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1. 高对比谐振器系统中的波导：理论与快速计算

Waveguiding in systems of high contrast resonators: Theory and fast computations

作者: Habib Ammari, Bowen Li, Borui Miao, Jiayu Qiu, Lara Vrabac

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26906v1

摘要翻译: 在这项工作中，我们研究高对比谐振器系统在非零内部Neumann频率附近、超亚波长区域之外的导波模式。在正则外部区域（参考波数处外部Dirichlet问题适定）中，我们通过将外部Helmholtz Dirichlet-to-Neumann映射压缩到内部共振Neumann特征空间的迹上，引入一个无穷维频率相关的电容算子。我们证明当对比度$δ\to0$时，连续问题按范数预解收敛到该离散有效算子，并推导了紧致缺陷频率和线缺陷带函数的一阶渐近公式。随后，我们通过Combes–Thomas论证建立电容系数的指数离对角衰减，从而得到离散算子的指数精确截断，并证明其保留系数可由局部Helmholtz问题计算。在物理频率处，该局部近似在增长有限簇问题的一致稳定性假设下指数收敛。通过引入消失复吸收以及Hermitian对称化，可以去除稳定性假设。特别地，阶为$\sqrtδ$的吸收参数，连同相互作用截断和修补半径阶为$|\logδ|$，足以保持缺陷本征频率的一阶高对比展开的$O(δ^2)$精度，从而产生一种快速计算方法。偶极和四极共振的数值实验说明了该离散模型对由材料或几何失谐产生的直波导和弯曲波导的精度、指数局部性和适用性。

Comment: 39 pages

链接: http://arxiv.org/abs/2608.26906v1

1. 研究对象和背景

本文研究高对比度谐振器系统中的波导现象，着重于非亚波长频率下的有效模型与快速计算。系统由无限周期阵列的谐振器构成，每个谐振器为$\mathbb{R}^d$（$d=2,3$）中的有界连通$C^{1,1}$域$D_j$，其内部波速为$v_j$，外部均匀介质波速为$v$。引入对比度参数$0<\delta\ll1$，定义$a_\delta=\delta\mathbf{1}_{\mathbb{R}^d\setminus D}+\mathbf{1}_D$与$m_\delta=\frac{\delta}{v^2}\mathbf{1}_{\mathbb{R}^d\setminus D}+\sum_{j\in\Lambda}\frac{1}{v_j^2}\mathbf{1}_{D_j}$，则本征问题$L_\delta u=\omega^2 u$等价于要求$u$在外部满足$\Delta u+\frac{\omega^2}{v^2}u=0$，在每个谐振器内满足$\Delta u+\frac{\omega^2}{v_j^2}u=0$，在边界$\partial D_j$上满足$u|^+=u|^-$以及$\delta\partial_\nu u|^+-\partial_\nu u|^-=0$。这里$L_\delta=-m_\delta^{-1}\nabla\cdot(a_\delta\nabla)$定义在$L^2(\mathbb{R}^d,m_\delta dx)$上，是自伴椭圆算子。研究的核心未知量是共振频率$\omega$（或$z=\omega^2$）和场$u$，希望在对比度参数$\delta\to0$时刻画它们在基准频率$\omega_0$附近的渐近行为。

本文选择$\omega_0\neq0$，并假设$\omega_0$是某个缺陷谐振器的内部Neumann本征频率，即存在非平凡函数$u_{j,\ell}$满足$-\Delta u_{j,\ell}=\frac{\omega_0^2}{v_j^2}u_{j,\ell}$于$D_j$内且$\partial_\nu u_{j,\ell}=0$于$\partial D_j$上。这种设定超出了经典亚波长（$\omega_0=0$）范围。在亚波长情形，每个谐振器上主导模式是常数函数，有效模型是经典电容矩阵；而在非亚波长情形，内部Neumann本征空间维数$m_j$可能大于1，外部场满足Helmholtz方程而非Laplace方程，辐射效应也可能进入一阶项。

这类问题的重要性在于，高对比度谐振器阵列通过局部共振的杂化可以引导电磁波或声波，形成缺陷模式、直线波导、弯曲波导以及拓扑保护边缘态。为了设计此类器件，需要定量预测当$\delta$很小时共振频率随系统参数变化的规律。亚波长的电容矩阵理论已经成熟，但只能处理低频。许多应用场景（例如偶极和四极谐振）要求更高的频率，此时必须发展非亚波长有效理论。

已有理论背景包括：Ammari等人在[9]中引入频率依赖电容矩阵，对周期系统刻画了Fabry–Pérot共振的一阶渐近，但未涉及非周期缺陷；随后[10]在亚波长机制下建立了一般非周期系统的范数-预解收敛与局部patch近似理论，但其主要工具Poincaré不等式在$\omega_0\neq0$时不适用。本文的目标正是填补这一空白：在外部谱隙条件（即$\omega_0^2/v^2$属于外部Dirichlet问题预解集）下，引入无穷维频率依赖电容算子$C(\omega_0)$，证明连续问题到该离散算子的范数-预解收敛，推导缺陷频率和线缺陷能带的一阶渐近，并建立指数型局域化从而支持快速局部计算。

2. 主要定理或主要结果

本文的主要结果分为三部分。

（1）范数-预解收敛与一阶渐近（定理3.1、3.8、3.12、3.13）。设$\omega_0$满足内部共振条件(3.2)、(3.4)与外部谱隙条件(3.5)。定义电容算子$C(\omega_0)$为$\ell^2(I(\omega_0))$上有界自伴算子，其核为$C_{(j,\ell),(j',\ell')}(\omega_0)=-\frac{v_jv_{j'}}{2\omega_0}\int_{\partial D_j}\partial_\nu U_{j',\ell'}|_+ u_{j,\ell}d\sigma$，其中$U_{j',\ell'}$是外部Helmholtz问题(3.8)的唯一解。令$K\subset\rho(2\omega_0C(\omega_0))$为紧集，$z_{\delta,\zeta}=\omega_0^2+\delta\zeta$，则对充分小的$\delta>0$，$A_\delta(z_{\delta,\zeta})$可逆，且缩放后的预解$\delta R(z_{\delta,\zeta},\delta)$按算子范数收敛于$(2\omega_0C(\omega_0)-\zeta)^{-1}$的嵌入（定理3.8）。因此，若$\lambda$是$C(\omega_0)$的特征值，则存在$L_\delta$的共振频率$\omega_\delta$满足

$$\omega_\delta=\omega_0+\delta\lambda+O(\delta^2).$$

对于紧致缺陷，此式在定理3.12中给出；对于直线缺陷$J_{def}=\{(j_1,0):j_1\in\mathbb{Z}\}$，利用部分Floquet–Bloch变换得到准周期电容矩阵$C_\alpha(\omega_0)$，其特征值$\lambda_j^\alpha$确定导带

$$\omega_j^\alpha(\delta)=\omega_0+\delta\lambda_j^\alpha+O(\delta^2),\qquad \text{对 }\alpha\in\widehat{Y}^*\text{ 一致}.$$

