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1. 神经富集有限元方法：一种用于强振荡问题或界面问题的混合框架
Neural enrichment finite element method: A hybrid framework for problems with strong oscillations or interface problems
作者: Shihan Guo, Thomas Richter
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09700v1
摘要翻译: 我们提出一种混合方法，即神经富集有限元方法（NEFEM），用于处理涉及强振荡或具有弱间断的界面问题。该方法基于稳定的广义有限元方法（SGFEM）框架，其中引入神经网络（NNs）作为自适应富集函数，并采用Ritz泛函进行训练。本工作有两项主要贡献。首先，该方法构造了具有优越逼近性质的局部子空间，显著减少了所需的自由度（DoFs）数目。其次，定义富集函数仅需极少的先验知识，因为神经网络在训练过程中以启发式方式演化。此外，对于光滑问题，我们给出了基于残差的误差估计子，并证明了其可靠性与有效性。对于界面问题，我们研究了SGFEM的最优收敛性理论分析，特别是不施加额外正则性假设。这些分析结果指导了网络架构设计和训练策略。所提方法的性能和有效性通过若干数值实验得到验证。
Comment: 无
链接: http://arxiv.org/abs/2605.09700v1
1. 研究对象和背景
本文研究两类典型的椭圆型偏微分方程数值求解问题：强振荡问题与含弱间断的界面问题。
第一类问题是经典散度型方程
$$ -\nabla\cdot(a(x)\nabla u) = f \quad \text{in }\Omega,\quad u|_{\partial\Omega}=0, $$其中扩散系数 $a\in W^{1,\infty}(\Omega)$ 对称正定且具有一致上下界，右端项 $f\in L^2(\Omega)$。在实际情形中，$a(x)$ 与 $f$ 可能呈现高频振荡，导致解 $u$ 呈现强烈的多尺度特征，标准有限元方法（FEM）需要极细的网格才能获得有意义的逼近，造成自由度（DoF）爆炸。
第二类问题是具有固定界面 $\Gamma$ 的扩散问题
$$ -\nabla\cdot(a_i\nabla u) = f \ \text{in }\Omega_0\cup\Omega_1,\quad [u]=0,\ [a\partial_\mathbf{n}u]=0\ \text{on }\Gamma, $$其中 $a_i>0$ 为分片常数，解在界面处仅满足 $u\in H^1_0(\Omega)$，且界面上法向通量连续但梯度间断。经典 FEM 在处理此类问题时，无论有限元多项式次数 $r$ 多高，能量范数误差均退化为 $O(h^{1/2})$（Babu\v{s}ka [4]），根本原因在于界面穿越单元导致全局正则性不足。
针对上述困难，一类重要方法是通用/扩展有限元方法（GFEM/XFEM）[26]，它通过单位分解（PUM）[24]在标准多项式空间之上引入自定义的增强函数来捕捉解的局部特征。然而，GFEM 面临两大挑战：(i) 增强函数的构造往往依赖强先验知识，在复杂或多尺度问题中难以获取；(ii) 经典的 GFEM 在界面问题中因“混合效应”导致次优收敛，且系统条件数恶化。稳定化广义有限元方法（SGFEM）[5] 通过将增强函数替换为 $\phi_i - I_h\phi_i$（$I_h$ 为标准线性插值）改善了条件数并消除了混合效应，但收敛分析通常需要附加额外的正则性假设。
近年来，深度神经网络（NN）在求解 PDE 方面展现出强大的逼近能力，代表性的工作包括 Deep Ritz 方法 [30]、物理信息神经网络（PINNs）[25] 等。将 NN 与 FEM 结合的混合方法受到关注，例如 Baek 等 [7] 提出的神经网络增强单位分解（NN-PU）方法，以及 Wang 等 [29] 的神经网络单元法。这些方法利用 NN 自适应地构造增强函数，降低了对先验知识的需求，但 NN-PU 中不规则支集导致精确数值积分困难，而 GFEM 中 NN 的引入仍然面临条件数恶化与缺乏严格后验误差分析的问题。
本文系统性地回答了以下开放问题：
- (Q1) 能否为每个特定问题“定制”逼近子空间以大幅减少自由度？
- (Q2) 如何设计自适应增强函数以最小化对先验知识的依赖？
- (Q3) 如何将 NN 有机融入 SGFEM 框架，并构造合适的损失函数、网络架构与训练策略？
- (Q4) 能否设计可靠且有效的误差估计子以动态监控并引导训练过程？
- (Q5) 界面问题的一般增强条件如何指导网络架构设计？
- (Q6) 能否仅需 $u\in H^1_0(\Omega)\cap H^2(\Omega_0\cup\Omega_1)$（而不附加额外的界面正则性）即证明 SGFEM 的最优收敛？
本文通过提出神经增强有限元方法（NEFEM）并建立相应的理论分析，对上述所有问题给出了肯定回答。
2. 主要定理或主要结果
本文的主要理论结果分为两部分：针对光滑问题（或强振荡问题）的后验误差估计子的可靠性与有效性，以及针对界面问题的 SGFEM 最优收敛性分析（无需额外正则性假设）。
残差估计子的可靠性（定理 3.1）：设 $u$ 为问题 (2.1) 的弱解，$u_h\in V_N(\mathcal{T}_h,\mathcal{I}^{\mathrm{enr}}_h;\boldsymbol{\theta})$ 为 NEFEM 近似解，局部误差指示子 $\eta_K$ 如式 (2.11) 定义，则存在正常数 $C$ 使得
$$ \|u-u_h\|^2_E \le C\sum_{K\in\mathcal{T}_h}\eta^2_K, $$其中能量范数 $\|\cdot\|_E = \sqrt{a(\cdot,\cdot)}$。该结论的证明依赖于 Galerkin 正交性与 Cl'ement 插值，并利用了 $P_1\subset V_N$ 的事实，从而将经典 FEM 的可靠性论证平移到神经增强框架。
残差估计子的有效性（隐于 §3.1，具体定理未在提供的摘录中完整给出，但文中明确宣称证明了有效性）：在假设 3.1（局部神经增强基函数在参考单元上一致线性独立）以及类似于单元边辅助假设下，可建立局部空间的逆不等式（引理 3.3）及 bubble 函数的等价范数不等式（引理 3.5），进而通过 Verf"urth 的标准技术得到局部有效性，即存在与网格尺寸无关的常数 $C$ 使得
$$ \eta_K \le C\|u-u_h\|_{E,\omega_K}, $$其中 $\omega_K$ 为与 $K$ 相邻的一批单元构成的局部片。
关键引理：
- 逆不等式（引理 3.3）：若元素 $K$ 上的所有神经增强函数 $\phi_j$ 满足假设 3.1，则对任意 $v_h\in V_N(K)$ 有 $$ \|\nabla v_h\|_{L^2(K)} \le C h^{-1} \|v_h\|_{L^2(K)}. $$ 该结果通过尺度化论证、线性独立性导出增强空间与 $P_1$ 的分离性，并利用紧嵌入与矛盾证法得到。
- Bubble 函数等价性与逆不等式（引理 3.5）：在相同假设下，对于单元 bubble 函数 $b_K$ 及任意 $v_h\in V_N(K)$，有 $$ \|v_h\|^2_{L^2(K)} \le C\int_K v_h^2 b_K\,dx,\qquad \|\nabla(b_K v_h)\|_{L^2(K)} \le C h^{-1} \|v_h\|_{L^2(K)}. $$ 这些估计是建立有效性的关键工具。
界面问题的最优收敛性分析：论文在 §3.2（文本未完全展示）中声称证明了 SGFEM 对界面问题的最优收敛性，且不需要除 $u\in H^1_0(\Omega)\cap H^2(\Omega_0\cup\Omega_1)$ 以外的额外正则性假设。该结论回答了 (Q6)，并表明“一般界面条件”(2.14) 足以保证 SGFEM 恢复 $O(h)$ 的能量误差。此外，这一分析直接指导了网络架构设计：在输入中显式引入满足条件 (2.14) 的拟距离函数 $D(x)$，使网络能够自然地表达界面处的通量间断，且该间断位置在训练过程中固定不移位。
方法本身的核心结果：NEFEM 算法（算法 2.1 与自适应版本算法 2.2）实现了：
- 利用 Ritz 能量泛函作为损失函数，并通过恒等式 $\frac{\partial L}{\partial\boldsymbol{\theta}} = \frac{\partial L}{\partial\mathbf{A}}\frac{\partial\mathbf{A}}{\partial\boldsymbol{\theta}} + \frac{\partial L}{\partial\mathbf{F}}\frac{\partial\mathbf{F}}{\partial\boldsymbol{\theta}}$ 避免了对线性系统解关于参数的梯度追踪，从而可无缝对接任意第三方线性求解器。
- 自适应的误差指示与 Doerfler 标记策略，动态选择需增强的区域并逐步停训已充分逼近的区域，在显著减少自由度的同时保持精度。
3. 研究方法、关键技术和核心工具
本文在技术上的核心创新体现在以下几个方面：
1. 神经网络与 SGFEM 的无缝衔接
传统 GFEM 中增强函数 $\phi_i$ 需预先指定（如裂纹问题的奇异函数），NEFEM 将局部神经网络 $\mathcal{N}_i(\cdot-\mathbf{p}_i;\boldsymbol{\theta}_i)$ 作为增强函数，并通过 SGFEM 的稳定化修正 $\phi_i - I_h\phi_i$ 纳入空间。这样既保留了 SGFEM 改善条件数与消除混合效应的优点，又赋予了增强函数强大的自适应逼近能力。每个神经网络的输入经过坐标平移 $\mathbf{x}-\mathbf{p}_i$，这不仅有利于优化，而且提升了不同节点增强函数之间的线性独立性，间接辅助满足假设 3.1。
