(\Omega)^3$）的弱解框架下建立了半离散格式的收敛性。本文进一步发展了全离散情形，证明了 Crank–Nicolson 时间步进与 Nédélec/RT 空间的结合不仅保持能量恒等式，而且在无任何额外光滑性假设下收敛到弱解。相比[1]的半离散分析，全离散需要处理时间离散引起的误差，以及时间重构函数的紧性，难度更大。此外，本文的稳定性估计 (10) 对 $\sigma\ge0$ 一致成立，这对控制问题的存在性证明至关重要。

二、与 Maxwell 方程最优控制的文献对比

据作者所知，文献[3] 是唯一处理全时变麦克斯韦系统最优控制的工作。本文与[3] 存在以下根本区别：

目标泛函的时间范围：[3] 考虑终端时刻目标，即 $J$ 中仅含 $t=T$ 的跟踪项；而本文的目标定义在整个时间区间 $(0,T)$，更适合如隐身等需要全程调控的应用。
控制变量的时空结构：[3] 假设控制可分离 $\mathbf{z}(x,t) = u(x)a(t)$，这种形式对于许多实际源项 $\mathbf{I}_{\mathrm{src}}(x,t)$ 无法有效隐身（本文图 1 即展示了分离控制不能掩蔽非分离源的情形）；本文允许完全非分离的 $\mathbf{z}(x,t)$，大大拓宽了可控范围。
电荷守恒的实施方式：[3] 通过显式添加 $\operatorname{div}\mathbf{z}=0$ 的约束并引入额外时间正则性来保证电荷守恒，这增加了问题的复杂性和正则性要求；本文利用控制量的旋度形式 $\nabla\times\mathbf{z}$ 自动满足 $\operatorname{div}(\nabla\times\mathbf{z})=0$，无需额外约束，且对 $\mathbf{z}$ 只需 $L^2(0,T;H(\operatorname{curl};\Omega_{\mathrm{ctrl}}))$ 的正则性，比添加 $\operatorname{div}\mathbf{z}=0$ 约束的条件更易处理。
数值分析的深度：本文针对所采用的 CN-Nédélec/RT 全离散格式给出了完整的收敛性证明（包括状态、伴随和最优控制的收敛），而[3] 没有提供针对此类结构保持离散的收敛理论。本文还证明了离散最优性系统与 Galerkin 形式的一致性，这在[3] 中未被讨论。

三、关于梯度计算与伴随离散

在许多 PDE 最优控制的数值分析中，常采用“先离散后优化”或“先优化后离散”两种策略。本文采用的是一种混合策略：先对连续系统推导最优性条件，然后对其中所有方程使用统一的 CN-Nédélec/RT 离散，并证明得到的离散梯度与直接离散目标函数得到的梯度一致。文献中，对于 Maxwell 方程，这种基于交换投影的一致性分析尚属首次。相比于直接对离散目标函数使用自动微分或有限差分，这种方法能确保离散最优系统保持连续系统的结构特征（如能量平衡、高斯定律），这对于粗网格和大时间步的计算稳定性至关重要。

四、代价与局限

作者也指出了这种方法的代价和未来方向。首先，控制空间 $Z$ 要求 $\mathbf{z}$ 满足 $\operatorname{curl}\mathbf{z}\in L^2$，这排除了完全不具旋度正则性的控制。但在电荷守恒非必须时，可放宽到 $L^2(0,T;L^2(\Omega)^3)$ 并直接注入 $\mathbf{z}$。其次，分析中假定了 $\sigma$ 对称半正定，这涵盖了非完美导体，但若 $\sigma$ 具有更一般的对称部分以外的结构（如反对称部分表示旋光效应），可能需额外处理。最后，数值实验仅限于源隐身，更一般的控制问题（如边界控制或参数控制）有待探索。此外，文中虽证明了收敛性，但未给出误差估计，这主要是因为低正则框架下难以获得最优收敛阶。

了解领域论文（摘要翻译）

1. 不可压霍尔-磁流体力学方程的散度保持混合有限元格式的误差分析

Error analysis of a divergence-preserving mixed finite element scheme for the incompressible Hall–magnetohydrodynamic equations

作者: Beniamin Goldys, Agus L. Soenjaya, Thanh Tran

类别: math.NA

发布日期: 2026-05-03

arXiv ID: 2605.01764v1

链接: http://arxiv.org/abs/2605.01764v1

摘要翻译

中文标题: 不可压霍尔-磁流体力学方程的散度保持混合有限元格式的误差分析

摘要翻译: 不可压霍尔-磁流体力学（Hall–MHD）系统由于其刚性、高度非线性的霍尔项以及磁场必须保持无散的严格要求，给分析和计算带来了巨大挑战。本文研究了一个沃伊特正则化的霍尔-磁流体力学系统，该系统具有独立的分析意义，并提供了原模型的一个物理上相容的适定正则化。针对该正则化问题，我们提出、分析并实现了一种保结构、线性、全离散的有限元方法。借助有限元外微分和混合变分形式，空间离散精确地施加了磁场的无散条件，而一个斜对称、线性隐式的时间离散则保证了无条件能量稳定性。我们建立了沃伊特正则化问题的最优收敛速率，此外，还推导了未正则化的霍尔-磁流体力学系统的误差估计，其中沃伊特正则化在无电阻情形中起到了关键作用。最后，2.5D和3D情况下的数值模拟证实了理论结果，并展示了该格式的物理保真性。

2. 连续力源浸入边界方法中超越一阶精度及其良态投影求解

Beyond first-order accuracy in continuous-forcing immersed boundary methods, and their well-conditioned projection-based solution

作者: Diederik Beckers, H. Jane Bae, Andres Goza

类别: math.NA, physics.flu-dyn

发布日期: 2026-04-30

arXiv ID: 2604.28152v1

链接: http://arxiv.org/abs/2604.28152v1

摘要翻译

中文标题: 连续力源浸入边界方法中超越一阶精度及其良态投影求解

摘要翻译: 我们提出一种改进的浸入边界（IB）方法，该方法在实际计算中能够达到优于一阶的精度，同时保持“连续力源”IB方法的关键性质，即在控制方程中保留奇异源项。受多相流和浸入层公式的启发，该方法利用光滑化的指示（Heaviside）函数，将IB解重新表述为内部场和外部场的复合。我们证明，当通过这种复合解的视角审视时，先前的连续力源IB方法可被视为忽略了控制方程和约束方程中的某些项，这些项将解限制在一阶精度。我们纳入这些项以系统地提高精度，而无需求助于启发式修正。在典型泊松问题中，我们通过实验展示了二阶收敛性；在不可压缩Navier-Stokes模拟中，该方法达到略低于二阶的性能。虽然目前的研究聚焦于这些情形，但该框架指向了进一步扩展以实现二阶或更高精度的路径。这一视角重新框定了通常归因于IB格式的精度限制。尽管连续力源IB方法常被报道仅为一阶精度，我们证明光滑化或界面插值本身并不固有地限制可达的精度阶数。此外，我们自然地将该高阶格式融入基于投影的求解过程。所得算法同时减轻了病态线性系统产生的虚假表面应力，并降低了对几何分辨率的敏感性，从而在一个统一的方法中同时处理了条件数和精度问题。

3. 有界、交换、保持离散迹的投影

Bounded, Commuting, Discrete-trace Preserving Projections

作者: Alexandre Ern, Johnny Guzmán, Pratyush Potu

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.28103v1

链接: http://arxiv.org/abs/2604.28103v1

摘要翻译

中文标题: 有界、交换、保持离散迹的投影

摘要翻译: 我们构造了三维de Rham复形的有界、交换的投影，并具有额外性质：如果迹是适当迹空间中的分段多项式，则投影保持函数/场的迹。这些投影是局部定义的，并且在图范数下稳定。更准确地说，图范数中涉及外导数的部分仅涉及该导数在接触边界且由局部网格尺寸加权的单元窄带内的振荡。此外，当作用于外导数是适当空间中的分段多项式的函数/场时，投影是局部$L^2$稳定的。我们给出了当前有界、交换、保持离散迹的投影的两个显著应用：分段多项式数据的稳定提升的构造，以及关于分段多项式数据的离散与连续延拓的最优性结果。

不相关论文（基本信息）

1. 基于Newton$\unicode{x2013}$Schulz的Stiefel流形上的二阶方法

A second-order method on the Stiefel manifold via Newton$\unicode{x2013}$Schulz

作者: Xinhui Xiong, Bin Gao, P. -A. Absil

类别: math.OC, cs.AI, cs.LG, math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02838v1

链接: http://arxiv.org/abs/2605.02838v1

摘要: Retraction-free approaches offer attractive low-cost alternatives to Riemannian methods on the Stiefel manifold, but they are often first-order, which may limit the efficiency under high-accuracy requirements. To this end, we propose a second-order method landing on the Stiefel manifold without invoking retractions, which is proved to enjoy local quadratic (or superlinear for its inexact variant) convergence. The update consists of the sum of (i) a component tangent to the level set of the constraint-defining function that aims to reduce the objective and (ii) a component normal to the same level set that reduces the infeasibility. Specifically, we construct the normal component via Newton$\unicode{x2013}$Schulz, a fixed-point iteration for orthogonalization. Moreover, we establish a geometric connection between the Newton$\unicode{x2013}$Schulz iteration and Stiefel manifolds, in which Newton$\unicode{x2013}$Schulz moves along the normal space. For the tangent component, we formulate a modified Newton equation that incorporates Newton$\unicode{x2013}$Schulz. Numerical experiments on the orthogonal Procrustes problem, principal component analysis, and real-data independent component analysis illustrate that the proposed method performs better than the existing methods.