并证明当$\delta$足够小时这些缺陷特征值与导带位于未扰动体算子谱隙中。

（2）指数局域化与算子压缩（定理3.15、4.1）。利用Combes–Thomas估计，存在$\beta>0$使得

$$|C_{(j,\ell),(j',\ell')}(\omega_0)|\lesssim e^{-\beta|j-j'|}.$$

从而截断到距离$N$的算子$C_N^{tr}$满足$\|C-C_N^{tr}\|=O(e^{-\mu N})$，其中$\mu>0$。达到算子精度$O(\epsilon)$只需$N=O(\log\epsilon^{-1})$。若缺陷集有内蕴维数$d_{\mathrm{act}}$，则每谐振器只需$O((\log\epsilon^{-1})^{d_{\mathrm{act}}})$个相互作用块；对含$N_{\mathrm{act}}$个谐振器的有限线缺陷，压缩矩阵非零块数为$O(N_{\mathrm{act}}\log\epsilon^{-1})$，而稠密矩阵为$O(N_{\mathrm{act}}^2)$。

（3）局部patch逼近与快速算法（定理4.2、4.8、4.11）。在实频率$\omega_0$处，用半径为$R$的局部Helmholtz问题代替全局问题，得到的截断电容$C_{N,R}$满足

$$\|C(\omega_0)-C_{N,R}\|=O(e^{-\beta N}+e^{-\mu R}),$$

但需要增长簇问题的一致可解性假设。若引入复吸收$z_\gamma=\omega_0^2+i\gamma$（$\gamma>0$）并通过Schwarz反射和Hermite对称化构造对称化局部逼近$C_{N,R,\gamma}$，则无条件成立

$$C(\omega_0)=C_{N,R,\gamma}+O(\gamma^2+e^{-\beta N}+\gamma^{-1}e^{-\mu R}).$$

取$\gamma R=e^{-\mu R/3}$且$N\asymp R$，可得到指数收敛稳定近似。特别地，当$\gamma=\sqrt{\delta}$、$N,R=O(|\log\delta|)$时，有限簇近似保持一阶模型的$O(\delta^2)$精度：

$$\omega_{\delta,j}=\omega_0+\delta\lambda_{j,N,R,\sqrt{\delta}}+O(\delta^2),$$

且对直线缺陷的准动量$\alpha$一致成立。文中还给出了数值实验，对偶极和四极谐振验证了直线和弯曲波导中模式重构的准确性。

3. 研究方法、关键技术和核心工具

证明的核心困难在于，非亚波长设定下内部共振本征值使补空间算子失去强制性。作者通过四步递进地建立理论。

（1）外部DtN压缩与电容算子。为了消除外部区域，引入外部Dirichlet-to-Neumann映射$T^z$：对边界数据$\phi\in H^{1/2}(\partial D)$，$T^z\phi=\partial_\nu u|^+$，其中$u$是$\mathbb{R}^d\setminus D$中满足$(-\Delta-z/v^2)u=0$且$u|^+=\phi$的有界能量解。在外部谱隙条件(3.5)下，$T^z$在$z=\omega_0^2$附近解析。将原问题化为定义在$H^1(D)$上的算子$A_\delta(z)$，其内积为

$$a(u,w;z,\delta)=\sum_{j\in\Lambda}\int_{D_j}(\nabla u\cdot\nabla w-\frac{z}{v_j^2}uw)dx-\delta\langle T^z[u|_{\partial D}],w|_{\partial D}\rangle.$$

随后在每个谐振器上选取内部Neumann本征空间$E_j(\omega_0)$，并构造离散系数空间$\ell^2(I(\omega_0))$。定义算子$G:a\mapsto\sum a_{j,\ell}u_{j,\ell}$和$M:a\mapsto v_j^{-2}a_{j,\ell}$，则电容算子写为$C(\omega_0)=-\frac{1}{2\omega_0}M^{-1/2}G^*T^{\omega_0^2}GM^{-1/2}$。$C(\omega_0)$的有界自伴性由迹定理和$T^{\omega_0^2}$的自伴性得到（命题3.3）。

（2）补空间的谱演算与Schur补消去。设$P$为从$L^2(D)$到$\bigoplus_j E_j(\omega_0)$的投影，$Q=I-P$。作者将$A_\delta$按$P^*\ell^2\oplus Q$分解为四块。在亚波长情形，$\omega_0=0$时$Q$块强制；这里$\omega_0^2/v_j^2$是内部Neumann本征值，$Q$块可能不定。定理3.7证明，若(3.4)和(3.5)成立，则存在与$j$无关的常数使得补块$A_{QQ}+$（含$\delta$修正）一致可逆，其逆关于$\delta$有界。这是通过谱演算和外部DtN的小ness估计获得的，取代了[10]中的Poincaré不等式。随后利用Schur补消去$Q$，得到有效算子$A_\delta^{eff}$，其展开的零阶项正是$-2\omega_0C(\omega_0)$，从而得到范数-预解收敛。

（3）Combes–Thomas权估计与指数衰减。电容系数$C_{(j,\ell),(j',\ell')}$可表示为边界上$T^{\omega_0^2}$的内积，其中外部函数$U_{j',\ell'}$由区域$D_{j'}$上的源$u_{j',\ell'}$通过外部Helmholtz问题传播而来。由于$\omega_0^2/v^2$属于外部Dirichlet谱隙，作者对预解$(-\Delta-\omega_0^2/v^2)^{-1}$应用Combes–Thomas估计：存在$\beta>0$使$\|e^{\beta|x|}(-\Delta-\omega_0^2/v^2)^{-1}e^{-\beta|x|}\|$有界。将其作用于单位点源，得到$U_{j',\ell'}$在距离$|j-j'|$处的迹指数衰减，进而得到(3.18)中的指数界。这个衰减不仅导出压缩定理，也决定了patch半径的尺度。

（4）局部patch逼近与复吸收对称化。由于每个电容系数涉及全局问题，作者提出用有界局部域$B_R(j)$上的外部Helmholtz问题近似。在实频率下，增长局部问题可能遇到散射共振，使预解无界；作者在频率中加入正虚部$i\gamma$，使局部问题预解界为$O(\gamma^{-1})$且与簇尺寸无关。直接使用复吸频率会改变电容系数一阶项；利用Schwarz反射关系$T^{\bar z}=\overline{T^z}$，作者构造$C_{N,R,\gamma}$为两个复频率$z_\gamma$和$\bar z_\gamma$下局部系数的Hermite平均，使得一阶项抵消，仅留下$O(\gamma^2)$偏差。这一技巧使得不需要任何实频率稳定性假设即可获得定理4.8的无条件误差估计。最后，在定理4.11中取$\gamma=\sqrt{\delta}$，平衡稳定化误差与截断误差，并证明局部解的频率移位与真实一阶渐近相差$O(\delta^2)$。