2. 损失函数设计与免梯度训练
NEFEM 的训练目标是极小化 Ritz 能量泛函（2.10）。其关键突破在于利用离散解 $\mathbf{c}$ 满足 $\mathbf{A}(\boldsymbol{\theta})\mathbf{c}=\mathbf{F}(\boldsymbol{\theta})$ 这一事实，通过链式法则消去了 $\frac{\partial\mathbf{c}}{\partial\boldsymbol{\theta}}$ 项，从而损失函数的梯度仅涉及 $\mathbf{A}$ 与 $\mathbf{F}$ 对参数的偏导。这意味着在每一次训练迭代中，可先调用外部线性求解器（直接法或迭代法，甚至预条件）求解刚度方程得到 $\mathbf{c}$，然后固定 $\mathbf{c}$ 计算梯度更新网络权重。这种解耦使得 NEFEM 能与传统有限元软件或高效求解库轻松集成，避免了深度学习自动微分框架与大规模线性系统求解之间的冲突。
3. 后验误差估计与局部逆不等式证明的难点突破
为证明残差估计子的有效性，必须建立局部神经增强空间的逆不等式 $\|\nabla v_h\|_{L^2(K)} \lesssim h^{-1}\|v_h\|_{L^2(K)}$，但这里的 $v_h$ 并非多项式，而是由神经网络构成。作者将问题转化为对参考单元上空间 $\hat V_N(\hat K)=\operatorname{span}\{\hat L_j \hat r_{j,h}^K\}$ 的分析，其中 $\hat r_{j,h}^K$ 是增强函数的局部插值误差。核心步骤是：
- 假设 3.1 的作用：为保证该空间不退化，要求基函数 $\{\hat L_j \hat r_{j,h}^K\}$ 的一致线性独立，即其 Gram 矩阵的最小特征值有一致的正下界。这在数值上可通过迭代过程中移除近似线性相关的冗余基实现。
- 三步证明：首先，利用假设 3.1 得到增强空间内部的 $L^2$ 范数控制；其次，利用紧性论证证明增强空间与 $P_1(\hat K)$ 在 $L^2$ 意义下一致分离（即存在不依赖于 $h,K$ 的常数 $\delta>0$，使得 $\operatorname{dist}_{L^2}(w,P_1(\hat K)) \ge \delta\|w\|_{L^2(\hat K)}$），这是避免增强函数“退化”为一次多项式从而失去高阶逼近能力的关键；最后，结合范数等价与尺度化技巧完成逆不等式的证明。类似的思路被用于构建 bubble 函数的等价范数与逆不等式，进而借助 Verf"urth 的 bubble 函数技术得到有效性。
4. 界面问题的网络架构与理论对齐
对于界面问题，直接使用光滑激活函数的神经网络难以表达界面处的梯度跳跃；若使用 ReLU 等非光滑激活，跳跃位置会在训练中漂移，破坏数值积分的精确性。本文提出将满足一般界面条件 (2.14) 的拟距离函数 $D(\mathbf{x})$ 作为网络的附加输入，即 $\phi_i(\mathbf{x},D(\mathbf{x})) = \mathcal{N}_i(\mathbf{x}-\mathbf{p}_i, D(\mathbf{x});\boldsymbol{\theta}_i)$。条件 (2.14) 包含：(1) $D\in W^{2,\infty}(\overline\Omega_r)$ 分片光滑；(2) $[D]=0$ 在界面上连续；(3) $0<\kappa_1\le |[\partial_\mathbf{n}D]|\le \kappa_2$ 法向导数跳跃有正上下界。由链式法则 $\nabla\phi_i = \partial_{\mathbf{x}}\mathcal{N}_i + (\partial_D\mathcal{N}_i)\nabla D$，第二项自然引入通量间断，且该间断被锚定在物理界面的位置，因为 $D$ 是固定的。这使得网络可以使用 Tanh 等光滑激活函数，同时稳定地表达弱间断。该网络结构（图 1）由收敛性分析（需 $u\in H^2(\Omega_0\cup\Omega_1)$ 且无额外界面正则性）所激发。
5. 自适应策略与训练干预
NEFEM 的自适应版本采用双层标记：首先通过百分比标记策略选择初始增强节点集 $\mathcal{I}^{\mathrm{enr}}_h$，构建增强空间；训练过程中周期性地利用残差估计子 $\eta_K$ 评估局部精度，并使用 D"orfler 标记策略筛选出仍需进一步训练的子集 $\mathcal{S}_t$，仅更新对应节点的网络参数，其余已收敛区域的网络则停止训练。这类似于 FEM 中的 $p$-自适应，但这里“$p$-提升”是由网络训练完成，且动态分配计算资源。
4. 与之前工作的比较
与标准 FEM 及传统 GFEM/SGFEM 的比较
- 自由度效率：对于强振荡或界面问题，标准 FEM 需要极细网格才能分辨局部特征，导致线性系统规模巨大。GFEM/XFEM 通过引入少量增强函数降低了自由度，但其增强函数的构造严重依赖问题的物理先验（如裂尖奇异函数、界面跳跃函数等）。NEFEM 用神经网络替代手工设计的增强函数，网络在训练中自动接近“最优”形状，大幅减少对先验知识的依赖（回答 Q2）。同时，数值实验表明 NEFEM 在同等精度下所需自由度显著低于标准 FEM，并可在粗网格上得到高精度逼近（回答 Q1）。
- 稳定性与收敛性：传统 GFEM 存在严重的条件数恶化问题，且对于界面问题因混合效应而导致次优收敛。SGFEM [5] 通过修正增强函数 $\phi_i\to\phi_i-I_h\phi_i$ 解决了这两个问题，但在理论分析中，现有工作（如 [28,13,32]）仍需假设解满足额外的界面正则性（如分片 $H^2$ 且界面上的某些协调性）。本文在 SGFEM 框架下添加 NN 增强，不仅继承了 SGFEM 的良好代数性质，还在 §3.2 中首次仅要求 $u\in H^1_0(\Omega)\cap H^2(\Omega_0\cup\Omega_1)$ 即证明了 SGFEM 对界面问题的最优收敛（回答 Q6），这是理论上的显著推进，放宽了长期以来的正则性限制。
与其他神经网络-有限元混合方法的比较
- 与 NN-PU（Baek 等 [7]）及神经单元法（Wang 等 [29]）：两者的共同点是利用 NN 作为单位分解的局部增强，但它们未采用 SGFEM 的稳定化，也未针对界面问题设计专门的输入变换。NN-PU 在数值积分时因核函数支集不规则而面临困难；而 NEFEM 完全在标准有限元网格上进行，数值积分与普通 FEM 无异。此外，NEFEM 的损失梯度推导使得训练与线性求解解耦，可直接使用成熟求解器，而上述工作往往依赖自动微分全流程。
- 与基于深度学习的 PDE 求解器（Deep Ritz、PINNs 等）：这类无网格方法全局构造网络逼近解，通常需要大量配点且训练可能不稳定。NEFEM 保留了 FEM 的严谨基底、单元积分与强制边界条件，因此数值解的稳定性和可修复性更强。同时，NEFEM 的后验误差估计子为自适应提供可靠指引，这在纯 NN 求解器中较难实现。
误差估计方面的贡献
- 在经典 FEM 中，残差型误差估计子的可靠性与有效性依赖于多项式逼近空间的性质（如逆不等式、逼近性质）。NEFEM 中由于增强函数由神经网络构成，传统证明中的关键工具——局部逆不等式不再平凡成立。本文通过引入假设 3.1（局部增强基的一致线性独立）并构造细密的分离性论证，严格证明了该框架下保持逆不等式，进而建立了可靠且有效的后验误差估计（回答 Q4）。这为神经网络增强有限元空间提供了首个同时具备可靠性与有效性的误差估计理论。
界面问题网络设计的独特性
- 以往用神经网络处理界面问题或间断问题，多采用分区定义或特殊激活函数。NEFEM 将满足一般界面条件的拟距离函数 $D$ 作为输入，巧妙地将通量跳跃结构化地编码进网络，使得训练过程不会破坏界面的物理位置。这一设计直接受 SGFEM 最优化收敛分析的启发，体现了理论指导网络结构的思路（回答 Q5）。相比纯数据驱动的破坏性跳跃捕获，这种嵌入几何与物理信息的方式更稳健、训练更高效。
局限性与条件
- 假设 3.1 要求局部增强基函数一致线性独立，这在理论上是一个较强的假设，尽管作者指出可通过数值检测和移除冗余基来近似满足，但在极端情况下（如两个神经元输出几乎重合）可能导致估计失效。此外，有效性的完整证明需补充对待跳项的处理细节（文中提及需类似假设，但文本未见完整叙述）。
- 论文的数值验证目前限于二维算例（§4 未展示），对三维效果、更复杂界面拓扑、大规模网络的性能尚待进一步考察。
- 训练过程依赖于能量泛函，故当前仅适用于变分形式可解释为能量极小化的对称椭圆问题，推广至对流占优或非对称问题仍需额外研究。
2. 通过局部子空间迭代和时间分裂解耦多尺度抛物方程的尺度
Decoupling scales via localized subspace iteration and temporal splitting for multiscale parabolic equations
作者: Eric T. Chung, Lijian Jiang, Mengnan Li, Yajun Wang
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.08634v1
摘要翻译: 模拟非均匀介质中的扩散具有显著的计算挑战，因为解析微观物理尺度传统上需要过于精细的计算网格。为克服这一障碍，我们将局部子空间迭代（LSI）框架推广到多尺度抛物方程。所提方法通过局部标准子空间迭代（LSSI）或局部Krylov子空间迭代（LKSI）迭代地逼近局部逆算子的主导特征空间，从而构造最优的低维试探空间。由于这些LSI基函数本质上专为捕获抛物解中衰减缓慢的低频模式而设计，它们自然地抑制了长时间积分过程中的误差累积。为进一步提高计算效率，我们将基的构造解耦为一个离线阶段，并在在线时间步进中实施一种与对比度无关的部分显式时间分裂格式。通过显式推进主导宏观模式，同时隐式处理高频微观修正，该格式在保证稳定性的同时，不会施加限制性的时间步长约束。我们建立了能量范数和 $L^2$ 范数下的严格先验误差估计。数值实验表明了LSI框架的精度和效率，尤其突显了LKSI方法在处理高对比度、复杂多尺度介质方面的优势。