2. 广义阿贝尔微分方程的存在性、渐近行为与数值分析及其在金融建模中的应用

Existence, Asymptotic Behavior, and Numerical Analysis of a Generalized Abel Differential Equation with Applications in Financial Modeling

作者: Dragos-Patru Covei

类别: math.NA, math.AP

发布日期: 2026-05-04

arXiv ID: 2605.02831v1

链接: http://arxiv.org/abs/2605.02831v1

摘要: We present a comprehensive investigation into a generalized class of Abel ordinary differential equations (ODEs), extending the classical cubic form to arbitrary polynomial nonlinearities of degree $n \geq 1$. This work provides a rigorous treatment of the existence and uniqueness of regular solutions on both bounded and unbounded domains. Utilizing a unified barrier-based approach, we derive sharp growth rates and prove the existence of exact asymptotic plateaus, establishing the first systematic treatment of such generalizations in the literature. The regularity of solutions is rigorously justified through the lens of Sobolev spaces and classical ODE theory. To complement our analytical findings, we implement a high-order numerical framework based on Radau IIA implicit Runge–Kutta schemes, providing detailed stability arguments and error analysis. The numerical results demonstrate exceptional consistency with our theoretical predictions, particularly in capturing the asymptotic behavior. Finally, we discuss the implications of our results for real-world models, including Merton-type credit risk analysis and Hamilton–Jacobi–Bellman stochastic control problems, bridging the gap between abstract nonlinear dynamics and applied science.

3. 逆双调和散射中不可穿透障碍物重构的因子分解与单调性方法

Factorization and monotonicity methods for reconstructing impenetrable obstacles in inverse biharmonic scattering

作者: Tielei Zhu, Zhihao Ge, Bangmin Wu

类别: math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02802v1

链接: http://arxiv.org/abs/2605.02802v1

摘要: The inverse scattering problem for biharmonic waves, governing flexural vibrations of elastic plates, presents fundamental analytical challenges distinct from acoustic inverse problems due to the fourth-order differential operator and higher-order boundary conditions. This paper addresses the reconstruction of impenetrable obstacles with Dirichlet or Neumann boundary conditions from far-field measurements. We establish new factorizations of the far-field operator by considering structures of the biharmonic fundamental solution and the boundary conditions. We rigorously prove that the factorizations satisfy the range identities and derive characterizations of the obstacle’s support by the factorization methods, valid for all wavenumbers except the associated transmission eigenvalues. Furthermore, we establish a monotonicity relation for the eigenvalues of the far-field operator, which yields an alternative characterization of the obstacle’s support that remains applicable for all wavenumbers. Numerical experiments for the Dirichlet obstacles with various shapes are presented to demonstrate the effectiveness and robustness of the proposed reconstruction schemes.

4. 多参数奇异摄动动力系统的双尺度神经网络

Two-scale Neural Networks for Singularly Perturbed Dynamical Systems with Multiple Parameters

作者: Qiao Zhuang, Taorui Wang, Rita Wanjiku, Majid Bani-Yaghoub, Zhongqiang Zhang

类别: math.NA, physics.comp-ph

发布日期: 2026-05-04

arXiv ID: 2605.02799v1

链接: http://arxiv.org/abs/2605.02799v1

摘要: We extend our two-scale neural-network method for scalar singularly perturbed problems with one small parameter to dynamical systems with multiple small parameters. To accommodate multiple small parameters, we use a single effective scale parameter defined as the geometric mean of all parameters. We thus augment the network input with a scale-aware feature, enabling it to capture sharp solution transitions intrinsically. Numerical experiments across a range of dynamical systems demonstrate that the proposed framework can handle coupled systems with multiple and high-contrast small parameters and obtain satisfactory accuracy in capturing solution features induced by small parameters.

5. 二维空间中蒸发驱动泪膜模型的参数估计

Parameter estimation for evaporation-driven tear film model in two space dimensions

作者: Qinying Chen, Tobin Driscoll

类别: math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02674v1

链接: http://arxiv.org/abs/2605.02674v1

摘要: The tear film (TF) plays a critical role in maintaining ocular surface health, and its disruption through tear breakup (TBU) is closely associated with dry eye disease. Evaporation-driven thinning is a primary mechanism underlying TBU, yet quantitative in vivo estimates of key physical parameters remain limited. In this work, we fit an evaporation-driven TF thinning model, originally developed by Braun et al. and extended to two dimensions using proper orthogonal decomposition (POD) by Chen et al., to experimental fluorescence (FL) imaging data from normal subjects. The use of dimension reduction enables efficient solution of the governing PDEs and facilitates parameter estimation from imaging data. Our results provide in vivo estimates of evaporation-related and thinning parameters within TBU regions. These findings enhance understanding of TF thinning and dry-spot formation and establish a quantitative baseline for comparison with dry eye patient data.

6. 通过区域分解和质量矩阵集中高效生成度量图上的高斯随机场

Efficient generation of Gaussian random fields on metric graphs via domain decomposition and mass matrix lumping

作者: Mihály Kovács, Gyula Molnár, Máté András Száraz

类别: math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02670v1

链接: http://arxiv.org/abs/2605.02670v1

摘要: We consider Gaussian Random Fields on metric graphs defined implicitly as the stationary solution to a fractional SPDE driven by Gaussian white noise. Sampling from the finite element approximation requires the Cholesky factorization of the mass matrix, causing non-linear execution time explosions and massive memory fill-in on large graphs. Hence, we combine Neumann-Neumann graph decomposition with mass matrix lumping and demonstrate empirically, that our approach preserves exact theoretical convergence rates established in [8] while achieving multi-order speedups and massive memory reductions.

7. 大规模和在线高斯过程预测问题的快速准确条件化

Fast and accurate conditioning for large-scale and online Gaussian process prediction problems

作者: Samanyu Arora, Christopher J. Geoga

类别: stat.CO, math.NA, stat.ME

发布日期: 2026-05-04

arXiv ID: 2605.02574v1

链接: http://arxiv.org/abs/2605.02574v1

摘要: Gaussian Process (GP) models provide a flexible framework for prediction and uncertainty quantification. For most covariance functions, however, exact GP prediction with $n$ points scales as $\mathcal{O}(n^3)$, making it prohibitively expensive for large datasets or large numbers of prediction points. While nearest neighbor-based prediction can work well in certain settings, non-pathological circumstances (for example measurement noise) can severely restrict its efficiency. This work presents a complementary approach where one conditions on carefully designed linear combinations of data, which is particularly effective in the setting of predicting many values in large connected regions of the data domain. For kernel functions that are smooth away from the origin, conditioning on a small number $r$ of such data contrasts can be machine-precision accurate for the full exact conditional distributions. These contrasts cost $\mathcal{O}(T r^2)$ work to compute where $T$ is the cost of solving a linear system with the data covariance matrix, and so in many cases can be computed in linear or near-linear cost by exploiting rank structure in well-behaved covariance matrices. At the cost of $\mathcal{O}(nr^2)$ additional precomputation work, this approach can also provide predictions at arbitrary points of a designated region in $\mathcal{O}(1)$ online work, making it particularly attractive for problems where prediction points are not known in advance.