4. 与之前工作的比较

本文与已有工作的比较可以从三个层面展开。

首先，与亚波长电容矩阵理论（例如[2,8,6,1,10]）相比，本文把有效模型的适用范围从$\omega_0=0$推进到任意非零内部Neumann频率。在亚波长机制中，每个谐振器的共振模式是常数，有效算子为经典电容矩阵/算子，外部场是调和的。本文在非亚波长情形下，内部模式可以是多个非平凡本征函数（偶极、四极等），外部场必须求解Helmholtz方程。从证明角度看，文献[10]的范数-预解收敛依赖补空间的强制性；本文通过谱演算建立补算子的一致可逆性，因此这是一个本质性推广。代价是引入了外部谱隙假设(3.5)，该假设在谐振器近紧贴的几何中可以通过谱隙开方获得，但并非任意几何都成立。

其次，与周期结构的频率依赖电容矩阵工作[9]相比，本文不再要求整体周期性。事实上，本文的构造允许任意非周期缺陷，包括紧致缺陷、直线缺陷和完全非周期的弯曲波导。当配置退化为直线缺陷时，部分Floquet–Bloch变换给出的准周期电容矩阵$C_\alpha(\omega_0)$与[9]的周期性Bloch电容矩阵一致，因而本文包含了周期结果。本文还补充了缺陷本征值和导带位于体谱隙中的证明，并建立了指数衰减，这是[9]未涉及的。

第三，与超晶胞方法和基于DtN映射的能带计算[25,26,22,19]相比，本文的计算方法具有明显的复杂度优势。传统方法需要在包含许多谐振器的大超晶胞上求解精细Helmholtz问题，而高对比度参数要求非常细的网格，导致计算量极大。本文利用电容算子的指数局域性，通过截断到$N=O(|\log\delta|)$和局部patch半径$R=O(|\log\delta|)$来近似每个矩阵元，仅需求解小尺寸局部问题。对于有限线缺陷，压缩矩阵的非零块数从$O(N_{\mathrm{act}}^2)$降到$O(N_{\mathrm{act}}\log\epsilon^{-1})$，因此可以处理长缺陷和宽频率范围。数值实验表明，在直波导和弯波导中均能重构偶极和四极局域模式。

需要指出，作者并未显式展开对所有已有方法的数值对比，其对计算复杂度的讨论是理论层面上的。此外，本文的严格结果依赖于外部谱隙条件，对于外部谱无隙的情形（例如孤立谐振子列阵），作者未做讨论。局部patch方法在实频率下需要一致可解性假设，虽然复吸收可以去除该假设，但引入的$\gamma$需要适当选取以保持精度，这增加了实现中的参数调节。总体而言，本文的贡献在于首次将高对比度谐振器系统的局部可计算性理论从亚波长扩展到非亚波长，并为弯曲波导等非周期结构提供了快速、可靠的数值方案。

2. 三维声散射中多极展开法的谱收敛

Spectral Convergence of the Multipole Expansion Method for Acoustic Scattering in Three Dimensions

作者: Jinrui Zhang, Jun Lai

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26884v1

摘要翻译: 多重散射是声学和电磁学中一种基本的波相互作用现象。多极展开法（MEM）是此类问题许多快速算法（如快速多极方法（FMM））的基础。然而，由于涉及无穷多次波反射，其在三维空间中的收敛性仍未得到探索。本文证明了三维空间中由有限多个分离良好的球体产生的时谐声散射的 MEM 的谱收敛性。利用对角预处理单层公式，我们在自然的球谐能量空间中分析 $N$ 阶截断系统。我们将相互作用截断分为目标侧和源侧的高阶部分，并为每一部分选择不同的表示：前者采用投影格林核表示，后者采用平移球波族的逐阶估计。由此得到的基于帕塞瓦尔恒等式和球谐加法定理的论证，为 MEM 的收敛性分析提供了一个通用框架，并揭示了收敛因子的几何和物理起源。我们还通过首次传递分析获得了更精细的估计。数值实验证实了预测的谱衰减和几何收敛因子。这为一大类多重散射快速算法的收敛性分析铺平了道路。

Comment: 无

链接: http://arxiv.org/abs/2608.26884v1

1. 研究对象和背景

本文研究三维时间调和声波在有限个互相分离的球体上的多体散射问题。设 $B_p=\{x\in\mathbb R^3: |x-c_p|a_p+a_q$ ($p\ne q$)。散射场 $u^s$ 在外区域 $\Omega^+=\mathbb R^3\setminus\bigcup_p\overline{B_p}$ 中满足 Helmholtz 方程 $\Delta u^s+k^2u^s=0$, 在边界上满足声软条件 $u^s=-u^{\mathrm{inc}}$ 于 $\Gamma=\bigcup_p\Gamma_p$, 以及 Sommerfeld 辐射条件 $\lim_{r\to\infty}r(\partial_r u^s-iku^s)=0$。入射波可取平面波 $u^{\mathrm{inc}}(x)=e^{ik\hat d\cdot x}$ 或点源 $u^{\mathrm{inc}}(x)=\Phi_k(x,x_0)=e^{ik|x-x_0|}/(4\pi|x-x_0|)$。本文采用单层位势表示 $u^s=S\phi$, $S\phi(x)=\int_\Gamma \Phi_k(x,y)\phi(y)\,ds_y$, 导出边界积分方程 $V\phi=-u^{\mathrm{inc}}|_\Gamma$。多极展开方法(MEM)将每个球面上的密度或出射多极系数按球谐函数 $Y_n^m$ 展开, 通过球谐加法定理把不同球心的展开平移耦合起来。实际计算必须截断到 degree $N$, 形成维数为 $M(N+1)^2$ 的代数系统。此前虽然已有针对单个平移、快速求和过程或约化 T-matrix 的误差估计, 但对三维全局耦合 MEM 截断系统的稳定性和收敛性, 严格理论是缺失的。二维分析 [9] 已经指出, 三维平移系数由 Gaunt 系数控制, 连带 Legendre 函数和超几何函数使逐项估计极其困难; 本文正是要填补这一空白。