Comment: 25
链接: http://arxiv.org/abs/2605.08634v1
1. 研究对象和背景
本文研究对象是扩散过程在非均匀介质中的多尺度抛物方程：
$$ \partial_t u - \nabla \cdot (\kappa(x) \nabla u) = f(x,t), \quad (x,t) \in \Omega \times (0,T], $$边界条件为 $u=0$ 在 $\partial \Omega$，初值 $u(x,0)=u_0(x)$。渗透系数 $\kappa(x)$ 满足 $0<\kappa_{\min}\le \kappa(x)\le \kappa_{\max}<\infty$，且可能呈现高对比度（high-contrast）和不可分离的多尺度变化。此类方程广泛见于多孔介质渗流、复合材料热传导、地下储能模拟等科学计算与工程问题。
直接采用标准有限元法求解需要在所有微小物理尺度上解析，导致网格过密、计算量与内存需求急剧增大。为克服这一障碍，多个数值降阶与多尺度方法被提出。经典均匀化理论为周期结构提供了严格的渐近极限，异质多尺度方法（HMM）通过耦合宏观和微观模型处理非均匀介质；变分多尺度方法（VMS）将解分解为粗尺度和细尺度分量；多尺度有限元法（MsFEM）在粗网格基函数中嵌入微观信息；局部正交分解（LOD）利用局部化正交校正子，即使没有尺度分离也能获得最优逼近阶。近年来，多尺度谱广义有限元法（MS-GFEM）和广义多尺度有限元法（GMsFEM）通过计算局部算子的主导特征向量有效捕捉介质的多尺度特征，其中约束能量最小化GMsFEM（CEM-GMsFEM）进一步结合了过采样与能量极小化技术。
针对时变问题，上述框架虽已推广至抛物方程，但往往未充分利用扩散算子的谱衰减本质。抛物方程的解具有谱展开：
$$ u(x,t)=\sum_{k=1}^{\infty}\left(c_k e^{-\lambda_k t}+\int_0^t e^{-\lambda_k (t-s)} f_k(s)\,ds\right)\varphi_k(x), $$其中 $(\lambda_k,\varphi_k)$ 是椭圆算子 $-\nabla\cdot(\kappa\nabla)$ 的特征对。由于高频分量对应的大特征值 $\lambda_k$ 随时间指数衰减，长时间积分的精度与稳定性完全取决于数值空间对低频特征子空间的逼近质量。因此，构造能够最优捕获低频模态的基函数成为关键。
作者在先前工作中针对椭圆问题提出了局部子空间迭代（LSI）框架，包括局部标准子空间迭代（LSSI）和局部Krylov子空间迭代（LKSI），通过迭代方式近似局部逆算子的主导特征空间，避免直接求解特征值问题。本文将该框架推广至抛物方程，自然利用了谱衰减特性，理论上提供长期积分的高精度。同时，为缓解高对比度介质对时间步长的严格限制，本文还设计了一种对比度无关的部分显式时间分裂方案：利用离线阶段导出的局部特征值，将多尺度空间正交分解为捕获低频宏观模式的显式子空间和处理高频微观校正的隐式子空间，从而将昂贵的隐含逆限制在低维子空间，实现了稳定性与大步长的兼顾。
2. 主要定理或主要结果
本文的主要理论贡献为局部基函数的逼近误差定量估计，以及由此导出的全局先验误差界，并配合时间分裂的稳定性分析。
定理 4.3（LSSI 局部逼近） 设 $\varphi_i^j$ 是局部谱问题 $\mathcal{L}_i^{-1}\varphi_i^j = \lambda_i^j \varphi_i^j$ 的第 $j$ 个特征函数，对应特征值 $\lambda_i^j$（按降序排列）。若初始子空间 $V_{S,i}^{(0)}$ 到主导特征子空间 $\text{span}\{\varphi_i^1,\dots,\varphi_i^{L_i}\}$ 的投影满秩，则对 $k$ 次迭代后得到的空间 $V_{S,i}^{(k)}$，存在 $\hat{\varphi}_i^{j,k}\in V_{S,i}^{(k)}$ 满足
$$ \|\hat{\varphi}_i^{j,k}-\varphi_i^j\|_a \le C\left(\frac{\lambda_i^{L_i+1}}{\lambda_i^j}\right)^k, \quad j=1,\dots,L_i, $$其中 $\|\cdot\|_a$ 是能量范数，$C$ 与 $k$ 无关。收敛速度由相邻特征值之比决定，当 $\lambda_i^{L_i+1}/\lambda_i^1<1$ 时呈几何收敛。
定理 4.4（LKSI 局部逼近） 设 $\varphi_i^j$ 同上，初始函数 $\psi_i^0$ 沿 $\varphi_i^j$ 的分量非零。对于 $k>j$ 的 Krylov 子空间 $V_{K,i}^{(k)} = \mathcal{K}_k(\mathcal{L}_i^{-1},\psi_i^0)$，存在 $\hat{\varphi}_i^{j,k}\in V_{K,i}^{(k)}$ 使得
$$ \|\hat{\varphi}_i^{j,k}-\varphi_i^j\|_a \le C\left(\frac{1}{1+4\gamma_i^j}\right)^k, $$其中相对谱间隙 $\gamma_i^j = (\lambda_i^j-\lambda_i^{j+1})/\lambda_i^{j+1}$，$C$ 与 $k$ 无关。该结果表明 LKSI 的收敛速率由谱间隙决定，当间隙较大时远超 LSSI 的速率。
基于上述局部结果，结合解的局部化分解（引理4.1：对任一 $u\in H_0^1(\Omega)$ 有分解 $u=\sum_i u_i$ 满足 $\sum_i \|\mathcal{L}u_i\|^2 \le M\|\mathcal{L}u\|^2$，其中 $M$ 为覆盖重叠数）和局部特征插值误差估计（引理4.2：$\|u - I_{\text{eig}}u\|_a \le M\sqrt{\lambda^{L+1}}\|\mathcal{L}u\|$，$\lambda^{L+1}=\max_i\lambda_i^{L_i+1}$），可以建立全局多尺度空间 $V_{\text{ms}}^{(k)}$ 的逼近性质。进一步，对半离散和全离散格式导出了能量范数与 $L^2$ 范数的先验误差估计（具体定理形式在论文第5节展开，此处提供材料未完全给出，但可推断误差界包含特征值截断项 $O(\sqrt{\lambda^{L+1}})$ 以及迭代收敛项）。
时间分裂方案：离线阶段获得的局部特征值信息被用于在线阶段将多尺度空间正交分解为低频显式子空间和高频隐式子空间。该方案被证明是对比度无关且无条件稳定的，即显式处理低频部分并不引入与 $\kappa$ 的对比度相关的 CFL 条件，时间步长仅受粗网格尺寸限制，大幅提升计算效率。
3. 研究方法、关键技术和核心工具
本文的核心技术路线是将多尺度基函数的构造转化为对局部逆算子 $\mathcal{L}_i^{-1}$ 主导特征子空间的迭代逼近，且不需要显式求解局部特征值问题。证明框架围绕局部谱逼近与抛物误差分解展开。
1. 局部化与解的分解 利用单位分解 $\{\chi_i\}$ 将右端项局部化，进而通过求解局部 Dirichlet 问题定义解的局部分量 $u_i$（见(4.4)）。引理4.1证明了该分解的稳定性：$\sum_i \|\mathcal{L}u_i\|^2 \le M \|\mathcal{L}u\|^2$，其中 $M$ 是子域覆盖的重叠常数。该引理使得全局误差可通过局部误差的平方和来控制，为局部特征逼近向全局收敛的传递提供了基础。
2. LSSI：子空间幂迭代的实现与分析 LSSI 并非直接显式作用 $\mathcal{L}_i^{-1}$，而是通过求解局部鞍点问题实现等价迭代：对 $j=1,\dots,L_i$，每一步求解
$$ a(\varphi_i^{j,k+1},v) + \sum_{r=1}^{L_i} \mu_r^{k+1} q_r^{k}(v) = 0 \quad \forall v\in V(\omega_i), $$附加约束 $q_r^{k}(\varphi_i^{j,k+1})=\delta_{rj}$，其中 $q_r^{k}$ 为上一步基函数定义的对偶泛函。这样得到的序列 $\{\varphi_i^{j,k}\}$ 张成的空间正是 $V_{S,i}^{(k)}$。
定理4.3的证明技巧：对每个目标特征函数 $\varphi_i^j$，构造初始空间中的函数 $s_j$ 使得其到主导特征子空间的投影恰好等于 $\varphi_i^j$（满秩假设保证存在性）。将 $s_j$ 分解为 $\varphi_i^j + w$，其中 $w$ 属于正交补子空间（对应特征值 $\le \lambda_i^{L_i+1}$）。经过 $k$ 次逆算子作用并适当缩放后，正交补分量被压缩因子 $(\lambda_i^{L_i+1}/\lambda_i^j)^k$ 抑制，能量范数误差界即由谱半径给出。最终通过 Galerkin 投影的最优性得到 $\hat{\varphi}_i^{j,k}$ 的误差估计。
3. LKSI：Krylov 子空间与多项式滤波 LKSI 构造 Krylov 子空间 $\mathcal{K}_k(\mathcal{L}_i^{-1},\psi_i^0)$，同样通过鞍点问题实现每一步的 $\mathcal{L}_i^{-1}$ 作用并施加正交化：