8. 浅水方程间断伽辽金离散的良好平衡子单元限制

Well-Balanced Subcell Limiting for Discontinuous Galerkin Discretizations of the Shallow-Water Equations

作者: Andrés M. Rueda-Ramírez, Patrick Ersing, Andrew R. Winters, Gregor J. Gassner

类别: math.NA, physics.comp-ph

发布日期: 2026-05-04

arXiv ID: 2605.02560v1

链接: http://arxiv.org/abs/2605.02560v1

摘要: High-order discontinuous Galerkin (DG) methods equipped with subcell finite-volume (FV) limiters provide an efficient framework for the simulation of nonlinear hyperbolic balance laws featuring shocks and complex flow structures. However, for systems with non-conservative terms, the design of hybrid DG/FV schemes that simultaneously guarantee high-order accuracy, robustness, and well-balancedness remains challenging. In particular, for the shallow water equations with variable bottom topography, standard flux-differencing formulations combined with node-wise subcell limiting generally destroy the well-balanced property, even if both the underlying DG and FV methods are individually well-balanced. In this work, we propose a novel flux-differencing formulation for non-conservative systems that enables node-wise subcell limiting while preserving steady states exactly. The key idea is to construct staggered DG fluxes whose non-conservative contributions are in local-times-jump form and vanish individually at equilibrium. To achieve this, we introduce a reformulation of the shallow water equations in which the source term is proportional to the gradient of the total water height. This reformulation allows the design of staggered fluxes that preserve equilibrium locally at the node level, thereby enabling arbitrary nodal blending with low-order FV fluxes. The resulting DG/FV method is high-order accurate, robust, and exactly well-balanced under node-wise limiting. Numerical experiments, including two-dimensional dam-break configurations with wet/dry fronts and complex obstacle interactions, demonstrate the improved stability and accuracy of the proposed approach. Although this work focuses on the shallow water equations, the well-balanced hybrid DG/FV methods developed here are applicable to a broader class of nonlinear systems of balance laws.

9. 具有梯度噪声的线性和非线性输运方程的结构保持LDG方法

Structure-preserving LDG methods for linear and nonlinear transport equations with gradient noise

作者: Thomas Christiansen, Kenneth H. Karlsen

类别: math.NA, math.AP

发布日期: 2026-05-04

arXiv ID: 2605.02460v1

链接: http://arxiv.org/abs/2605.02460v1

摘要: We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the Itô formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-Itô correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.

10. 非各向同性非均匀半渗透介质中椭圆半变分不等式的有限元方法

A Finite Element Method for Elliptic Hemivariational Inequalities in Non-isotropic and Heterogeneous Semipermeable Media

作者: Ban Li, Bangmin Wu

类别: math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02451v1

链接: http://arxiv.org/abs/2605.02451v1

摘要: This work investigates finite element approximations for a general class of elliptic hemivariational inequalities arising in semipermeable media. The proposed model incorporates non-isotropic and heterogeneous diffusion coefficients, alongside both interior and boundary semipermeability terms, extending the isotropic and homogeneous framework examined by Han (2019). The existence and uniqueness of solutions are rigorously established. An optimal a priori error estimate for the linear finite element approximation is derived under appropriate solution regularity assumptions. Numerical experiments are presented to corroborate the theoretical analysis and to confirm the optimal convergence rates for the non-isotropic and heterogeneous case.

11. ZNO: Z 域中离散时间动态的稳定有理神经算子

ZNO: Stable Rational Neural Operators in the Z-Domain for Discrete-Time Dynamic

作者: Xianli Zhu, Jia Yin

类别: cs.LG, math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02356v1

链接: http://arxiv.org/abs/2605.02356v1

摘要: We introduce the Z-Domain Neural Operator (ZNO), a causal neural operator whose layers are stable low-rank multiple-input multiple-output (MIMO) rational filters parameterized directly in the $z$-plane. ZNO addresses a limitation of existing operator learning methods, many of which are primarily tailored for continuous-time problems, while a large class of system-identification problems is intrinsically discrete-time. The $z$-domain form expresses stability as a unit-disk pole constraint and makes learned discrete-time poles directly readable. The model combines low-rank channel mixing, smooth stable pole reparameterization, causal recurrence, and an optional short finite impulse response (FIR) branch in a single $z$-domain rational recurrent layer. Across controlled discrete system-identification experiments, ZNO’s advantage is most evident when the target dynamics are stable rational systems with lightly damped poles near the unit circle. Under matched parameter budgets, ZNO is not uniformly dominant; however, with validation-selected configurations, the same architecture can achieve the lowest mean error across the controlled tasks. A five-bin difficulty sweep over near-unit-circle / long-memory dynamics shows that ZNO has the lowest mean error across memory regimes, from short (approximately 10 steps) to long (approximately 100-200 steps). On five public nonlinear system-identification benchmarks, ZNO is competitive with neural operator and state-space baselines, achieving the lowest mean error on benchmarks whose dynamics align with stable rational discrete-time filters, while classical or state-space baselines remain preferable on some systems. These results position ZNO as a strong model for stable rational discrete-time dynamics, especially in near-unit-circle and long-memory regimes, but not as a universal replacement for specialized system-identification methods.

12. 用于参数化多物理场代理模型的变分矩阵学习傅里叶网络

Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates

作者: Xinyu Li, Jianhua Zhang, Liang Chen

类别: cs.LG, math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02280v1

链接: http://arxiv.org/abs/2605.02280v1

摘要: Multiphysics simulation is critical for system-technology co-optimization (STCO) in chiplet-based design, but repeated finite-element solutions of PDE-governed problems are computationally expensive in parametric design exploration. This paper proposes a variational matrix-learning Fourier network (VMLFN) for efficient parametric multiphysics surrogate modeling. VMLFN constructs a log-space sine neural representation with randomly sampled spectral frequencies, frequency-dependent decay regulation, and embedded Dirichlet boundary conditions. With fixed hidden-layer parameters, the output-layer weights are determined by reformulating the governing PDEs into variational weak forms and enforcing the stationarity condition of the resulting energy functional. This converts physics-informed training into a linear matrix-solving problem, requiring only first-order derivatives and avoiding both high-order automatic differentiation and penalty-coefficient tuning. A heuristic frequency-scanning algorithm is further introduced to select a problem-adaptive maximum frequency that covers the dominant spectral range of the target problem. The proposed method is validated on heat conduction, solid mechanics, and Helmholtz wave propagation problems. Results from five benchmark cases demonstrate that VMLFN delivers accurate full-field predictions with substantial speedup over conventional physics-informed neural networks and repeated finite-element simulations.

13. 面向黎曼优化的黎曼几何基础：含详细推导的专著

Foundations of Riemannian Geometry for Riemannian Optimization: A Monograph with Detailed Derivations

作者: Benyamin Ghojogh

类别: math.DG, cs.LG, math.NA, math.OC

发布日期: 2026-05-04

arXiv ID: 2605.02279v1

链接: http://arxiv.org/abs/2605.02279v1

摘要: Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures – including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics – emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields.

14. 渐近常反馈机制下二维非局部MEMS方程的一种保结构分阶段重缩放算法

A Structure-Preserving Stagewise Rescaling Algorithm for a Two-Dimensional Nonlocal MEMS Equation in an Asymptotically Constant-Feedback Regime

作者: Takiko Sasaki, Tetsuji Tokihiro

类别: math.NA

发布日期: 2026-05-04

arXiv ID: 2605.02237v1

链接: http://arxiv.org/abs/2605.02237v1

摘要: Nonlocal MEMS equations exhibit finite-time quenching, or touchdown, which is difficult to capture numerically. We study a stagewise rescaling algorithm for a two-dimensional nonlocal MEMS equation in an asymptotically constant-feedback touchdown regime. The equation is not exactly invariant under the $A^{3/2}$–$A^3$ scaling used here; the scaling is justified when the reciprocal-integral feedback $K(t)=1+\int_Ω(1-u)^{-1}dx$ remains bounded and converges to a finite positive limit, as in the single-point touchdown profiles of Duong–Zaag. In this regime the leading-order core dynamics reduce to a local MEMS equation with an asymptotically constant coefficient. Using a fixed-stage scaling of the deficit variable, we obtain a gradient flow for a rescaled energy at frozen amplitude and prove an exact energy dissipation identity within each stage. We introduce a minimizing-movement stage solver and derive a discrete energy inequality. Since strict monotonicity need not hold across stage transitions, we separate the switch and outer-update defects and prove an exact defect balance. Under a uniform switch-defect estimate, this yields quantitative almost monotonicity and a defect-aware criterion for nonexistence of a global admissible continuation. The numerical section is organized around reproducible two-dimensional reference computations: a full-domain stagewise run showing trigger detection, fixed-stage energy decay, and geometric accumulation of physical time, and a direct fixed-domain energy check. These tests are not used as proof of the bounded-window criterion; instead, they report finite-feedback diagnostics and identify the ideal-transfer switch-energy diagnostics required for a posteriori verification.