2. 主要定理或主要结果

在假设 2.1 成立时, 即对所有 $p,n$ 有 $j_n(ka_p)\ne0$ (等价地 $k^2$ 不是任意球 $B_p$ 的内部 Dirichlet 特征值), 作者建立了 MEM 截断系统的谱收敛。记 $V_{pp}$ 在球谐基下对角化, $V_{pp}b_{p,nm}=\lambda_n^{(p)}b_{p,nm}$, 其中 $\lambda_n^{(p)}=ika_p^2j_n(ka_p)h_n^{(1)}(ka_p)=a_p/(2n+1)(1+O(n^{-1}))$。令 $D=\mathrm{diag}(V_{11},\dots,V_{MM})$, $W=D^{-1}V=I+A$, $G=-D^{-1}u^{\mathrm{inc}}|_\Gamma$, 则无穷系统为 $W\Phi=G$。设 $P_N$ 为保留每个球上 $0\le n\le N$ 所有 order 的投影, $Q_N=I-P_N$, 截断系统为 $P_N(I+A)P_N\Phi_N=P_NG$。作者证明: 存在 $N_0$, 当 $N\ge N_0$ 时 $W_N:=I+P_NAP_N$ 在 $H^{-1/2}:=\bigoplus_p H^{-1/2}(\Gamma_p)$ 上可逆, 且零延拓后的截断解满足精确误差恒等式

$$\Phi-\Phi_N=W_N^{-1}\bigl[Q_NG-(A-P_NAP_N)\Phi\bigr],$$

从而

$$\|\Phi-\Phi_N\|_{H^{-1/2}}\le \|W_N^{-1}\|_{H^{-1/2}\to H^{-1/2}}\bigl(\|Q_NG\|_{H^{-1/2}}+\|A-P_NAP_N\|_{H^{-1/2}\to H^{-1/2}}\|\Phi\|_{H^{-1/2}}\bigr).$$

对交叉块, 令 $r_q(x)=|x-c_q|$, $\hat r_q=(x-c_q)/r_q$, 作者在 Lemma 3.3 中对 $r_q\in[R_-,R_+]$, $R_->a_q$ 建立了 degree-wise 估计:

$$\sum_{m=-n}^n\|j_n(ka_q)h_n^{(1)}(kr_q)Y_n^m(\hat r_q)\|_{H^{1/2}(\Gamma_p)}^2\le C\langle n\rangle^\mu\eta^{2n},$$

$$\sum_{m=-n}^n\left\|\frac{h_n^{(1)}(kr_q)}{h_n^{(1)}(ka_q)}Y_n^m(\hat r_q)\right\|_{H^{1/2}(\Gamma_p)}^2\le C\langle n\rangle^\mu\eta^{2n},$$

其中 $\eta$ 可任意接近 $a_q/R_-$。由 Lemma 3.2, 这类尾项满足 $\sum_{n>N}\langle n\rangle^\alpha\theta^{2n}\le C\eta^N$ 对任意 $\eta>\theta$ 成立, 故得到 $\|Q_NG\|+\|A-P_NAP_N\|\lesssim \eta^N$。最终收敛因子由几何决定, 其典型形式为 $a_p/(d_{pq}-a_q)$, 即目标球半径与目标球心到源球最近点距离之比。文中还通过 first-transfer 分析得到更尖锐的估计, 该因子由孤立粒子场可解析延拓的更大区域决定。数值实验验证了谱衰减和几何收敛因子。

3. 研究方法、关键技术和核心工具

证明的核心障碍是三维球谐 mode 的多重性: degree $n$ 有 $2n+1$ 个 order, 平移耦合通过 Gaunt 系数将 $(n,m)$ 与 $(\ell,r)$ 混合, 若像二维那样逐个估计平移系数, 将面对连带 Legendre 函数和超几何函数的不可控渐近。作者的真正创新在于不估计单个 Gaunt 系数, 而是先把截断误差分解为目标端(target-side)和源端(source-side)的高阶部分, 再分别用不同的积分表示处理。具体框架如下:

对角预处理与能量空间。单层算子 $V$ 的分块 $V_{pq}$ 中, 自块 $V_{pp}$ 在球谐基下对角, 特征值渐近为 $\lambda_n^{(p)}=a_p/(2n+1)(1+O(n^{-1}))$。定义 $D=\mathrm{diag}(V_{11},\dots,V_{MM})$, 则 $W=D^{-1}V=I+A$, 其中 $A_{pq}=V_{pp}^{-1}V_{pq}$ 对 $p\ne q$ 因核光滑而紧。自然能量空间取 $H^{-1/2}=\bigoplus_p H^{-1/2}(\Gamma_p)$, 其范数 $\|\Phi\|^2=\sum_{p,n,m}\langle n\rangle^{-1}|\Phi_{p,nm}|^2$ 与球谐系数权重匹配。
截断误差恒等式。Proposition 3.1 给出 $\Phi-\Phi_N=W_N^{-1}[Q_NG-(A-P_NAP_N)\Phi]$。因此只需分别控制入射数据尾部 $Q_NG$ 和相互作用算子尾部 $A-P_NAP_N$。
目标端估计。对固定输出模, 将 Helmholtz Green kernel 投影后, 自块中的正则 Bessel 因子被消去, 留下由 Hankel 比控制的核。利用球谐加法定理可先对 order 求和, 例如 $\sum_{m=-n}^n|h_n^{(1)}(kr)Y_n^m(\hat r)|^2=(2n+1)/(4\pi)|h_n^{(1)}(kr)|^2$, 以及梯度版本的公式 (33)–(34), 从而把高阶重数约化为关于 degree 的 Bessel 函数比值估计。
源端估计。固定输入模后得到一族平移出射球波 $j_n(ka_q)h_n^{(1)}(kr_q)Y_n^m(\hat r_q)$, 其 $H^{1/2}$ 范数通过物理值能量和梯度能量控制。Lemma 3.3 在 $r_q\in[R_-,R_+]$, $R_->a_q$ 的假设下给出 degree-wise 估计, 其中常数含多项式因子 $\langle n\rangle^\mu$ 和几何因子 $\eta^n$。这里的关键是先用 Parseval 恒等式和加法定理把 order 全部求和, 再做 large-degree 估计; 因此三维空间中的 $2n+1$ 重数只贡献多项式因子, 不改变第 $N$ 次根的几何速率。
尾部求和与可逆性。Lemma 3.2 证明 $\sum_{n>N}\langle n\rangle^\alpha\theta^{2n}$ 的 $2N$ 次根收敛到 $\theta$, 所以多项式因子不影响谱衰减。结合 $A$ 的紧性和小尾估计, 可证明 $W_N$ 对充分大 $N$ 一致可逆, 最后代入误差恒等式得到谱收敛。

此外, 作者使用单层密度的系数而非传统出射多极振幅作为变量; 在任意有限截断度下两者由对角关系联系, 因此截断系统本质仍是同一 MEM 离散。这一选择让 Sobolev 结构和 interaction tails 更加透明。作者还指出, 该 target/source 分解框架在二维同样适用, 可绕过 [9] 中的双指标求和。

4. 与之前工作的比较

与最相关的二维工作 [9] 相比, [9] 从显式圆柱平移系数出发, 估计耦合双指标和, 将其化为超几何函数并用大阶渐近处理; 本文证明不必逐个估计平移系数。通过目标/源分解和 Parseval-加法定理, 先在所有 order 上求和, 再对 degree 取渐近, 因而三维 Gaunt 系数的复杂性不会进入估计。作者还指出该框架在二维同样适用, 可绕过 [9] 的 double-index sums, 说明这不是三维特有的技巧。