$$ a(\psi_i^{k+1},v) + \mu^{k+1} q^{k}(v) = 0 \quad \forall v\in V(\omega_i), $$约束 $q^{k}(\psi_i^{k+1})=1$，其中 $q^{k}=\langle\psi_i^k,\cdot\rangle$。该过程生成一列共轭方向。
定理4.4的分析核心是“多项式滤波”：任意 $v\in V_{K,i}^{(k)}$ 可写成 $v=p(\mathcal{L}_i^{-1})\psi_i^0$，其中 $p$ 是次数 $\le k-1$ 的多项式。通过选择特定的多项式 $p^*$，使其满足 $p^*(\lambda_i^j)=1$ 且在其余特征值处取值尽量小，从而极小化误差 $\|p^*(\mathcal{L}_i^{-1})\psi_i^0 - \varphi_i^j\|_a$。最优多项式逼近的经典结果给出收敛因子 $1/(1+4\gamma_i^j)$，其中 $\gamma_i^j$ 是相对谱间隙，反映了特征值的分离程度。该估计表明 LKSI 在特征值分离良好时收敛远快于 LSSI。
4. 时间分裂方案的理论基础 在线阶段，将离线构造的多尺度空间 $V_{\text{ms}}$ 按局部特征值大小分解为 $V_{\text{ms}} = V_{\text{ms}}^{\text{exp}} \oplus V_{\text{ms}}^{\text{imp}}$，使得 $V_{\text{ms}}^{\text{exp}}$ 由对应特征值较小的基函数张成（低频模式），$V_{\text{ms}}^{\text{imp}}$ 由对应较大特征值的基函数张成（高频校正）。对半离散方程
$$ \langle (u_{\text{ms}})_t, v\rangle + a(u_{\text{ms}}, v) = \langle f, v\rangle \quad \forall v\in V_{\text{ms}} $$引入显隐分裂：对 $v\in V_{\text{ms}}^{\text{exp}}$ 采用显式时间步，对 $v\in V_{\text{ms}}^{\text{imp}}$ 采用隐式。这种做法的稳定性基于以下事实：显式部分对应的特征值小，因而 CFL 条件 $\Delta t \le C/\max\lambda_{\text{exp}}$ 中 $\max\lambda_{\text{exp}}$ 远小于全显式时的 $\lambda_{\max}(\kappa)$，且 $\max\lambda_{\text{exp}}$ 仅与粗网格尺寸有关，与介质对比度无关。隐式部分则因只涉及高频、快速衰减的模式，其刚性可通过局部小规模系统高效处理。整体上，该方案在理论分析中被证明是无条件稳定的，且时间步长限制不依赖于 $\kappa$ 的振荡幅度。
4. 与之前工作的比较
本文直接延续了作者在椭圆问题上的 LSI 工作[21]，将其拓展至抛物情形并加入时间分裂，构成了一套完整的离线-在线解耦方法。与[21]相比，进展体现在：(1) 提供了更精密的全局误差分解（引理4.1、4.2），消除了[21]中对局部正则性的额外假设，使理论更普适；(2) 首次给出了 LSSI 和 LKSI 对局部特征函数的迭代收敛速率（定理4.3、4.4），这是[21]未曾量化的结果；(3) 从抛物方程谱展开的角度论证了 LSI 基函数的长时最优性，并将该性质纳入了误差分析；(4) 提出并理论分析了对比度无关的部分显式时间分裂，使方法在实际模拟中具备竞争力。
与现有多尺度抛物方程方法的比较：
与 MsFEM、LOD MsFEM 和 LOD 通过求解局部椭圆问题构造基函数，虽隐含利用了谱信息，但未显式对准低频子空间。LSI 迭代地逼近主导特征空间，理论上能在相同基函数数量下获得更高的低频逼近精度，尤其适合长期积分。LOD 方法通过局部正交校正子获得近似最优收敛率，即使无尺度分离也有效，但其基函数个数常需随宏观网格自由度平方增长；LSI 可独立选择每个子域的基函数个数，灵活性更强。此外，LOD 的校正子构造需要求解多个局部问题，而 LSI 通过迭代线性方程组生成基函数，可能更易实现。
与 GMsFEM、CEM-GMsFEM GMsFEM 系列方法直接求解局部特征值问题来提取基函数，在高维或极度复杂介质中计算量很大。LSI 回避了特征分解，仅需反复求解带约束的椭圆问题，大大降低了离线成本。CEM-GMsFEM 通过过采样和能量最小化进一步提高精度，但需要求解更多的局部问题和全局耦合系统，而 LSI 在局部子域上独立迭代，天然并行。在抛物问题中，CEM-GMsFEM 常需在每个特征值变化处重新构建基函数（如时间步参数变化），而 LSI 的基函数仅依赖于空间算子，离线计算一次即可用于任意长的时间积分，效率显著。此外，本文的时间分裂方案基于 LSI 自身的谱信息自动分离尺度，无需额外构造或识别尺度，这是 GMsFEM 方法未具备的。
与 HMM、VMS HMM 和 VMS 通常要求介质具有尺度分离或已知宏观模型。LSI 不依赖尺度分离假设，仅利用局部逆算子的谱性质，适用于不可分离的多尺度问题。其在时间推进上的部分显式处理也与经典 IMEX 方法有本质区别：IMEX 一般针对同一空间分解，而本文的分解是基于粗尺度低频模式和高频校正的物理分离，更具针对性。
局限性 定理4.3和4.4表明，当特征值密集（$\lambda_i^{L_i+1}\approx \lambda_i^{L_i}$）或谱间隙 $\gamma_i^j$ 很小时，LSI 的收敛可能变慢，此时需增加迭代次数或增大 $L_i$。时间分裂的阈值选择目前缺乏自适应策略，如何在精度与效率间自动平衡是未来工作方向。此外，从提供的材料中未见到与[29]（CEM-GMsFEM 用于抛物方程）或[14]（时空 GMsFEM）等方法的详细数值对比，但理论分析已表明 LSI 在基函数构造成本和长时间表现上的优势。
了解领域论文（摘要翻译）
1. 基于保守通量重构的椭圆界面问题的界面降维
Interface Reduction for Elliptic Interface Problems with Conservative Flux Reconstruction
作者: C. Attanayake, So-Hsiang Chou
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09800v1
链接: http://arxiv.org/abs/2605.09800v1
摘要翻译
中文标题: 基于保守通量重构的椭圆界面问题的界面降维
摘要翻译: 我们提出了一种基于保守通量重构的椭圆界面问题的低维界面降维方法。该方法结合了贴体的 $P_1$ 有限元离散和文献 \cite{ChouTang2000} 中的通量恢复过程，产生局部守恒的通量，并满足界面条件至机器精度。一个主要结果表明，降维解的误差完全由界面数据的逼近误差控制。对连续和间断界面条件的数值实验证实，一旦界面数据被精确表示，全解被恢复到舍入精度。这些结果表明，椭圆界面问题的本质复杂性集中在界面上。
不相关论文（基本信息）
1. 基于优化的理想MHD不变域保持限制器中的高效容许集投影
Efficient Admissible Set Projection in Optimization-based Invariant-Domain-Preserving Limiters for Ideal MHD
作者: Chen Liu, Chi-Wang Shu, Xiangxiong Zhang
类别: math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10929v1
链接: http://arxiv.org/abs/2605.10929v1
摘要: Preserving the admissible set of the ideal magnetohydrodynamics (MHD) equations is important not only for producing physically meaningful numerical solutions, but more importantly for achieving robust computations. In this paper, we develop an optimization-based limiter to enforce admissibility while preserving global conservation and accuracy. For an easy and efficient projection, we decompose the admissible set into slices parameterized by the magnetic energy, so that the MHD projection reduces to a one-dimensional minimization, which can be solved efficiently by the Brent method. The splitting method can be used to efficiently solve the global minimization problem of the optimization-based limiter, which can be used to enforce cell average admissibility in discontinuous Galerkin (DG) schemes, and pointwise admissibility can be further enforced by the Zhang-Shu positivity-preserving limiter. We apply the limiter to high-order DG schemes and present numerical results for a few representative MHD problems.