15. 论局部线性不稳定性在熵稳定格式中的实际影响

On the Practical Impact of Local Linear Instabilities in Entropy-Stable Schemes

作者: Alex Bercik, David W. Zingg

类别: math.NA

发布日期: 2026-05-03

arXiv ID: 2605.02081v1

链接: http://arxiv.org/abs/2605.02081v1

摘要: Local linear instability refers to the linearized discrete operator exhibiting perturbation growth exceeding that of the corresponding continuous linearized problem. In the context of nonlinear entropy-stable discretizations, we argue that local linear instabilities should be interpreted as a source of numerical error whose practical impact is often negligible compared with other discretization errors. For split-form discretizations of the variable-coefficient linear advection equation, such as those resulting from linearizations of entropy-stable discretizations of the Burgers equation, perturbations can indeed exhibit unphysical modal growth. However, we demonstrate that this growth satisfies physically interpretable bounds and is typically small. Furthermore, through modified-equation analysis and numerical experiments, we show that the growth is dominated by highly oscillatory and boundary-localized unphysical modes, and can therefore be readily controlled by small amounts of numerical dissipation. More generally, this modal perturbation growth does not extend directly to nonlinear two-point-flux discretizations of the type used in entropy-stable discretizations of the Euler equations. Floquet analysis demonstrates that unstable spectra of frozen-baseflow Jacobians need not lead to unstable perturbation growth. Using the geometric flux for the variable-coefficient linear advection equation, we derive a sharp perturbation growth bound predicting negligible growth, then show analogous behaviour for the logarithmic flux numerically. Finally, we argue that robustness issues observed for entropy-stable schemes in density-wave problems are better attributed to poor near-vacuum behaviour of the logarithmic mean than to local linear instabilities. Overall, our results suggest that local linear instabilities do not pose a practical obstacle to the use of high-order entropy-stable schemes.

16. 标量守恒律的神经网络方法及其激波解的收敛速率

A Neural Network Method for Scalar Conservation Laws with Convergence Rates for Shock-Wave Solutions

作者: Jiachuan Cao, Buyang Li, Hao Li

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.27458v2

链接: http://arxiv.org/abs/2604.27458v2

摘要: We introduce an entropy-compatible neural network method for scalar hyperbolic conservation laws based on a computable surrogate of the Kružkov entropy residual. The proposed loss is designed to enforce weak consistency and entropy admissibility while remaining amenable to rigorous analysis. For piecewise smooth entropy solutions, we construct explicit neural network competitors with provably small loss by combining shock-adapted continuous piecewise linear approximations with representability results for neural networks. These approximation bounds, together with entropy-based stability estimates, yield rigorous (L^1) error estimates for minimizers of the resulting loss functional. In particular, when the network size scales comparably to the number of degrees of freedom of a space time mesh of size $h$, we recover the classical (O(h^{1/2})) convergence rate in shock-dominated regimes. The analysis also covers solutions containing rarefaction waves and regular shock interactions; in this extended version, we further treat smooth initial data that develop a shock in finite time, for which an (L^1) estimate of order (O(h^{1/2}|\ln h|)) is obtained. Numerical experiments in one and two space dimensions illustrate the effectiveness of the method and its ability to resolve nonsmooth wave structures accurately.

17. 具有记忆效应的非局部平衡律系统的稳定性估计

Stability estimates for systems of nonlocal balance laws with memory

作者: Aekta Aggarwal, N. K. Aswini, Sarvesh Kumar, Ganesh Vaidya

类别: math.AP, math.NA

发布日期: 2026-05-03

arXiv ID: 2605.01709v1

链接: http://arxiv.org/abs/2605.01709v1

摘要: In this work, we investigate entropy solutions for a class of systems of nonlocal {balance laws in which the convective flux and the source involves terms where the state variable convolved with kernels} in both spatial and temporal variables. This formulation captures the dependence of the flux on the solution within its spatial neighborhood (spatial nonlocality) as well as on its past states (temporal nonlocality), thereby incorporating memory effects. The resulting systems are coupled through these nonlocal interactions. We establish stability estimates for entropy solutions with respect to perturbations in the flux, the spatial and temporal kernels, and the initial data for the corresponding initial value problems. Finally, we present numerical experiments to illustrate the theoretical results and to highlight the influence of memory and source terms on the solution dynamics.

18. 稳态 Navier-Stokes 方程的无正则化最优控制问题的误差估计

Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

作者: Francisco Fuica, Nicolai Jork

类别: math.NA, math.OC

发布日期: 2026-05-02

arXiv ID: 2605.01633v1

链接: http://arxiv.org/abs/2605.01633v1

摘要: We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and – necessary and sufficient – second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure.

19. 非局部泊松方程的间断分析与半解析谱逼近

Discontinuity Analysis and Semi-Analytic Spectral Approximation for the Nonlocal Poisson Equation

作者: Thinh Dang, Bacim Alali, Nathan Albin

类别: math.NA, math.AP

发布日期: 2026-05-02

arXiv ID: 2605.01617v1

链接: http://arxiv.org/abs/2605.01617v1

摘要: We study a nonlocal Poisson problem with discontinuous source term and analyze how the regularity of the integral kernel determines the discontinuity structure of the corresponding solution. Under general assumptions on compactly supported integrable kernels, we show that jump discontinuities in the source term are inherited by the solution. We then identify two principal mechanisms governing higher-order regularity: singular behavior of the kernel at the origin and jump discontinuities of the kernel, or of its derivatives, at the horizon endpoints. Singularities at the origin lead to blow-up of certain derivatives of the solution at the source discontinuity, while jumps at the horizon generate cascades of derivative discontinuities at translated locations. These phenomena occur for kernels commonly used in peridynamic-type models. By contrast, compactly supported (C^\infty) kernels do not generate derivative blow-up or cascading losses of regularity, and in this case the source term and the solution have equivalent piecewise smooth regularity. Motivated by this analysis, we develop a semi-analytic spectral method for the accurate numerical treatment of discontinuous nonlocal problems. The method uses successive smoothing transformations and explicitly constructed correction functions to convert the original problem into an auxiliary problem with improved regularity. A spectral solver is then applied to the smoothed problem, and the approximation to the original solution is recovered by adding back the analytic corrections. Numerical experiments show substantial gains in accuracy and convergence, demonstrating that the method effectively mitigates the loss of accuracy caused by discontinuities and Gibbs oscillations while retaining the efficiency of spectral methods.

20. 林德布拉德方程的完全正且保迹格式结合张量火车压缩

Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

作者: Peter DelMastro, Daniel Appelö, Yingda Cheng

类别: math.NA, quant-ph

发布日期: 2026-05-02

arXiv ID: 2605.01494v1

链接: http://arxiv.org/abs/2605.01494v1

摘要: We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

21. 用于低秩张量补全的四元数非线性变换诱导核范数

Quaternion Nonlinear Transform-Induced Nuclear Norm for Low-Rank Tensor Completion

作者: Biswarup Karmakar, Ratikanta Behera

类别: math.OC, cs.CV, math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01467v1

链接: http://arxiv.org/abs/2605.01467v1

摘要: Tensor completion has emerged as a powerful framework for recovering missing data in multidimensional signals by exploiting low-rank tensor structures. Among existing approaches, linear transform-based tensor nuclear norm (TNN) methods have achieved considerable success by enforcing low-rankness on transformed frontal slices. However, the low-rank structure revealed by linear transforms remains inherently limited. To better capture intrinsic correlations, nonlinear transform-based TNN (NTTNN) models have been proposed, significantly enhancing low-rank representation through composite transforms. Despite their effectiveness, existing NTTNN methods are restricted to real-valued tensors and fail to model quaternion-valued data, which are essential for preserving inter-channel dependencies in color images and videos. Extending nonlinear TNN models to the quaternion domain is challenging due to the non-commutativity of quaternion multiplication and the complexity of quaternion singular value decomposition. To address the limitations encountered in prior works, we propose a quaternion nonlinear transform-induced tensor nuclear norm (QNTTNN) via a real embedding of quaternions, enabling tractable nuclear norm definitions and efficient optimization. Building upon QNTTNN, we formulate a quaternion tensor completion model and develop a proximal alternating minimization algorithm with rigorous convergence guarantees. Extensive experiments on benchmark color video inpainting datasets validate the superior performance of the proposed method over existing approaches.