与截断球谐加法定理和 FMM 平移估计 [1,7,21,26,28] 相比, 那些工作主要针对单个平移或快速求和过程, 不涉及全局耦合 degree-$N$ MEM 系统; 本文处理的是经过任意多次反射传播后的截断算子 $A-P_NAP_N$, 因此属于系统级误差分析。与约化基 T-matrix 近似 [12] 相比, [12] 针对单个散射体的 T-matrix 做约化基逼近, 本文分析的是粒子间多极平移的截断误差; 文章提出可将两者结合以获得一般形状粒子的联合收敛率, 但未在本文中实现, 属于未来方向。与计算研究 [16,24] 相比, 本文提供了严格证明; 与低频多体散射的谱和条件数估计 [2,32] 相比, 本文给出了 $H^{-1/2}$ 中的稳定性与显式收敛阶。

推进之处包括: 证明对任意固定球面 Sobolev 尺度 $H^s$ 成立, 额外的 Sobolev 权重只影响多项式因子; 改变边界条件(声硬或阻抗)或积分表示只改变亚指数因子, 只要对应的 modal self block 估计可用; 收敛因子的几何来源被揭示, 全收敛因子由目标半径与目标球心到源粒子最近点距离之比决定, 例如 $a_p/(d_{pq}-a_q)$; 更尖锐的 first-transfer 因子由孤立粒子场的解析延拓区域决定。作者未在摘要或节选部分给出 first-transfer 因子的显式公式, 从提供内容中无法确认其具体闭式表达。

中文标题: 基于固定应力分裂与能量自然梯度下降的Biot模型物理信息神经网络

摘要翻译: 物理信息神经网络（PINNs）近年来作为求解偏微分方程的无网格框架受到了广泛关注。然而，当应用于强耦合多物理场系统（如Biot固结模型）时，由于优化景观严重病态，其性能会下降。在这项工作中，我们提出了一种基于PINN的稳健求解器，称为FS-ENGD-PINN，它协同整合了基于物理的解耦与几何感知优化。具体而言，采用固定应力（Fixed-Stress, FS）分裂格式将耦合的孔隙弹性系统分解为收缩的力学子问题和流动子问题，从而显著提高训练稳定性和收敛性。为了进一步加速优化，我们采用能量自然梯度下降（Energy Natural Gradient Descent, ENGD），该方法在函数空间中近似牛顿方向，有效缓解了刚度引起的慢收敛问题。此外，为了解决近不可压缩区域中出现的体积锁死问题，我们将带有附加总压力变量的三场混合公式纳入PINN框架。大量数值实验表明，所提出的FS-ENGD-PINN在精度和稳健性方面始终优于标准PINN公式，为广泛材料参数范围内的孔隙弹性问题提供了一种统一且可靠的基于学习的求解器。

不相关论文（基本信息）

1. 采用Mortar单元方法的高阶稳定化无矩阵旋转混合设备模拟

High-order stabilized matrix-free simulation of rotating mixing devices using the Mortar Element Method

作者: B. Campos, P. Munch, V. O. Ferreira, D. C. Boffito, X. Banquy, B. Blais

类别: physics.flu-dyn, math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27423v1

链接: http://arxiv.org/abs/2608.27423v1

摘要: We present a finite element framework to simulate rotating mixing devices using the Mortar Element Method as a domain decomposition strategy. The model is implemented within a matrix-free Navier-Stokes framework which uses a high-order Continuous Galerkin method. The discretized domain is subdivided into rotor and stator parts. An Arbitrary Lagrangian-Eulerian approach accounts for the relative rotor-stator motion, and stabilization is ensured through the Streamline-Upwind/Petrov-Galerkin and Pressure-Stabilizing Petrov-Galerkin methods. The rotor-stator domains are connected by an interface composed of mortar cells, and continuity is weakly enforced in a Discontinuous Galerkin fashion by accounting for boundary integrals at the rotor-stator interface. Verifications of the convergence order in two-dimensional steady and transient examples report optimal rates. The geometric non-conformity created at the mortar interface due to rotor rotation does not introduce significant error in the solution. A three-dimensional example is used to investigate the model’s scalability, which yields ideal strong scaling for large problems. A two-dimensional Rushton impeller example uses a torque analysis to showcase the mesh convergence, and the corresponding velocity profile is in agreement with existing numerical results. In a three-dimensional pitched blade turbine case, the power number curve (Np vs Re) shows good agreement with experimental data for Reynolds number values from 1 to 2000. An energy balance analysis reports a numerical dissipation of 1% for Re=200 and of 10% for Re=2000. By exploiting modern hardware capabilities through matrix-free methods, the proposed model is a robust, accurate, and efficient framework suitable for simulating flows with rotating geometries.

2. 用于失效概率估计的多层交互粒子系统方法

A Multilevel Interacting Particle System Method for the estimation of Failure Probabilities

作者: Rubén Aylwin, José Pinto

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27275v1

链接: http://arxiv.org/abs/2608.27275v1

摘要: We propose a novel method that combines a multilevel decomposition with the framework of Interacting Particle Systems to compute the probability that given quantities of interest take values above or below a certain pre-specified value. Through the application of a sequential sampling scheme, the algorithm is able to achieve convergence rates faster than those achieved by the classical Multilevel Monte Carlo algorithm, and competes with the current state of the art techniques, while also being dimension independent. Furthermore, the sample points generated by the algorithm form a sequence which concentrates near the critical value of the quantity of interest. Our results are rigorously established and later verified through numerical experiments.

3. QH-GEM：量子流体动力学生成建模

QH-GEM: Quantum-Hydrodynamic Generative Modeling

作者: Harbir Antil, Alex Kaltenbach, Sarswati Shah

类别: math.NA, math-ph, math.OC, quant-ph

发布日期: 2026-08-27

arXiv ID: 2608.27216v1

链接: http://arxiv.org/abs/2608.27216v1

摘要: In this paper, we develop a deterministic, physically constrained generative framework based on the Madelung formulation of the free-particle Schrödinger equation. A reference Born probability density and a controllable initial phase function serve as initial data for the free Madelung system, which couples the Born probability density and phase function through the Bohm quantum potential, while the phase function determines the hydrodynamic velocity field. Provided that the Born probability density remains positive and the hydrodynamic velocity field generates a unique global characteristic flow, samples drawn from the reference density and transported along the characteristic flow are distributed according to the evolving Born probability density at every time. As a consequence, randomness enters only through the initial sampling; the subsequent generation is deterministic and involves neither stochastic dynamics nor an independently parameterized time-dependent velocity field. We formulate terminal-time distribution matching as a PDE-constrained phase-identification problem and derive the underlying Hamiltonian and Fisher-information structure. For isotropic Gaussian wave packets, we obtain explicit dynamics and a necessary and sufficient condition for exact reachability of isotropic Gaussian targets by quadratic initial phase functions, together with the corresponding sampling map. For a smooth prescribed potential initial velocity field, we further establish that the characteristic flow approximates the associated first-order transport map with an O(T^2) error, both uniformly and in the 1- and 2-Wasserstein distances. A numerical Gaussian benchmark validates the fully discrete forward solver, while full-grid PDE-constrained phase identification is demonstrated for asymmetric bimodal targets.