2. 基于可分离估计器的松弛：算法与实现
Relaxation via Separable Estimators: Arithmetic and Implementation
作者: Yanlin Zha, Mario Eduardo Villanueva, Boris Houska, Benoît Chachuat
类别: math.NA, math.OC
发布日期: 2026-05-11
arXiv ID: 2605.10854v1
链接: http://arxiv.org/abs/2605.10854v1
摘要: This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation’s tightness with its computational cost.
3. Unitaria：基于块编码的量子线性代数
Unitaria: Quantum Linear Algebra via Block Encodings
作者: Matthias Deiml, Oliver Hüttenhofer, Ram Mosco, Jakob S. Kottmann, Daniel Peterseim
类别: quant-ph, cs.ET, cs.SE, math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10768v1
链接: http://arxiv.org/abs/2605.10768v1
摘要: We introduce Unitaria, a Python library that brings the simplicity of classical linear algebra toolkits such as NumPy and SciPy to the implementation of quantum algorithms based on block encodings, a general-purpose abstraction in which a matrix is embedded as a sub-block of a larger unitary operator. Their implementation has so far required deep knowledge of low-level circuit construction, which Unitaria aims to eliminate. The library provides a composable, array-like interface through which users can define block encodings of matrices and vectors, combine them through standard operations such as addition, multiplication, tensor products, and the Quantum Singular Value Transformation, and extract the resulting quantum circuits automatically. A key feature is a matrix-arithmetic evaluation path in which every operation can be computed directly on encoded vectors and matrices without dependence on ancilla qubits or circuit simulation. This enables correctness verification and classical simulation that scale well beyond what state vector simulation permits and also allows resource estimation, including gate counts, qubit counts, and normalization constants, without executing any circuit. Together, these capabilities allow researchers to develop, verify, and analyze quantum linear algebra algorithms today, ahead of the availability of error-corrected hardware. Unitaria is open source and available at https://github.com/tequilahub/unitaria.
4. 仿射追踪：概率线性求解器的一种新范式
Affine Tracing: A New Paradigm for Probabilistic Linear Solvers
作者: Disha Hegde, Marvin Pförtner, Jon Cockayne
类别: stat.ML, cs.LG, math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10566v1
链接: http://arxiv.org/abs/2605.10566v1
摘要: Probabilistic linear solvers (PLSs) return probability distributions that quantify uncertainty due to limited computation in the solution of linear systems. The literature has traditionally distinguished between Bayesian PLSs, which condition a prior on information obtained from projections of the linear system, and probabilistic iterative methods (PIMs), which lift classical iterative solvers to probability space. In this work we show this dichotomy to be false: Bayesian PLSs are a special case of non-stationary affine PIMs. In addition, we prove that any realistic affine PIM is calibrated. These results motivate a focus on (non-stationary) affine PIMs, but their practical adoption has been limited by the significant manual effort required to implement them. To address this, we introduce affine tracing, an algorithmic framework that automatically constructs a PIM from a standard implementation of an affine iterative method by passing symbolic tracers through the computation to build an affine computational graph. We show how this graph can be transformed to compute posterior covariances, and how equality saturation can be used to perform algebraic simplifications required for computation under specific prior choices. We demonstrate the framework by automatically generating a probabilistic multigrid solver and evaluate its performance in the context of Gaussian process approximation.
5. 面向建筑瞬态CO₂-温度网络模型的数据驱动移动窗口贝叶斯推断
Data-driven moving-window Bayesian inference for transient CO2-temperature network models of buildings
作者: Zhijian Wang, Stein K. F. Stoter, Clemens V. Verhoosel, Idoia Cortes Garcia
类别: math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10562v1
链接: http://arxiv.org/abs/2605.10562v1
摘要: In this work, we proposes a CO2-temperature network model that links multi-zone mass transport and thermal dynamics through shared latent drivers, airflow and occupancy. The thermal component is formulated as a resistance-capacitance (RC) network augmented with airflow-driven convective exchange, while the CO2 component is governed by inter-zonal convective transport. To calibrate the model and track time-varying operating conditions based on sparse sensing, we introduce a moving-window Bayesian inference procedure that jointly estimates thermal parameters, airflow and occupancy trajectories. The estimation also provides room-specific sensor noise levels, yielding posterior predictive forecasts with credible intervals. The framework is assessed using a controlled synthetic benchmark, and a scaled physical validation experiment using CO2 and temperature sensing. In both settings, the posterior accurately reconstructs trajectories within windows and delivers low forecast errors. When inference windows overlap abrupt regime transitions, the widened uncertainty bands and increased inferred noise levels provide an interpretable diagnostic of model-data mismatch, followed by rapid recovery once the new regime is observed. Overall, coupling CO2-informed airflow with thermal dynamics yields a robust approach for conductive and advective temperature prediction, supporting practical monitoring and energy-performance assessment under limited sensing.
6. 分子动力学中输运系数计算的数学分析与数值方法
Mathematical analysis and numerical methods for the computation of transport coefficients in molecular dynamics
作者: Noe Blassel, Louis Carillo, Shiva Darshan, Raphael Gastaldello, Alessandra Iacobucci, Elisa Marini, Regis Santet, Xiaocheng Shang, Gabriel Stoltz, Urbain Vaes
类别: math.NA, cond-mat.stat-mech, math.PR
发布日期: 2026-05-11
arXiv ID: 2605.10507v1
链接: http://arxiv.org/abs/2605.10507v1
摘要: We review various numerical approaches to compute transport coefficients in molecular dynamics. These approaches can be broadly classified into three groups: (i) nonequilibrium methods based on applying an external driving field to the system, measuring the average response in the system, and evaluating the related linear response coefficient; (ii) approaches reformulating the transport coefficient of interest through a time correlation function for the equilibrium dynamics (the most popular instances being Green–Kubo and Einstein formulas); (iii) transient techniques, where the transport coefficient can be computed by monitoring the return to the steady state of a dynamics perturbed off its stationary distribution. For all three classes of methods, we provide elements of numerical analysis, allowing to estimate or at least quantify the level of numerical errors in the estimator of the transport coefficient; and also briefly present recent attempts to more efficiently compute transport coefficients with variance reduction approaches such as control variates, importance sampling and coupling methods. The computation of transport coefficients remains nonetheless challenging and will continue requiring research efforts in the foreseeable future.
7. 不固定基，而学之：面向偏微分方程的自适应基学习谱表示
Don’t Fix the Basis – Learn It: Spectral Representation with Adaptive Basis Learning for PDEs
作者: Xuxiang Zhao, Angelica I. Aviles-Rivero
类别: cs.LG, math.FA, math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10451v1
链接: http://arxiv.org/abs/2605.10451v1
摘要: Spectral neural operators achieve strong performance for PDE learning, but rely on fixed global bases that limit their ability to represent spatially heterogeneous and multiscale dynamics. We propose Adaptive Basis Learning (ABLE), a framework that learns data-dependent spectral representations instead of relying on predefined bases. ABLE constructs a spatially adaptive Parseval frame via a learned ancillary density, enabling the operator to act in a lifted spectral space while preserving invertibility and maintaining $O(N\log N)$ complexity through FFT-based implementation. This shifts the source of expressivity from spectral coefficients to the representation itself, allowing the model to capture localized structures and non-translation-invariant interactions more efficiently. ABLE integrates seamlessly into existing neural operator architectures as a drop-in replacement for spectral layers. Across a range of benchmarks ABLE improves accuracy over strong baselines, with the largest gains in regimes characterized by sharp gradients and multiscale behavior. Moreover, augmenting existing models (e.g., U-FNO, HPM) with ABLE further enhances their performance, demonstrating its role as a general and complementary spectral refinement. Our results highlight that the data-driven choice of representation, rather than operator complexity alone, is a key bottleneck in neural operator design. By learning the basis itself, ABLE provides a principled and efficient framework for improving spectral methods in PDE learning.