22. 一族计算四元数矩阵广义逆的迭代方法及其应用

A Family of Iterative Methods for Computing Generalized Inverses of Quaternion Matrices and its Applications

作者: Biswarup Karmakar, Neha Bhadala, Ratikanta Behera

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01464v1

链接: http://arxiv.org/abs/2605.01464v1

摘要: The computation of generalized inverses of quaternion matrices is a fundamental problem in quaternion linear algebra, with wide-ranging applications in signal processing, image restoration, and multidimensional data analysis. This paper presents three efficient quaternion iterative algorithms for computing the Moore-Penrose pseudoinverse: (i) the quaternion rapid iterative method (QRAPID), (ii) the quaternion strong approximate inverse (QSAI), and (iii) the quaternion hyperpower iterative method of order nineteen (QHPI19). Convergence theorems and perturbation bounds are established to ensure numerical stability and robustness. The QSAI method is further employed as a preconditioner for quaternion Krylov subspace solvers, resulting in substantial reductions in the iteration count and runtime for large-scale linear systems. Comprehensive numerical experiments demonstrate that the proposed algorithms achieve an accuracy comparable to or better than existing approaches, including quaternion SVD, quaternion Newton-Schulz, and classical hyperpower schemes, while offering significant computational savings. The practical utility of the framework is illustrated through two representative applications: image completion via CUR decomposition and signal filtering, which confirm its scalability and effectiveness in real-world multidimensional data applications.

23. 求解心脏电生理反问题的神经潜变量动力学方法

A Neural Latent Dynamics Approach for Solving Inverse Problems in Cardiac Electrophysiology

作者: Edoardo Centofanti, Giovanni Ziarelli, Simone Scacchi, Luca Franco Pavarino

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01463v1

链接: http://arxiv.org/abs/2605.01463v1

摘要: Solving inverse problems in cardiac electrophysiology consists in the recovery of physiological parameters from surface electrocardiogram (ECG) measurements, a task which is often computationally unfeasible due to the severe ill-posedness and the prohibitive computational complexity of PDE-constrained optimization. In this work, we introduce a data-driven framework leveraging Latent Dynamics Networks (LDNets) to construct efficient surrogate models of the forward operator. By mapping low-dimensional parameters, representing ectopic activation sites or ischemic region descriptors, to the ECG signals via latent dynamics governed by neural ordinary differential equations, our approach circumvents the computational burden of evaluating high-fidelity cardiac models during iterative parameter estimation. The surrogate is trained offline on high-fidelity data, enabling rapid and robust inversion. We validate the proposed framework through rigorous numerical experiments with synthetic data across both 2d and 3d geometries. Results show that the LDNet-based surrogate achieves precise reconstruction of cardiac parameters while drastically reducing computational overhead, thereby enabling near real-time clinical applications.

24. 面向带 MAPE 损失和样本依赖箱约束的 $\varepsilon$-SVR 的序列最小优化

Sequential Minimal Optimization for $\varepsilon$-SVR with MAPE Loss and Sample-Dependent Box Constraints

作者: Pablo Benavides-Herrera, Riemann Ruiz-Cruz, Juan Diego Sánchez-Torres

类别: math.NA, stat.ML

发布日期: 2026-05-02

arXiv ID: 2605.01446v1

链接: http://arxiv.org/abs/2605.01446v1

摘要: We derive a Sequential Minimal Optimization (SMO) algorithm for the quadratic dual problem arising from $\varepsilon$-SVR~\cite{Vapnik1995, Drucker1997, Smola2004} modified to minimize the Mean Absolute Percentage Error (MAPE)~~\cite{Makridakis1993, Hyndman2006} directly in the loss function~~\cite{benavides2025support}. This formulation is part of a broader family of SVR models with percentage-error losses that also includes least-squares variants~\cite{Suykens2002} and symmetric-kernel extensions~\cite{Espinoza2005}, whose unified structure is studied in~\cite{benavides2026unified}. The key structural difference from standard $\varepsilon$-SVR is that the box constraints become \emph{sample-dependent}: $α_k, α_k^* \in [0,\, 100C/y_k]$. We show that this modification affects only (i) the feasibility sets $\Iup$ and $\Idown$ in the working-set selection and (ii) the clipping bounds in the analytic two-variable update, while leaving the curvature formula and gradient update structurally identical to the standard SMO~\cite{Platt1998, Platt1999, Fan2005}. A shrinking heuristic adapted to the sample-dependent bounds is derived and shown to introduce an asymmetry between $α$ and $α^*$ variables controlled by the gap $2y_k\varepsilon/100$. The same solver applies to the symmetric-kernel variant (m2) by replacing $Ω$ with $Ω_s = \tfrac{1}{2}(Ω+ aΩ^*)$~~\cite{Espinoza2005}. An implementation is available in the open-source \texttt{psvr} R package~~\cite{BenavidesHerrera2026Rpsvr}.

25. 球面上浅水方程半隐式半拉格朗日方法中的谱插值

Spectral interpolation in semi-implicit semi-Lagrangian methods for shallow water equations on the sphere

作者: Michael Chiwere, Daniel Fortunato, Grady B. Wright

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01313v1

链接: http://arxiv.org/abs/2605.01313v1

摘要: Semi-implicit semi-Lagrangian (SISL) methods are commonly used for the shallow water equations (SWE) because they allow for larger time steps than those permitted by the Courant-Friedrichs-Lewy (CFL) stability condition in Eulerian schemes. In these methods, the semi-Lagrangian treatment of advection is typically performed using lower-order interpolation, such as tensor-product Lagrange interpolation with cubic or quintic polynomials. However, operational SISL schemes routinely employ spectrally accurate spatial discretizations, such as spherical harmonics or the double Fourier sphere (DFS) method, for computing horizontal derivatives of the prognostic variables. This creates a mismatch in numerical accuracy, making the use of low-order interpolation less clearly justified. In this work, we present the first numerical investigation of spectrally accurate interpolation in SISL schemes for the SWE. Our approach builds upon the recently developed DFS-based SWE model, incorporating a spectral interpolation scheme that is accelerated using the nonuniform fast Fourier transform (NUFFT) to maintain the same overall computational complexity as the original model. Using several standard SWE test cases, we evaluate the accuracy, conservation, and numerical diffusion of the new model, particularly over long integration times. Compared to an equivalent SISL model with low-order interpolation, the new model achieves higher accuracy, improved mass and energy conservation, and reduced numerical diffusion, demonstrating the potential benefits of incorporating spectrally accurate interpolation into SISL schemes.

26. Alikhanov-XfPINNs: 非均匀网格上非线性分数阶偏微分方程的自适应物理信息学习

Alikhanov-XfPINNs: Adaptive Physics-Informed Learning for Nonlinear Fractional PDEs on Nonuniform Meshes

作者: Himanshu Kumar Dwivedi, Matthias Ehrhardt, Rajeev

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01305v1

链接: http://arxiv.org/abs/2605.01305v1

摘要: To address the initial singularity inherent in solutions to fractional partial differential equations (fPDEs), we propose an accelerated Alikhanov discretization formulation implemented on nonuniform time grids. Based on the physics-informed neural networks (PINNs) framework, we introduce an Alikhanov-extended fractional PINNs (XfPINNs) architecture that combines high-order temporal discretization and deep learning. The nonlocal memory term in fPDEs leads to high computational cost, while the weak singularity near $t\to 0^+$ can deteriorate accuracy on uniform meshes. To separate temporal discretization effects from optimization and sampling errors, we further develop an auxiliary time-marching configuration that enables auditable temporal-convergence studies under controlled training tolerances. This architecture can solve general nonlinear fPDEs. The XfPINNs approach is designed for forward and inverse problems, allowing for data-driven solution reconstruction and parameter estimation. First, the neural network approximates the solution of nonlinear fPDEs; then, an adaptive activation function accelerates convergence and enhances training efficiency. The optimization framework embeds a variational loss function constructed from the Alikhanov scheme, where the initial and boundary conditions are imposed using a combination of hard and soft constraints. Numerical experiments, including cases with known and unknown exact solutions which demonstrate the robustness, computational efficiency, and significant CPU time savings of the Alikhanov-XfPINNs method.

27. 一类非对称线性矩阵方程的低秩短递推

A class of low-rank short recurrences for nonsymmetric linear matrix equations

作者: Davide Palitta, Catherine E. Powell, Valeria Simoncini

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01276v1

链接: http://arxiv.org/abs/2605.01276v1

摘要: We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local subspace projection to speed up convergence with rank truncation strategies and randomization procedures to limit memory consumption. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.