4. 多类动理学交通流模型：离散速度公式与扩散修正的宏观极限

A multi-class kinetic traffic flow model: discrete-velocity formulation and diffusively-corrected macroscopic limits

作者: Carmen Mezquita-Nieto, Paola Goatin, Axel Klar

类别: math.NA, math.AP

发布日期: 2026-08-27

arXiv ID: 2608.27159v1

链接: http://arxiv.org/abs/2608.27159v1

摘要: This paper introduces a multi-class extension of a discrete-velocity kinetic traffic flow model based on a non-local Prigogine-Herman framework. We derive a hyperbolically scaled system of equations from a continuous kinetic formulation describing interactions between different vehicle classes through braking and relaxation terms. The model is then discretized with respect to the velocity variable for an arbitrary number of vehicle classes, and the structural properties of the resulting formulation are analyzed. In particular, we prove hyperbolicity and total linear degeneracy. Due to the non-conservative structure of the model, we employ a path-conservative finite volume scheme for the numerical approximation of the system. Finally, we derive the corresponding diffusively-corrected macroscopic multi-class model, investigate its stability and present numerical simulations on a single-lane road to illustrate the theoretical findings.

5. 原始-对偶方法与加速用于Morozov和等式约束正则化

Primal-dual methods and acceleration for Morozov and equality constrained regularization

作者: Diana-Elena Mirciu, Martin Benning, Elena Resmerita

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27106v1

链接: http://arxiv.org/abs/2608.27106v1

摘要: This work develops a regularization analysis of non-accelerated and accelerated primal-dual methods for solving linear inverse problems in the presence of noisy data. We investigate a Condat-Vũ algorithm and an accelerated primal-dual hybrid gradient method in Hilbert spaces, with focus on quantifying the effect of data perturbations on the reconstruction error. For the non-accelerated scheme, we derive error estimates in terms of Bregman distances, whereas for the accelerated scheme we establish error estimates in norm. The study accommodates a general class of convex data fidelities satisfying suitable perturbation conditions, which are verified explicitly for equality constrained and Morozov regularization. For the non-accelerated method, the analysis is further extended to Banach spaces, taking into account non-Euclidean geometries and including a particular non-reflexive setting tailored to nonnegative solution reconstruction. The results recover known behavior in classical settings while extending the regularization analysis to these more general frameworks. Numerical experiments with representative regularizers, including sparsity and entropic models, support the theoretical findings and illustrate practical performance under noise.

6. 随机拟高斯-牛顿法求解一般非线性方程组

Randomized Quasi-Gauss–Newton Methods for Solving General Nonlinear Equations

作者: Chengchang Liu, Luo Luo

类别: math.OC, math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27084v1

链接: http://arxiv.org/abs/2608.27084v1

摘要: This paper considers the local convergence for solving general nonlinear equations. We establish randomized quasi-Gauss–Newton methods based on the approximation of the Gram matrix, addressing both the underdetermined and the overdetermined settings. For the underdetermined case, we show that our methods achieve local superlinear convergence to the optimal solution. For the overdetermined case, we show that our methods achieve local condition-number-free convergence to the stationary point of the nonlinear least-square formulation of the problem. In contrast, existing quasi-Newton methods mostly focus on square systems and additionally require the initial Jacobian estimate to be sufficiently close to the exact one.

7. 正则化谱近似的精确低维重构

Exact low-dimensional reformulations for regularized spectral approximation

作者: Shengxiang Deng, Xudong Li, Yangjing Zhang

类别: math.OC, math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27052v1

链接: http://arxiv.org/abs/2608.27052v1

摘要: We study exact low-dimensional reformulations for a regularized spectral approximation problem and its weighted variant. In the unweighted setting, weak-majorization monotonicity of the fidelity term enables an exact reformulation over singular values. In the weighted setting, we establish such a reformulation under compatibility and simultaneous diagonalization conditions. We further show, via a counterexample, that this exact reformulation can fail in the absence of simultaneous diagonalization.

8. 基于偏微分方程的贝叶斯反问题的量子辅助框架

A quantum-assisted framework for PDE-based Bayesian inverse problems

作者: Dong An, Yinan Li, Pucheng Tang, Yunfeng Xiong

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.27028v1

链接: http://arxiv.org/abs/2608.27028v1

摘要: Quantum computing offers potential advantages for solving partial differential equations (PDEs). However, most existing quantum PDE solvers primarily focus on preparing quantum states for solutions, while the efficient recovery of classical information from these states remains less explored. Motivated by the readout limitation, we propose a quantum-classical hybrid framework for Bayesian PDE inversion problems: The quantum processor evolves the PDE and evaluate the loss function with sampling noises, while the classical computer tunes the hyper-parameters in the Gaussian Process Regression to explore the next trial candidate. To match the quantum solvers for linear and semi-linear autonomous evolution PDEs, we suggest to use a normalized quantum-state loss as the data-misfit function and evaluate the new misfit by combining quantum PDE solvers with the Hadamard test, thereby allowing us to extract useful classical information using only a limited number of quantum state copies without reconstructing the full solution vector. The analysis of error propagation and overall complexity of loss evaluation under a prescribed accuracy shows that the new data-misfit function outperforms the conventional L2-loss under quantum measurements. Quantum circuit simulations of 1D and 2D linear convection diffusion equations under approximate and finite sampling loss evaluations, together with classical numerical experiments on a nonlinear forced viscous Burgers equation, demonstrate the feasibility of the proposed approach for parameter inversion even when the loss evaluations are affected by sampling noise. This framework may provide a viable quantum-assisted scheme for PDE-based inverse problems and elucidate the potential of quantum PDE algorithms in addressing a complete quantum-to-end optimization stack.

9. 为何不使用高斯核

Why not to use the Gaussian kernel

作者: Toni Karvonen, Chris J. Oates

类别: stat.ML, cs.LG, math.NA, math.ST

发布日期: 2026-08-27

arXiv ID: 2608.26974v1

链接: http://arxiv.org/abs/2608.26974v1

摘要: Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density.