8. PCELM：用于Stefan问题的扰动校正极限学习机
PCELM: Perturbation-Correction Extreme Learning Machine for the Stefan problem
作者: Wenjie Liu, Siyuan Lang, Zhiyue Zhang
类别: math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10417v1
链接: http://arxiv.org/abs/2605.10417v1
摘要: For Stefan problems, characterized by moving boundaries and discontinuous coefficients due to phase changes, the inherent nonconvexity of the objective functional frequently causes optimization difficulty in randomized neural network approximations; to address this, we propose a Perturbation-Correction Extreme Learning Machine (PCELM) framework, built upon the extreme learning machine framework. This method first establishes a basic approximation during an initialization step by minimizing the original nonconvex residual, typically achieving only moderate accuracy, and then, in a subsequent correction step, determines a correction term by solving a subproblem based on a perturbation expansion around this basic approximation, thereby transforming it into a convex optimization problem for the output coefficients that ensures rapid convergence. We further provide a rigorous a convexity analysis, demonstrating that PCELM method solves a convex sub-problem. Numerical experiments on various Stefan problems, including multi-phase and multi-dimensional Stefan problems, confirm that the proposed PCELM method successfully overcomes optimization plateaus, with the correction step consistently delivering a significant improvement of 2-6 orders of magnitude in the relative L2 accuracy.
9. 弹塑性问题的hp有限元法
hp-Finite Elements for Elastoplasticity
作者: Patrick Bammer, Lothar Banz, Miriam Schönauer, Andreas Schröder
类别: math.NA
发布日期: 2026-05-11
arXiv ID: 2605.10392v1
链接: http://arxiv.org/abs/2605.10392v1
摘要: This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational formulation is introduced to resolve the non-differentiablility of the so-called plasticity functional appearing in the weak formulation of the model problem as a variational inequality of the second kind. The discretization of the mixed formulation is then represented as a system of decoupled nonlinear equations which allows the application of an efficient semismooth Newton solver. Finally, an a priori and a posteriori error analysis is given.
10. PowerStep：基于$\ell_p$-范数最速下降的内存高效自适应优化
PowerStep: Memory-Efficient Adaptive Optimization via $\ell_p$-Norm Steepest Descent
作者: Yao Lu, Dengdong Fan, Shixun Zhang, Yonghong Tian
类别: cs.LG, cs.AI, cs.CL, math.NA, math.OC
发布日期: 2026-05-11
arXiv ID: 2605.10335v1
链接: http://arxiv.org/abs/2605.10335v1
摘要: Adaptive optimizers, most notably Adam, have become the default standard for training large-scale neural networks such as Transformers. These methods maintain running estimates of gradient first and second moments, incurring substantial memory overhead. We introduce PowerStep, a memory-efficient optimizer that achieves coordinate-wise adaptivity without storing second-moment statistics. Motivated by steepest descent under an $\ell_p$-norm geometry, we show that applying a nonlinear transform directly to a momentum buffer yields coordinate-wise adaptivity. We prove that PowerStep converges at the optimal $O(1/\sqrt{T})$ rate for non-convex stochastic optimization. Extensive experiments on Transformer models ranging from 124M to 235B parameters demonstrate that PowerStep matches Adam’s convergence speed while halving optimizer memory. Furthermore, when combined with aggressive \texttt{int8} quantization, PowerStep remains numerically stable and reduces optimizer memory by $\sim\!8\times$ compared to full-precision Adam. PowerStep thus provides a principled, scalable and resource-efficient alternative for large-scale training. Code is available at https://github.com/yaolubrain/PowerStep.
11. jNO：用于神经算子与基础模型训练的 JAX 库
jNO: A JAX Library for Neural Operator and Foundation Model Training
作者: Leon Armbruster, Rathan Ramesh, Georg Kruse, Christopher Straub
类别: cs.LG, math.NA, physics.comp-ph
发布日期: 2026-05-11
arXiv ID: 2605.10159v1
链接: http://arxiv.org/abs/2605.10159v1
摘要: jNO (jax Neural Operators) is a JAX-native library for neural operators and foundation models with unified support for both data-driven and physics-informed training. Its core design is a tracing system in which domains, model calls, residuals, supervised losses, and diagnostics are written in one symbolic language and compiled into one optimization pipeline. This allows users to move between operator regression, mesh-aware residual evaluation, and PDE-constrained training without restructuring the surrounding code. jNO also supports multi-model compositions, fine-grained control at parameter level (model, optimizer, and learning rate), hyperparameter tuning, and JAX-native workflows for translated PDE foundation-model families. The source repository is available at https://github.com/FhG-IISB/jNO.
12. 总广义变差正则化弥合了地震走时层析成像中神经场方法与经典方法之间的差距
Total Generalized Variation regularization closes the gap between neural-eld and classical methods in seismic travel-time tomography
作者: Isao Kurosawa
类别: physics.geo-ph, cs.LG, math.NA
发布日期: 2026-05-11
arXiv ID: 2605.09960v1
链接: http://arxiv.org/abs/2605.09960v1
摘要: Travel-time tomography forces a trade-off between mesh resolution and stability in which the regularizer choice dominates what can be recovered. We introduce MIMIR, a differentiable framework that represents the 2D velocity field as a Fourier-feature neural network, replacing the grid-based slowness vector with a continuous, infinitely differentiable function. Prior neural-field tomography has staircased smooth fields under total-variation (TV) priors or oscillated near interfaces under $L^2$ Laplacian smoothing. We adopt second-order total generalized variation (TGV$^2$) and parametrize its auxiliary vector field as a second neural network jointly optimized with the velocity field, eliminating the inner Chambolle-Pock primal-dual loop that classically dominates TGV computation. On three synthetic benchmarks (Gaussian, horizontally layered, curved-fault inspired by OpenFWI) using cross-well acquisition, 5% travel-time noise, and five seeds, MIMIR-TGV$^2$ ties a classical FMM-LSMR baseline with auto-tuned hyperparameters on the Gaussian ($p=0.134$, paired $t$-test) and significantly outperforms it on layered ($p<0.0001$, 44% RMSE reduction) and curved-fault ($p=0.0002$, 33% reduction). Replacing TGV$^2$ with TV degrades performance on Gaussian ($p=0.004$) and layered ($p=0.003$); curriculum-annealed TV improves Gaussian RMSE by only 5.4%, confirming that TV’s staircase bias is intrinsic to the regularizer rather than a scheduling artifact. The results empirically validate the Bredies-Kunisch-Pock prediction that piecewise-affine priors are better suited to subsurface velocity recovery than piecewise-constant TV priors. We argue that the central design choice in physics-informed neural-field inversion is not the network architecture but the regularizer. The full pipeline reproduces in under one hour on consumer hardware.
13. 三维亥姆霍兹格林函数及其导数的方位角傅里叶模态的快速求值
Fast Evaluation of the Azimuthal Fourier Modes of the 3D Helmholtz Green’s Function and Their Derivatives
作者: Hanwen Zhang
类别: math.NA, physics.comp-ph
发布日期: 2026-05-11
arXiv ID: 2605.09952v1
链接: http://arxiv.org/abs/2605.09952v1
摘要: We introduce an $O(M)$ algorithm for evaluating the azimuthal Fourier modes $G_{k,m}$, $m = 0, 1, ..., M$, of the three-dimensional Helmholtz Green’s function with real wavenumber $k$, together with all their first- and second-order derivatives with respect to the cylindrical source and target coordinates. The cost is independent of both the wavenumber and the source-target separation, and high relative accuracy is retained even for modes whose magnitude is exponentially small. The method combines contour deformation at a few boundary modes with a boundary-value formulation of the five-term recurrence in the mode index. Derivative quantities are obtained from stable recurrences, adding only a small constant factor to the cost of $G_{k,m}$ alone. Numerical experiments demonstrate high relative accuracy, linear scaling in $M$, and applications to modal boundary integral equation solvers for axisymmetric acoustic scattering, where the $k$-independent kernel evaluator makes dense per-mode linear algebra the dominant cost.
14. 部分观测时间变换随机微分方程的参数估计
Parameter Estimation for Partially Observed Time-Changed SDEs
作者: Ke Zhao, Ajay Jasra
类别: math.NA, stat.CO, stat.ME
发布日期: 2026-05-11
arXiv ID: 2605.09880v1
链接: http://arxiv.org/abs/2605.09880v1
摘要: In this paper we consider the parameter estimation problem associated to partially-observed time changed SDEs, with observations that are given at discrete times. In particular we consider both likelihood and Bayesian estimation. We develop new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method to provide likelihood-type parameter estimators. We also use a variant of this MCMC algorithm to perform multilevel-based Bayesian parameter estimation. We prove that this latter method achieves a mean square error of $\mathcal{O}(ε^2)$ ($ε>0$) with a cost of $\mathcal{O}(ε^{-2}\log(ε)^2)$. Our methodologies are tested numerically on both simulated and real data.