28. 波导阵列中产生的Dirac方程的逆散射

Inverse Scattering for Dirac Equations Arising in Waveguide Arrays

作者: John C. Schotland, Shenwen Yu

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01230v1

链接: http://arxiv.org/abs/2605.01230v1

摘要: We investigate inverse scattering problems for Dirac equations that arise as continuum models of waveguide arrays. We first establish the well-posedness of the forward models. For the associated inverse problems, we develop the inverse Born series and the reduced inverse Born series, providing analysis of convergence and rigorous error estimates. Numerical experiments are presented to validate the proposed algorithms and demonstrate their effectiveness.

29. Pfaffian点过程的采样与辛Arnoldi方法

Sampling Pfaffian point processes and the symplectic Arnoldi method

作者: Alan Edelman, Sungwoo Jeong, Simeon Schaub

类别: math.NA

发布日期: 2026-05-02

arXiv ID: 2605.01202v1

链接: http://arxiv.org/abs/2605.01202v1

摘要: We present an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a wide range of statistics arising in random matrix theory and combinatorics. For instance, we can sample eigenvalues of the orthogonal and symplectic ensembles ($β= 1,4$). In addition, we introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials associated with a general weight function. This method can be used to efficiently construct the $2 \times 2$ matrix valued skew-symmetric kernels that arise in $β= 1,4$ polynomial ensembles. We illustrate our approach with several numerical examples and experiments, including the symmetric corner growth model, the finite-$N$ Gaussian (Hermite) orthogonal and symplectic ensembles, and the $β= 1,4$ Airy point processes and Tracy-Widom distributions.

30. 人类多能干细胞来源心肌细胞移植物与宿主心室组织之间电生理相互作用的建模

Modelling the electrophysiological interactions between human pluripotent cell-derived cardiomyocite grafts and host ventricular tissue

作者: Suran Galappaththige, Vadim N Biktashev, Faisal J Alibhai, Michael Laflamme

类别: q-bio.TO, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.01083v1

链接: http://arxiv.org/abs/2605.01083v1

摘要: Human pluripotent stem cell-derived cardiomyocytes (hPSC-CMs) are a promising therapy for regenerating myocardium after infarction, but their use is limited by graft-related arrhythmias that frequently occur shortly after transplantation. Experimental studies indicate that these arrhythmias can originate within the graft, which may act as an ectopic pacemaker, yet the mechanisms governing successful excitation of host tissue remain poorly understood. In particular, the role of electrical coupling at the graft-host interface is important, but difficult to measure directly or control. Computer modelling can help here. Here, we present a computational framework that enables systematic investigation of graft-host electrical interactions using a physiologically interpretable parameterisation. We model the graft-host interface as an internal boundary with a defined specific conductance, allowing direct control over coupling strength in units that correspond to measurable tissue properties. We formulate the governing equations and implement the computations using both finite-difference and finite-element discretisations in established cardiac modelling platforms. Using representative anatomical and physiological configurations, we demonstrate how variations in interface conductance influence the ability of spontaneous graft activity to initiate propagating excitation in host tissue. This framework provides a reproducible, mechanistically transparent tool for studying graft-related arrhythmogenesis and lays a foundation for evaluating strategies to mitigate arrhythmic risk in cardiac cell therapy.

31. 一种适用于非界面贴合网格的移位粘聚区方法及其在晶体塑性中的应用

A Shifted Cohesive-Zone Method for Non-Interface-Fitted Meshes with Applications to Crystal Plasticity

作者: Cheng-Hau Yang, Mark C. Messner, Tianchen Hu

类别: math.NA, math.AP

发布日期: 2026-04-30

arXiv ID: 2604.27407v2

链接: http://arxiv.org/abs/2604.27407v2

摘要: The accurate simulation of interface-dominated solid mechanics problems on complex microstructures remains challenging, particularly when interface-fitted quadrilateral or hexahedral meshes are difficult to generate. We extend the shifted boundary method (SBM) to cohesive-zone formulations and introduce the Shifted Cohesive Zone Method (SCZM), with applications to crystal plasticity on non-interface-fitted meshes. By shifting the enforcement of traction-separation laws from the true interface to a nearby surrogate interface, SCZM enables the use of standard finite element spaces while avoiding the meshing burden associated with interface-conformal discretizations. We present a simplified SCZM weak form defined on the surrogate interface, leading to a straightforward implementation of the nonlinear residual and consistent tangent matrix. The method is implemented in the open-source MOOSE framework and coupled with constitutive models from NEML2, enabling simulations with linear elasticity, multiple traction-separation laws, and history-dependent crystal plasticity. We further develop a geometry-aware, PCA-enhanced point classification algorithm to accelerate surrogate-domain construction. Verification and benchmark studies in two and three dimensions demonstrate that SCZM achieves first-order convergence for non-interface-fitted interface problems and closely matches interface-fitted reference solutions in terms of reaction forces, surface energy release, deformation, stress fields, and damage evolution. These results indicate that SCZM provides an accurate and efficient framework for modeling interface mechanics in complex microstructures without requiring interface-fitted meshes.

32. 先进空中机动运营中监视系统的可靠性、鲁棒性与弹性建模

Reliability, Robustness, and Resilience Modeling for Surveillance System in Advanced Air Mobility Operations

作者: Esrat Farhana Dulia, Caleb Adams, Syed Arbab Mohd Shihab, Ruben Del Rosario

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00823v1

链接: http://arxiv.org/abs/2605.00823v1

摘要: Ensuring the safe and efficient operation of Advanced Air Mobility (AAM) in low-altitude airspace requires a reliable, robust, and resilient surveillance system capable of continuously detecting, identifying, and tracking aircraft under both normal and off-nominal conditions. To address this need, this study develops a comprehensive 3R modeling framework, reliability, robustness, and resilience, for the Surveillance for Advanced Air Mobility (SAM) system, with a focus on the optimal design and operation of a multi-type sensor network. Under normal operating conditions, the reliability model determines the baseline sensor types, quantities, and locations required to satisfy surveillance coverage and detection requirements. To address external perturbations, such as adverse weather conditions or sudden increases in AAM traffic demand, the robustness model identifies additional sensor requirements needed to maintain system performance. Furthermore, for surveillance outages caused by primary sensor failures, the resiliency model develops backup sensor deployment and dispatch strategies to provide temporary surveillance coverage, minimize operational disruptions, and support the safe continuation of AAM operations.

33. HyCOP：用于偏微分方程可解释学习的混合组合算子

HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

作者: Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, Clint Dawson

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00820v1

链接: http://arxiv.org/abs/2605.00820v1

摘要: We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

34. 模形式与对$ζ(3)$的有理逼近的数值探索

Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$

作者: Cynthia Bortolotto, Lucas Oliveira

类别: math.NT, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00673v1

链接: http://arxiv.org/abs/2605.00673v1

摘要: We revisit Beukers’ modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers’ choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers’ irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.

35. 多面体网格上间断伽辽金方法的连续三维有限元子网格基函数

Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes

作者: Sixtine Michel, Lorenzo Diazzi, Walter Boscheri

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00619v1

链接: http://arxiv.org/abs/2605.00619v1

摘要: We present a novel high-order accurate nodal discontinuous Galerkin (DG) method for solving nonlinear hyperbolic systems of partial differential equations (PDEs) on fully unstructured three-dimensional polyhedral meshes. A mesh generator is firstly discussed in detail, which ensures the generation of admissible control volumes. For the first time, we then extend the concept of agglomerated finite element (AFE) basis functions to polyhedral grids. In this context, the discrete solution is represented within each polyhedral element using piecewise continuous polynomials of degree N, defined on an internal tetrahedral subgrid. The AFE basis functions are therefore constructed by agglomerating standard finite element basis functions on each sub-tetrahedron of the computational cell. This allows for the precomputation of universal local matrices (mass and stiffness) on the reference element given by the unit tetrahedron, enabling a quadrature-free implementation that remains efficient even on highly irregular polyhedral meshes. High-order of accuracy in time is achieved using a local spacetime Galerkin predictor as part of the ADER approach, applied independently within each polyhedral element. To ensure robustness in the presence of discontinuities such as shocks, an artificial viscosity limiter is embedded into the numerical scheme, allowing for controlled dissipation and stabilization without compromising the overall accuracy in smooth regions. To demonstrate the robustness and accuracy of the method, we validate it through different three-dimensional benchmark problems for the compressible Euler and Navier-Stokes equations.