10. 变步长 Peer 两步方法的刚性范数估计

Stiff Norm Estimates for Variable-Stepsize Peer Two-Step Methods

作者: Jens Lang, Bernhard A. Schmitt

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26913v1

链接: http://arxiv.org/abs/2608.26913v1

摘要: Peer two-step methods are attractive for the numerical solution of stiff initial value problems because they combine favorable features of Runge-Kutta and multi-step methods, including high-order accuracy, strong stability properties, and the avoidance of order reduction. This paper develops rigorous norm estimates for the propagation operators of variable-stepsize diagonally-implicit $L(α)$-stable Peer methods applied to stiff semi-linear systems. The analysis relies on a class of Peer methods whose propagation matrices possess a grid-independent left eigenvector, enabling the construction of suitable matrix norms for entire coefficient families. We show that this class contains $L(α)$-stable methods of stage order $q=s$ and super-convergence of order $s+1$ with $s$ stages, exceeding the classical stage-order barrier $q\le 2$ of irreducible $s$-stage diagonally-implicit Runge-Kutta methods, and construct such methods up to order 5. Achieving high stage order is one of the principal advantages of Peer methods, especially for stiff differential equations where it helps mitigate order reduction. The resulting theory provides uniform stability bounds for stiff semi-linear problems on variable grids and establishes a foundation for further applications to optimal control and related boundary value problems. Comparative numerical results are presented for three classical stiff benchmark problems, including three novel Peer two-step methods of order 3, 4, 5, and four singly diagonally implicit Runge-Kutta methods with a first explicit step of stage order 2 and order 3, 4, 5, and 6.

11. TOPIQ：有损压缩下感兴趣量预测的统计误差传播

TOPIQ: Statistical Error Propagation for Quantity-of-Interest Prediction under Lossy Compression

作者: Youyuan Liu, Bo Jiang, Taolue Yang, Sheng Di, Robert Underwood, Sian Jin

类别: cs.PF, cs.IT, math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26912v1

链接: http://arxiv.org/abs/2608.26912v1

摘要: Lossy compression is essential for managing massive scientific data, but per-element error bounds do not translate into bounds on downstream quantities of interest (QoIs) such as regional averages, neural network predictions, or multi-field derived quantities. We present TOPIQ, a statistical error-propagation framework that predicts QoI-level bias and uncertainty from compact compression metadata (less than 0.1% of original data). TOPIQ decomposes QoIs into primitive operators with closed-form propagation rules accounting for spatial error correlation and data-error coupling; new QoIs are supported by composition at runtime with no per-QoI derivation or retraining. Across 552 evaluations spanning 4 datasets, 3 compressors, 4 QoI families, and 8 error bounds, 93.1% of configurations achieve well-calibrated predictions. Pre-computed metadata enables post-hoc uncertainty quantification for arbitrary query regions at 56x-402x speedup over direct computation. A case study demonstrates integration into an AI-driven analysis pipeline with end-to-end confidence intervals for dynamically composed queries.

12. 分层介质中多粒子弹性散射的快速求解器

A fast solver for many-particle elastic scattering in layered media

作者: Jinrui Zhang, Yixiao He, Jun lai

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26875v1

链接: http://arxiv.org/abs/2608.26875v1

摘要: This paper proposes a fast solver for time-harmonic elastic scattering by multiple particles embedded in layered media, with either Dirichlet or Neumann boundary conditions imposed on the particle surfaces. Such problems arise in many important applications, including composite material optimization, nondestructive testing, and subsurface imaging. They are computationally challenging because of strong multiple scattering interactions among the particles and the layered interface. The proposed method represents the layered medium contribution through Sommerfeld integrals and couples this representation with a well-posed boundary integral formulation for the particle scattering problem. High-order integral equation discretization and scattering matrix are used to handle particles of general shape, while multiple scattering theory provides an efficient description of particle interactions. To reduce the cost for large particle systems, the resulting multiple scattering computation is further accelerated by the fast multipole method. The main formulation is developed in both two and three dimensions. Numerical experiments for rigid and traction-free particles validate the accuracy of the formulation, and demonstrate its flexibility in both direct scattering simulations and inverse scattering applications.

13. 朗道-利夫希茨-吉尔伯特方程的线性质量集中有限元方法：无条件能量耗散与长度保持

A linear mass-lumped finite element method for the Landau-Lifshitz-Gilbert equation: unconditional energy dissipation and length preservation

作者: Qiumei Huang, Binghong Li, Xiaoli Li, Cheng Wang, Jiang Yang

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26687v1

链接: http://arxiv.org/abs/2608.26687v1

摘要: We develop a linear, unconditionally energy-dissipative, mass-lumped finite element method for the highly nonlinear Landau–Lifshitz–Gilbert (LLG) equation on quasi-uniform triangular meshes. The method is built on a projection strategy for enforcing the nonconvex pointwise constraint $|\mathbf{m}| = 1$, whose simultaneous preservation with unconditional energy stability remains challenging for standard finite element discretizations. The key innovation is a unified hybrid finite element-finite difference framework that underlies both the design and the analysis of the proposed method. In the scheme construction, we exploit the weak formulation and nodal structure of mass-lumped finite element method, while incorporating suitable interpolation operators and a node-wise length-preserving mechanism inspired by finite difference discretizations. This combination yields a linear scheme that preserves the node-wise unit-length constraint and satisfies a discrete energy dissipation law. The same hybrid framework also plays a central role in the error analysis, where the weak formulation and quasi-uniform mesh structure of finite elements are combined with interpolation-based and nodewise finite difference method to control the strongly nonlinear damping term and to establish an optimal-order error estimate. More importantly, the proposed method provides a unified framework that systematically integrates the geometric flexibility of finite element method with the pointwise constraint-preserving property of finite difference method, and thus offers a general strategy for designing and analyzing structure-preserving discretizations of constrained dissipative systems. Numerical experiments, including a classical blow-up simulation, confirm the predicted accuracy, energy dissipation, and robustness of the method.

14. 动态线性粘弹性的一种稳定的DG-POD降阶方法

A stable DG-POD reduced-order method for dynamic linear viscoelasticity

作者: Byeong-Chun Shin, Yongseok Jang

类别: math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26625v1

链接: http://arxiv.org/abs/2608.26625v1

摘要: We present a reduced-order modeling framework based on proper orthogonal decomposition for dynamic linear viscoelasticity governed by generalized Maxwell models. The hereditary constitutive law is reformulated using internal variables, and the resulting problem is discretized in space by a symmetric interior penalty discontinuous Galerkin method and in time by either the backward Euler or Crank–Nicolson scheme. The reduced model is obtained by projecting the fully discrete system onto low-dimensional spaces constructed from displacement snapshots. For the full-order discontinuous Galerkin formulation, we show well-posedness and derive an a priori stability bound without using Gronwall’s inequality. Consequently, the bound grows at most linearly, rather than exponentially, with the final time. We further derive spatial error estimates and an error decomposition that separates the effects of spatial discretization, temporal discretization, and proper orthogonal decomposition truncation. The reduced formulation replaces the large full-order systems with low-dimensional problems, enabling efficient long-time simulations. Numerical experiments verify the predicted convergence behavior, assess different inner products for constructing the reduced basis, and demonstrate accurate recovery of transient oscillations and viscoelastic relaxation with substantial computational savings, even after accounting for snapshot generation and basis construction.