15. 利用快速素描加速幂法以实现更强的低秩近似
Accelerating Power Method with Fast Sketching for Stronger Low-Rank Approximation
作者: Shabarish Chenakkod, Michał Dereziński
类别: math.NA, cs.DS, cs.LG, stat.ML
发布日期: 2026-05-10
arXiv ID: 2605.09755v1
链接: http://arxiv.org/abs/2605.09755v1
摘要: The power method is one of the most fundamental tools for extracting top principal components from data through low-rank matrix approximation. Yet, when the target rank is large, the cost of matrix multiplication associated with this procedure becomes a major bottleneck. We develop an algorithmic and theoretical framework for accelerating the power method using fast sketching, which is a popular paradigm in randomized linear algebra. Our framework leads to simple and provably efficient methods for singular value decomposition, low-rank factorization, and Nyström approximation, which attain strong numerical performance on benchmark problems. The key novelty in our analysis is the use of regularized spectral approximation, a property of fast sketching methods which proves more flexible in generalizing power method guarantees than traditional arguments.
16. 增强主动通量平流格式的耗散行为
On Enhancing the Dissipative Behavior of Active Flux Advection Schemes
作者: Christian Klingenberg, Simon Krotsch, Philip Roe
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09669v1
链接: http://arxiv.org/abs/2605.09669v1
摘要: In this work, the traditional third-order Active Flux advection scheme is modified by reformulating the method and introducing additional parameters. The effect of these parameters is studied, leading to schemes with improved dissipative properties. These improvements are validated by numerical experiments.
17. 偏微分方程解算子的核学习
Kernel Learning of PDE Solution Operators
作者: Jianyu Hu, Juan-Pablo Ortega
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09643v1
链接: http://arxiv.org/abs/2605.09643v1
摘要: A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a kernel ridge regression framework, and employs a regularization-based formulation to construct an operator learner. This yields a closed-form estimator that is independent of the input functions that determine the underlying PDE. From the perspective of regularization theory, the resulting estimator induces a well-defined operator that links input and output spaces, which contain the functions that define a Dirichlet problem and its solution, respectively. Consequently, it effectively shifts from a PDE solver to an operator-based solver. In contrast to standard supervised learning methods, it does not rely on paired input–output training data and enables systematic extrapolation beyond observed regimes. A full error analysis is conducted, providing convergence rates for the operator-based solver under suitable choices of regularization parameters. Extensive numerical experiments, including Darcy flow and Helmholtz equations, demonstrate that the proposed method achieves high accuracy and efficiency across a range of problem settings, and compares favorably with operator learning approaches in both approximation quality and computational cost.
18. 理查兹方程有限元离散的离散正性与最大值原理
Discrete positivity and maximum principles for a finite element discretization of the Richards equation
作者: Abderrahmane Benfanich, Yves Bourgault, Abdelaziz Beljadid
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09615v1
链接: http://arxiv.org/abs/2605.09615v1
摘要: Standard finite element discretizations of the Richards equation may violate the discrete minimum principle, producing unphysical negative saturations. While existing bound-preserving methods typically rely on computationally expensive fully implicit solvers, we propose a novel semi-implicit finite element framework utilizing a bounded continuous auxiliary variable. Our approach treats the gravity-driven advective term using a linearly implicit technique, which improves the time-step restrictions required by explicit gravity methods near the degenerate limit. We provide rigorous mathematical proofs establishing sufficient geometric and algebraic constraints for discrete positivity and the discrete maximum principle, specifically a local Péclet condition and a discrete row-sum condition. When both conditions are satisfied on weakly acute meshes with mass lumping, our framework ensures that numerical solutions strictly respect physical bounds across highly degenerate conditions and initially dry soil regimes. Comprehensive numerical validation demonstrates the method across multiple flow regimes, including cases where algebraic conditions are satisfied, violated, and recovered through mesh refinement.
19. HS-FNO：面向非马尔可夫偏微分方程的历史-空间傅里叶神经算子
HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations
作者: Lennon J. Shikhman
类别: cs.LG, cs.CE, math.NA, physics.comp-ph, stat.ML
发布日期: 2026-05-10
arXiv ID: 2605.09523v1
链接: http://arxiv.org/abs/2605.09523v1
摘要: Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(θ,x)=u(t+θ,x)$, $θ\in[-τ,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction–diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.
20. 非均匀介质中含移动障碍物的波散射边界积分方法
A boundary integral method for wave scattering in a heterogeneous medium with a moving obstacle
作者: Raaghav Ramani
类别: math.NA, physics.comp-ph
发布日期: 2026-05-10
arXiv ID: 2605.09500v1
链接: http://arxiv.org/abs/2605.09500v1
摘要: We propose a time-domain boundary integral method to model linear wave propagation with refractive, focusing, and Doppler effects arising from medium heterogeneities and moving obstacles. In contrast to existing techniques, our method avoids volumetric discretization and yields a formulation posed only on the boundary of the obstacle. We combine two classical ingredients: a geometric–optics parametrix to capture leading-order behavior at propagating wavefronts, and a ray-based characterization of the distorted causal geometry. The former provides a framework for defining layer potentials and deriving the associated boundary integral equations, while the latter enables a pure boundary-only formulation. Taken together, these ingredients extend existing numerical techniques for the homogeneous, fixed-boundary case to the heterogeneous, moving-boundary setting, with appropriate modifications to capture the discrete causal structure arising from the intersection of distorted light cones with the worldsheet of the moving boundary. Numerical experiments demonstrate the ability of the method to resolve Doppler effects from moving obstacles, including a rotating turbine configuration, with stable performance up to Mach 0.9. In heterogeneous media, the method captures strong refractive effects from spherical inclusions: wave propagation wrapping around a gas bubble in water, and defocusing from a hot fireball rising through a stratified atmosphere.
21. 一种用于一般二维域上函数逼近的分片局部傅里叶延拓方法
A Patchwise Local Fourier Extension Method for Function Approximation on General Two-Dimensional Domains
作者: Zhenyu Zhao, Yanfei Wang
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09484v1
链接: http://arxiv.org/abs/2605.09484v1
摘要: We propose a patchwise local Fourier extension method for approximating smooth functions on general two dimensional domains with curved boundaries. The domain is embedded into a Cartesian background grid and decomposed into rectangular interior patches and one-side curved trapezoidal boundary patches. After local data transfer, all patches are converted into fixed-size tensor-product arrays and approximated by a truncated-SVD stabilized local Fourier extension procedure. Unlike global Fourier frame approximations, the proposed method localizes both the geometry and the ill-conditioned extension process. For fixed local parameters, the local algebraic operations are performed on fixed-size systems, and the reference Fourier extension matrices and their singular value decompositions are reused across patches. Boundary patches require additional one-dimensional transfer or completion steps, but their costs remain uniformly bounded by the local resolution. Consequently, the online complexity is (O(N)), where (N) denotes the total number of retained output points for fixed local resolution. Numerical experiments on smooth curved domains and on a mildly rough boundary domain demonstrate that the method achieves high accuracy with a fixed set of local parameters. The smooth-cover correction reduces the boundary-induced error by several orders of magnitude in the full-domain rough-boundary test, without changing the underlying scan-based partition.
22. MC$^2$: 用于快速椭圆型偏微分方程求解的蒙特卡罗校正
MC$^2$: Monte Carlo Correction for Fast Elliptic PDE Solving
作者: Ethan Hsu, Hong Meng Yam, Ivan Ge
类别: cs.LG, cs.AI, cs.CE, cs.CV, math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09288v1
链接: http://arxiv.org/abs/2605.09288v1
摘要: Partial differential equation (PDE) solvers underpin scientific computing, but real-world deployment is bounded by compute. Classical Monte Carlo solvers such as Walk-on-Spheres (WoS) are unbiased and geometry-agnostic but are slow. Learned solvers are fast but biased and brittle under distribution shift. We present \textbf{MC$^2$}, a hybrid WoS-Neural Network (WoS-NN) PDE solver that treats a low-budget Monte Carlo solution as a structured estimator of the true field and learns a single-pass neural correction to recover a high-fidelity solution. MC$^2$ matches the accuracy of solutions using over $1000\times$ more Monte Carlo compute, outperforming all evaluated classical, denoising, and neural-operator baselines. To enable reproducible study of finite-compute PDE solving, we additionally release \textbf{PDEZoo}, the largest standardized elliptic PDE benchmark to date: 2M PDEs spanning five elliptic families and unlimited geometric compositions, with analytic ground truth and multi-budget Monte Carlo trajectories. Together \textbf{MC$^2$} and \textbf{PDEZoo} (1) empirically establish that finite-sample Monte Carlo error is structured, learnable, and correctable in a single forward pass, (2) show that we can solve PDEs $\sim$\textbf{1000x} faster than with just WoS, and (3) provide the evaluation infrastructure the field has so far lacked.