36. 参数依赖字典下信号解混的最小二乘局部几何

Local Geometry of Least Squares for Unmixing Signals with Parameter-Dependent Dictionaries

作者: Santos Michelena, Maxime Ferreira Da Costa, José Picheral

类别: eess.SP, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00585v1

链接: http://arxiv.org/abs/2605.00585v1

摘要: Modeling signals as linear combinations of atoms from a dictionary is ubiquitous in modern signal processing. In the finite-dimensional setting, whenever atoms depend nonlinearly upon unknown parameters, the signal model is said to be separable. In this work, we study least-squares reconstruction of separable signals and establish a unified theoretical framework for their analysis. We introduce the unmixing metric, a distance that captures the distinct roles and sensitivities of linear and nonlinear parameters, and establish local convergence and stability guarantees under its topology. We then analyze variable projection from a geometric perspective, showing that it corresponds to restricting the optimization to the manifold of optimal linear parameters. This viewpoint provides a principled explanation for the improved algorithmic behavior of variable projection observed in practice, and produces sharp theoretical guarantees. The generic theory for separable problems is specialized to the case of point spread function (PSF) unmixing. We introduce a parametric notion of coherence and show that support separation directly controls both the size of the convergence region and the stability of recovery. Numerical experiments corroborate the theoretical predictions and demonstrate the practical relevance of the proposed framework.

37. Linking PageRank, Time Reversal, and Policy Evaluation

Linking PageRank, Time Reversal, and Policy Evaluation

作者: Konstantin Avrachenkov, Lorenzo Gregoris, Nelly Litvak

类别: math.OC, math.NA, math.PR

发布日期: 2026-05-01

arXiv ID: 2605.00532v1

链接: http://arxiv.org/abs/2605.00532v1

摘要: We establish a connection between policy evaluation in Markov decision processes and PageRank in network analysis. For a fixed policy, we show that the value function of a discounted Markov decision process can be obtained, up to an explicit rescaling, from the PageRank vector of a suitably defined time-reversed Markov chain. In this correspondence, the discount factor plays the role of the teleportation parameter, while rewards induce the restart distribution. Beyond the irreducible case, invoking quasi-stationary distributions and Doob $h$-transforms, we prove a general decomposition theorem showing that policy evaluation for arbitrary finite MDPs reduces to a collection of PageRank problems on the recurrent and transient components of the policy-induced Markov chain. This framework naturally extends to undiscounted MDPs with terminal states and to transition-dependent rewards. We conclude by showing efficiency of our approach on a numerical example of a sticky random walk on large deterministic and random graphs.

38. 时空分数阶扩散偏微分方程的深度皮卡迭代

Deep-Picard Iteration for Space-time Fractional Diffusion PDEs

作者: Zhijun Zeng, Zhitong Chen, Ling Qin, Yi Zhu

类别: math.NA, math.PR

发布日期: 2026-05-01

arXiv ID: 2605.00456v1

链接: http://arxiv.org/abs/2605.00456v1

摘要: We propose a Deep-Picard iteration framework for high-dimensional nonlinear space-time fractional diffusion equations.The method is based on a nonlinear fractional Feynman–Kac fixed-point formulation, which replaces direct discretization of the Caputo memory term and the nonlocal fractional Laplacian by Monte Carlo simulation of the associated fractional dynamics. Each Picard update is approximated by stochastic label generation and realized through supervised neural-network regression, thereby avoiding residual minimization involving fractional differential operators. The fractional trajectories are generated by coupling a discretized beta-stable subordinator with a walk-on-spheres-type simulation of the rotationally symmetric alpha-stable Lévy process. Numerical experiments on two-dimensional and high-dimensional test problems ddemonstrate stable Picard convergence and accurate approximation, with tests reported up to dimension d=100.

39. 利用观测位移估计岩浆库状态的变分方法

A variational approach to estimating the state of a magma reservoir from observed displacement

作者: Shungo Kun Tonoyama, Atsushi Suzuki, Takemasa Miyoshi

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00409v1

链接: http://arxiv.org/abs/2605.00409v1

摘要: We propose a numerical procedure to solve an inverse problem that estimates the state of a magma reservoir from observed surface displacement of a volcano. Our variational approach aims to find the minimizer of a cost function consisting of a norm concerning both data and derivative, which evaluates the misfit between the estimated and observed displacement. The extremal of the cost function leads to a linear system, to find the stress distribution on the reservoir surface, has very high condition number, but it is feasible to get appropriate solution by using high precision arithmetic.

40. 超越独立性：论贝叶斯反演中的联合正态先验

Beyond Independence: on Jointly Normal Priors in Bayesian Inversion

作者: Ruanui Nicholson, Matti Niskanen, Oliver J. Maclaren, Jari P. Kaipio

类别: stat.ME, math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00332v1

链接: http://arxiv.org/abs/2605.00332v1

摘要: We consider joint inversion for two or more unknown parameters from observational data in the Bayesian framework. Standard approaches often either treat the parameters as independent or impose structural similarity through regularisation terms that can be difficult to interpret statistically. We instead construct jointly Gaussian prior models with prescribed Gaussian marginals, so that correlation between the parameters can be incorporated without altering the marginal prior distributions. We propose a joint covariance construction that preserves the marginals, allows spatially varying cross-correlation, and supports uncertainty and inference in the correlation itself. The construction is valid for any strict contraction encoding the desired cross-correlation and is optimal in a canonical correlation sense under the principal square root factorisation. We demonstrate the method using prior sampling and several inference examples: a low-dimensional illustrative example and two higher-dimensional examples, including a PDE-constrained problem. The examples highlight both the potential pitfalls of ignoring or neglecting uncertainty in the correlation as well as reinforcing a key principle of the Bayesian paradigm: unknown quantities included in a model should be treated as random variables.

41. $(\star, ε)$-回文矩阵多项式的谱分解及其应用

Spectral decomposition of $(\star, ε)$-palindromic matrix polynomial and its applications

作者: Kang Zhao, Xin Wang, Xiaoxiao Ma

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00328v1

链接: http://arxiv.org/abs/2605.00328v1

摘要: This paper provides the spectral decomposition of $(\star,ε)$-palindromic quadratic matrix polynomial $P(λ)$ by a standard pair and a parameter matrix. When $J$ is assumed to be a block diagonal matrix, the parameter matrix $Γ$ has a special structure. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.

42. 用于神经网络PDE近似中残差最小化的自适应各向异性复合求积法

Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations

作者: Santiago Badia, Kishore Nori

类别: math.NA

发布日期: 2026-05-01

arXiv ID: 2605.00308v1

链接: http://arxiv.org/abs/2605.00308v1

摘要: We study the role of numerical quadrature in residual-minimisation methods for neural network approximation of partial differential equations. We first present an abstract error framework that separates approximation, quadrature and optimisation errors, and derive a nonlinear Strang-type estimate quantifying how inaccuracies in the discrete loss affect the final approximation. Motivated by this analysis, we propose an anisotropic adaptive composite quadrature strategy that controls the relative quadrature error of the residual loss using richer reference quadratures and bisection-based refinement. We then introduce a refresh-based training methodology that rebuilds the quadrature only when an online error indicator exceeds a prescribed threshold, balancing accuracy and computational cost. Numerical experiments on a range of benchmark problems show that the proposed approach narrows the gap between training and reference losses, uses quadrature points more efficiently and delivers strong approximation accuracy relative to non-adaptive quadrature strategies.

43. 狄拉克-弗伦克尔-昂萨格原理：关于偏微分方程解的非线性参数化的带有规范动量的瞬时残差最小化

A Dirac-Frenkel-Onsager principle: Instantaneous residual minimization with gauge momentum for nonlinear parametrizations of PDE solutions

作者: Matteo Raviola, Benjamin Peherstorfer

类别: cs.LG, math.NA, stat.ML

发布日期: 2026-04-30

arXiv ID: 2605.00284v1

链接: http://arxiv.org/abs/2605.00284v1

摘要: Dirac-Frenkel instantaneous residual minimization evolves nonlinear parametrizations of PDE solutions in time, but ill-conditioning can render the parameter dynamics non-unique. We interpret this non-uniqueness as a gauge freedom: nullspace directions that leave the time derivative unchanged can be used to select better-conditioned parameter velocities. Building on Onsager’s minimum-dissipation principle, we introduce a history variable – interpretable as momentum – and inject it only along the nullspace directions. The resulting Dirac-Frenkel-Onsager dynamics preserve instantaneous residual minimization, in contrast to standard regularization that can introduce bias, while promoting temporally smooth parameter evolutions. Examples demonstrate that the approach leads to increased robustness in singular and near-singular regimes.

44. 具有边界条件不完全信息的Stokes方程的一种近最优恢复算法

A near-optimal recovery algorithm for the Stokes equations with incomplete information on the boundary conditions

作者: Andrea Bonito, Diane Guignard

类别: math.NA

发布日期: 2026-04-30

arXiv ID: 2604.28051v1

链接: http://arxiv.org/abs/2604.28051v1

摘要: We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.