15. 超立方体上随机tanh神经网络的高概率导数界

High Probability Derivative Bounds for Random tanh Neural Networks on a Hypercube

作者: Josef Dick, Michael Feischl, Fabian Zehetgruber

类别: cs.LG, math.NA

发布日期: 2026-08-27

arXiv ID: 2608.26526v1

链接: http://arxiv.org/abs/2608.26526v1

摘要: We establish high-probability bounds for mixed input derivatives of wide random neural networks whose activation derivatives satisfy a factorial growth bound. Our main result specializes these estimates to $\tanh$ networks with Xavier initialization. A direct deterministic analysis based on Euclidean operator norms of the weight matrices yields derivative bounds that generally grow exponentially with the depth. We show that this growth can be substantially improved for sufficiently wide Gaussian networks by isolating the term that is linear in the highest-order derivative and controlling the corresponding tangent directions by measurable finite nets. For scalar-output $\tanh$ networks with Gaussian weights and Xavier initialization, we prove that there exist constants $C,C_0,C_1>0$ such that, whenever the common hidden width satisfies $n \geq C\left(L^3n_0^2(1+\log n_0)+L^2\left(1+\log(L/η)\right)\right)$, then, with probability at least $1-η$, the estimate $\left|D^u\mathcal{R}_{Φ^{(L)}}(x)\right| \leq C_0 |u|! (C_1L)^{|u|-1}\prod_{j\in u}β_j(η,n_0)$ holds simultaneously for every non-empty $u\subseteq[n_0]$ and every $x\in[0,1]^{n_0}$. Thus, the first-order derivative bound is independent of the depth, while a square-free mixed derivative of order $|u|$ grows at most polynomially as $L^{|u|-1}$, apart from the coordinate factors. As consequences, we obtain high-probability bounds for the Euclidean Lipschitz constant and for weighted Sobolev norms of the network realization. The latter connect the derivative estimates to quasi-Monte Carlo integration and indicate how such regularity can enter the analysis of QMC-based training.

16. 非等温相场系统的广义耗散解：存在性、弱-强唯一性与长时间行为

Generalised dissipative solutions for a non-isothermal phase-field system: existence, weak-strong uniqueness, and long-time behaviour

作者: Aaron Brunk, Marvin Fritz

类别: math.AP, math.NA

发布日期: 2026-08-26

arXiv ID: 2608.26297v1

链接: http://arxiv.org/abs/2608.26297v1

摘要: We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $ω$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.

17. 一种简单的交错级数求和算法

A simple algorithm for the summation of alternating series

作者: Eleonora Denich, Paolo Novati, Alvise Sommariva

类别: math.NA

发布日期: 2026-08-26

arXiv ID: 2608.26287v1

链接: http://arxiv.org/abs/2608.26287v1

摘要: This paper deals with the computation of the sum of alternating series, whose general terms can be expressed by means of analytic functions. After rewriting the series as a weighted integral with the Abel weight, we employ the sinc (trapezoidal) rule and analyze the remainder term with respect to the number of quadrature points. We provide some numerical experiments to show the reliability of the derived error estimate and to test the algorithm for automatic summation with prescribed accuracy. All the Matlab codes used in the present paper can be found as open-source software at the authors’ homepage.

重点关注论文（完整分析）#

1. 高对比谐振器系统中的波导：理论与快速计算#

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. 三维声散射中多极展开法的谱收敛#

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. 在算子学习中强制实施Dirichlet边界条件#

摘要翻译#

2. B-稳定DIRK格式的Bochner稳定性#

摘要翻译#

3. 缩放核空间的尺度一致逆不等式#

摘要翻译#

4. 海冰动力学的间断伽辽金离散#

摘要翻译#

5. 物理信息随机配置机：一种用于非线性微分方程的免反向传播快速训练神经网络#

摘要翻译#

6. 基于固定应力分裂与能量自然梯度下降的Biot模型物理信息神经网络#

摘要翻译#

不相关论文（基本信息）#

1. 采用Mortar单元方法的高阶稳定化无矩阵旋转混合设备模拟#

2. 用于失效概率估计的多层交互粒子系统方法#

3. QH-GEM：量子流体动力学生成建模#

4. 多类动理学交通流模型：离散速度公式与扩散修正的宏观极限#

5. 原始-对偶方法与加速用于Morozov和等式约束正则化#

6. 随机拟高斯-牛顿法求解一般非线性方程组#

7. 正则化谱近似的精确低维重构#

8. 基于偏微分方程的贝叶斯反问题的量子辅助框架#

9. 为何不使用高斯核#

10. 变步长 Peer 两步方法的刚性范数估计#

11. TOPIQ：有损压缩下感兴趣量预测的统计误差传播#

12. 分层介质中多粒子弹性散射的快速求解器#

13. 朗道-利夫希茨-吉尔伯特方程的线性质量集中有限元方法：无条件能量耗散与长度保持#

14. 动态线性粘弹性的一种稳定的DG-POD降阶方法#

15. 超立方体上随机tanh神经网络的高概率导数界#

16. 非等温相场系统的广义耗散解：存在性、弱-强唯一性与长时间行为#

17. 一种简单的交错级数求和算法#

重点关注论文（完整分析）

1. 高对比谐振器系统中的波导：理论与快速计算

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. 三维声散射中多极展开法的谱收敛

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. 在算子学习中强制实施Dirichlet边界条件

摘要翻译

2. B-稳定DIRK格式的Bochner稳定性

摘要翻译

3. 缩放核空间的尺度一致逆不等式

摘要翻译

4. 海冰动力学的间断伽辽金离散

摘要翻译

5. 物理信息随机配置机：一种用于非线性微分方程的免反向传播快速训练神经网络

摘要翻译

6. 基于固定应力分裂与能量自然梯度下降的Biot模型物理信息神经网络

摘要翻译

不相关论文（基本信息）

1. 采用Mortar单元方法的高阶稳定化无矩阵旋转混合设备模拟

2. 用于失效概率估计的多层交互粒子系统方法

3. QH-GEM：量子流体动力学生成建模

4. 多类动理学交通流模型：离散速度公式与扩散修正的宏观极限

5. 原始-对偶方法与加速用于Morozov和等式约束正则化

6. 随机拟高斯-牛顿法求解一般非线性方程组

7. 正则化谱近似的精确低维重构

8. 基于偏微分方程的贝叶斯反问题的量子辅助框架

9. 为何不使用高斯核

10. 变步长 Peer 两步方法的刚性范数估计

11. TOPIQ：有损压缩下感兴趣量预测的统计误差传播

12. 分层介质中多粒子弹性散射的快速求解器

13. 朗道-利夫希茨-吉尔伯特方程的线性质量集中有限元方法：无条件能量耗散与长度保持

14. 动态线性粘弹性的一种稳定的DG-POD降阶方法

15. 超立方体上随机tanh神经网络的高概率导数界

16. 非等温相场系统的广义耗散解：存在性、弱-强唯一性与长时间行为

17. 一种简单的交错级数求和算法