重点关注论文（完整分析）#

1. 带有先验界的椭圆型低维拟周期解的数值构造#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

2. Maxwell à la Helmholtz：基于 Helmholtz 算子的完美电导体三维散射直接边界积分方程#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

3. 结构保持的Maxwell方程最优控制及其在源隐身中的应用#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

了解领域论文（摘要翻译）#

1. 不可压霍尔-磁流体力学方程的散度保持混合有限元格式的误差分析#

摘要翻译#

2. 连续力源浸入边界方法中超越一阶精度及其良态投影求解#

摘要翻译#

3. 有界、交换、保持离散迹的投影#

摘要翻译#

不相关论文（基本信息）#

1. 基于Newton$\unicode{x2013}$Schulz的Stiefel流形上的二阶方法#

2. 广义阿贝尔微分方程的存在性、渐近行为与数值分析及其在金融建模中的应用#

3. 逆双调和散射中不可穿透障碍物重构的因子分解与单调性方法#

4. 多参数奇异摄动动力系统的双尺度神经网络#

5. 二维空间中蒸发驱动泪膜模型的参数估计#

6. 通过区域分解和质量矩阵集中高效生成度量图上的高斯随机场#

7. 大规模和在线高斯过程预测问题的快速准确条件化#

8. 浅水方程间断伽辽金离散的良好平衡子单元限制#

9. 具有梯度噪声的线性和非线性输运方程的结构保持LDG方法#

10. 非各向同性非均匀半渗透介质中椭圆半变分不等式的有限元方法#

11. ZNO: Z 域中离散时间动态的稳定有理神经算子#

12. 用于参数化多物理场代理模型的变分矩阵学习傅里叶网络#

13. 面向黎曼优化的黎曼几何基础：含详细推导的专著#

14. 渐近常反馈机制下二维非局部MEMS方程的一种保结构分阶段重缩放算法#

15. 论局部线性不稳定性在熵稳定格式中的实际影响#

16. 标量守恒律的神经网络方法及其激波解的收敛速率#

17. 具有记忆效应的非局部平衡律系统的稳定性估计#

18. 稳态 Navier-Stokes 方程的无正则化最优控制问题的误差估计#

19. 非局部泊松方程的间断分析与半解析谱逼近#

20. 林德布拉德方程的完全正且保迹格式结合张量火车压缩#

21. 用于低秩张量补全的四元数非线性变换诱导核范数#

22. 一族计算四元数矩阵广义逆的迭代方法及其应用#

23. 求解心脏电生理反问题的神经潜变量动力学方法#

24. 面向带 MAPE 损失和样本依赖箱约束的 $\varepsilon$-SVR 的序列最小优化#

25. 球面上浅水方程半隐式半拉格朗日方法中的谱插值#

26. Alikhanov-XfPINNs: 非均匀网格上非线性分数阶偏微分方程的自适应物理信息学习#

27. 一类非对称线性矩阵方程的低秩短递推#

28. 波导阵列中产生的Dirac方程的逆散射#

29. Pfaffian点过程的采样与辛Arnoldi方法#

30. 人类多能干细胞来源心肌细胞移植物与宿主心室组织之间电生理相互作用的建模#

31. 一种适用于非界面贴合网格的移位粘聚区方法及其在晶体塑性中的应用#

32. 先进空中机动运营中监视系统的可靠性、鲁棒性与弹性建模#

33. HyCOP：用于偏微分方程可解释学习的混合组合算子#

34. 模形式与对$ζ(3)$的有理逼近的数值探索#

35. 多面体网格上间断伽辽金方法的连续三维有限元子网格基函数#

36. 参数依赖字典下信号解混的最小二乘局部几何#

37. Linking PageRank, Time Reversal, and Policy Evaluation#

38. 时空分数阶扩散偏微分方程的深度皮卡迭代#

39. 利用观测位移估计岩浆库状态的变分方法#

40. 超越独立性：论贝叶斯反演中的联合正态先验#

41. $(\star, ε)$-回文矩阵多项式的谱分解及其应用#

42. 用于神经网络PDE近似中残差最小化的自适应各向异性复合求积法#

43. 狄拉克-弗伦克尔-昂萨格原理：关于偏微分方程解的非线性参数化的带有规范动量的瞬时残差最小化#

44. 具有边界条件不完全信息的Stokes方程的一种近最优恢复算法#

重点关注论文（完整分析）

1. 带有先验界的椭圆型低维拟周期解的数值构造

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

2. Maxwell à la Helmholtz：基于 Helmholtz 算子的完美电导体三维散射直接边界积分方程

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

3. 结构保持的Maxwell方程最优控制及其在源隐身中的应用

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

了解领域论文（摘要翻译）

1. 不可压霍尔-磁流体力学方程的散度保持混合有限元格式的误差分析

摘要翻译

2. 连续力源浸入边界方法中超越一阶精度及其良态投影求解

摘要翻译

3. 有界、交换、保持离散迹的投影

摘要翻译

不相关论文（基本信息）

1. 基于Newton$\unicode{x2013}$Schulz的Stiefel流形上的二阶方法

2. 广义阿贝尔微分方程的存在性、渐近行为与数值分析及其在金融建模中的应用

3. 逆双调和散射中不可穿透障碍物重构的因子分解与单调性方法

4. 多参数奇异摄动动力系统的双尺度神经网络

5. 二维空间中蒸发驱动泪膜模型的参数估计

6. 通过区域分解和质量矩阵集中高效生成度量图上的高斯随机场

7. 大规模和在线高斯过程预测问题的快速准确条件化

8. 浅水方程间断伽辽金离散的良好平衡子单元限制

9. 具有梯度噪声的线性和非线性输运方程的结构保持LDG方法

10. 非各向同性非均匀半渗透介质中椭圆半变分不等式的有限元方法

11. ZNO: Z 域中离散时间动态的稳定有理神经算子

12. 用于参数化多物理场代理模型的变分矩阵学习傅里叶网络

13. 面向黎曼优化的黎曼几何基础：含详细推导的专著

14. 渐近常反馈机制下二维非局部MEMS方程的一种保结构分阶段重缩放算法

15. 论局部线性不稳定性在熵稳定格式中的实际影响

16. 标量守恒律的神经网络方法及其激波解的收敛速率

17. 具有记忆效应的非局部平衡律系统的稳定性估计

18. 稳态 Navier-Stokes 方程的无正则化最优控制问题的误差估计

19. 非局部泊松方程的间断分析与半解析谱逼近

20. 林德布拉德方程的完全正且保迹格式结合张量火车压缩

21. 用于低秩张量补全的四元数非线性变换诱导核范数

22. 一族计算四元数矩阵广义逆的迭代方法及其应用

23. 求解心脏电生理反问题的神经潜变量动力学方法

24. 面向带 MAPE 损失和样本依赖箱约束的 $\varepsilon$-SVR 的序列最小优化

25. 球面上浅水方程半隐式半拉格朗日方法中的谱插值

26. Alikhanov-XfPINNs: 非均匀网格上非线性分数阶偏微分方程的自适应物理信息学习

27. 一类非对称线性矩阵方程的低秩短递推

28. 波导阵列中产生的Dirac方程的逆散射

29. Pfaffian点过程的采样与辛Arnoldi方法

30. 人类多能干细胞来源心肌细胞移植物与宿主心室组织之间电生理相互作用的建模

31. 一种适用于非界面贴合网格的移位粘聚区方法及其在晶体塑性中的应用

32. 先进空中机动运营中监视系统的可靠性、鲁棒性与弹性建模

33. HyCOP：用于偏微分方程可解释学习的混合组合算子

34. 模形式与对$ζ(3)$的有理逼近的数值探索

35. 多面体网格上间断伽辽金方法的连续三维有限元子网格基函数

36. 参数依赖字典下信号解混的最小二乘局部几何

37. Linking PageRank, Time Reversal, and Policy Evaluation

38. 时空分数阶扩散偏微分方程的深度皮卡迭代

39. 利用观测位移估计岩浆库状态的变分方法

40. 超越独立性：论贝叶斯反演中的联合正态先验

41. $(\star, ε)$-回文矩阵多项式的谱分解及其应用

42. 用于神经网络PDE近似中残差最小化的自适应各向异性复合求积法

43. 狄拉克-弗伦克尔-昂萨格原理：关于偏微分方程解的非线性参数化的带有规范动量的瞬时残差最小化

44. 具有边界条件不完全信息的Stokes方程的一种近最优恢复算法