23. 高度非均质空间网络模型的高效多尺度方法
Efficient Multiscale Methods for Highly Heterogeneous Spatial Network Models
作者: Yingjie Zhou, Xiang Zhong, Changqing Ye, Eric T. Chung
类别: math.NA
发布日期: 2026-05-10
arXiv ID: 2605.09280v1
链接: http://arxiv.org/abs/2605.09280v1
摘要: Modeling complex spatial networks with multiscale heterogeneity poses significant mathematical and computational challenges. Lacking explicit PDE discretizations and facing excessive degrees of freedom, conventional methods often become computationally prohibitive. To address these challenges, we propose an efficient multiscale modeling for highly heterogeneous spatial networks. We construct multiscale basis functions tailored to spatial network models with heterogeneous edge weights and node degrees. A key novelty is that the proposed method doesn’t introduce geometric parameters (such as Dirichlet nodes, distances, or mesh sizes), thereby preserving its purely algebraic nature and ensuring broad applicability. By incorporating a subgraph-wise estimate, we define a Poincaré constant $C_{\mathrm{po}}$ that renders the method independent of the underlying graph geometry. Then through an appropriate choice of the number of graph oversampling layers, we establish an $O(C_{\mathrm{po}})$ convergence independent of the local heterogeneity contrast. Notably, our scheme operates entirely within an algebraic framework, eliminating the need for Dirichlet nodes and positive-definiteness on specific matrices arising in the model. This flexibility enables the simulation of a wider range of physical models and accommodates various boundary conditions. Rigorous theoretical analyses are provided under suitable assumptions, and extensive numerical experiments validate the effectiveness of the proposed approach.
24. 线性最小二乘后向误差的一个变分方程与下界
A Variational Equation and Lower Bound for the Linear Least-Squares Backward Error
作者: Eric Hallman
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09211v1
链接: http://arxiv.org/abs/2605.09211v1
摘要: This paper derives a new variational equation for the linear least-squares backward error by expressing the backward error in terms of a generalized eigenvalue problem and using results from indefinite linear algebra. For problems with multiple right-hand sides, the variational equation also shows that the backward error can be decomposed as a sum of smaller backward error problems. Applications to stopping criteria for iterative methods are considered, and a new sketching-based lower bound is proposed which is provably of quality comparable to the sketched Karlson-Waldén estimate.
25. 一种用于主联合SVD型块对角化的NPDo方法
An NPDo Approach for Principal Joint SVD-type Block Diagonalization
作者: Ren-Cang Li, Li Wang, Mei Yang
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09202v1
链接: http://arxiv.org/abs/2605.09202v1
摘要: This paper is concerned with partial Joint SVD-type Block Diagonalization of several matrices so that the extracted diagonal parts collectively optimally assume part of the total mass of all given matrices. For that reason, it will be referred also as Principal Joint SVD-type Block Diagonalization. When each block-size is 1-by-1, it is about finding a dominant partial joint SVD decomposition for the matrices of interests. An NPDo approach is proposed for maximizing the common dominant block-diagonal parts collectively. It is shown that the NPDo approach combined with Gauss-Seidel-type updating is globally convergent to a stationary point while the objective increases monotonically. Numerical experiments are presented to illustrate the efficiency of the NPDo approach.
26. Rao-Blackwell 化的马尔可夫链蒙特卡罗光传输
Rao-Blackwellized Markov chain Monte Carlo Light Transport
作者: Sascha Holl, Gurprit Singh, Hans-Peter Seidel
类别: cs.CE, math.FA, math.NA, math.PR
发布日期: 2026-05-09
arXiv ID: 2605.09117v1
链接: http://arxiv.org/abs/2605.09117v1
摘要: In light transport simulation, Markov chain Monte Carlo methods are particularly effective at exploring regions with complex lighting characteristics. However, estimator variance is a central concern across Monte Carlo methods in general. In light transport, high variance directly manifests as increased noise or, equivalently, longer rendering times at fixed image quality. Variance reduction techniques based on Rao-Blackwellization have proven particularly effective. In practice, however, the RB approach traditionally used in light transport, waste-recycling, can yield little to no measurable variance reduction, a fact we empirically confirm in this work. Motivated by this lack of effective variance reduction, we introduce a novel RB technique for the general-purpose Metropolis-Hastings algorithm that is computationally efficient and achieves substantial variance reduction. We show that this method consistently outperforms waste-recycling in terms of both variance reduction and convergence speed. Building on this result, we adapt the proposed RB approach to the recently introduced general-purpose Jump Restore algorithm, where it similarly achieves substantial variance reduction and accelerated convergence. Through extensive experiments in light transport simulation, we demonstrate that our \gls{rb} technique significantly outperforms the traditional approaches for both MH-based light transport algorithms and Jump Restore Light Transport, under both equal-time and equal-sample-count comparisons.
27. 接触哈密顿系统的局部通用分裂积分器
Local Universal Splitting Integrators for Contact Hamiltonian Systems
作者: George A Kevrekidis
类别: math.DG, math.DS, math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09103v1
链接: http://arxiv.org/abs/2605.09103v1
摘要: Contact Hamiltonian systems extend symplectic Hamiltonian mechanics to dissipative settings while retaining geometric structure. We develop a structure-preserving splitting framework for contact Hamiltonian systems on $J^1(\mathbb{R}^n)$ based on two tractable classes of exact-contact subflows: strict contactomorphisms and prolonged diffeomorphisms. Our main theoretical result is that the Lie algebra generated by the corresponding strict and prolonged Hamiltonians contains all polynomial-in-$p$ Hamiltonians and is therefore dense, in the $C^r$ topology on compact sets, in the Lie algebra of smooth contact Hamiltonians. This yields a local universality result and contact splitting integrators built from exact strict and prolonged subflows. We then show how these subflows can be realized numerically by lifting symplectic integrators on $T^*\mathbb{R}^n$ and ODE integrators on $\mathbb{R}^n\times\mathbb{R}$. Finally, we illustrate the framework on a sequence of low-dimensional examples.
28. 非线性界面问题的标量界面约化
A scalar interface reduction for nonlinear interface problems
作者: So-Hsiang Chou
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09102v1
链接: http://arxiv.org/abs/2605.09102v1
摘要: We study finite element approximations of elliptic and parabolic interface problems with discontinuous coefficients and nonlinear jump conditions. We introduce a scalar interface reduction in which the solution is decomposed into a continuous component and a unit-jump response mode. This representation isolates the interface nonlinearity into a single scalar variable while the bulk problem remains linear. From this perspective, the nonlinear interface condition is reduced to a scalar nonlinear equation, which may be interpreted as a nonlinear Schur complement associated with the interface degree of freedom. The resulting formulation leads to a simple computational procedure consisting of linear solves combined with a low-dimensional nonlinear update. Numerical results for representative elliptic and parabolic problems confirm second-order accuracy for interface quantities and demonstrate the effectiveness of the proposed approach.
29. 物质点法的重叠施瓦茨时空细化框架
An Overlapping Schwarz Space-Time Refinement Framework for Material Point Method
作者: Zhaofeng Luo, Minchen Li, Yupeng Jiang
类别: cs.CE, math.NA, physics.comp-ph
发布日期: 2026-05-09
arXiv ID: 2605.09097v1
链接: http://arxiv.org/abs/2605.09097v1
摘要: We propose an overlapping Schwarz space-time refinement framework for the material point method (OS-MPM) to improve computational efficiency in problems with strongly localized deformation, contact, and large geometric nonlinearity. The method decomposes the domain into overlapping coarse and fine subdomains with heterogeneous spatial and temporal resolutions, while retaining standard MPM discretizations within each subdomain. Coarse-fine coupling is achieved through an MPM-specific Schwarz iteration combining mass-weighted spatial transmission and temporal interpolation for sub-cycling. In contrast to refinement strategies based on modified basis functions, transition kernels, or strongly enforced interface constraints, the proposed approach preserves the modular structure of standard MPM and shifts the coupling complexity to nonmatching-grid interface operators within the Schwarz alternating procedure. Numerical examples, including a gravity-driven cantilever beam, Hertzian contact, and an elastic inclusion problem, show that the method reproduces analytical or fine-resolution reference solutions with good accuracy and convergence behavior. In the inclusion benchmark, the proposed framework achieves comparable or slightly lower error than single-domain fine simulations at the finest tested resolutions, while reducing computational cost by up to 9.15 times. A three-dimensional folding example further demonstrates the generality of the framework. These results indicate that the proposed method provides an accurate, modular, and efficient route for local space-time refinement in MPM.
30. 从等距数据的局部 Legendre 框架逼近
Local Legendre Frame Approximation from Equispaced Data
作者: Benxue Gong, Zhenyu Zhao, Chenyang Wang
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09057v1
链接: http://arxiv.org/abs/2605.09057v1
摘要: We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence of the Runge phenomenon, the method partitions the interval into subintervals, maps each subinterval to a common reference interval, and computes local coefficients by a truncated singular value decomposition (TSVD) regularization. Since all subintervals share the same local sampling matrix, the method admits a natural offline–online implementation. We establish a quasi-optimal estimate for the regularized reconstruction and discuss practical parameter selection. Numerical results show that LLF attains high accuracy for relatively smooth and moderately oscillatory functions, while it remains applicable to highly oscillatory functions, although comparable accuracy generally requires more sampling points. For continuous piecewise smooth functions with derivative singularities, the method also provides an effective detect–localize–correct strategy based on one-sided coefficient-energy indicators. These results indicate that LLF provides a stable and flexible local approximation framework for equispaced data.
31. 具有常数扩散矩阵的时间依赖分数阶扩散方程
A time dependent fractional order diffusion equation with constant diffusivity matrix
作者: T. Catoe, V. J. Ervin
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09026v1
链接: http://arxiv.org/abs/2605.09026v1
摘要: Of primary interest in this paper is the numerical approximation of a time dependent fractional, in space, diffusion equation where the domain is assumed to be nonhomogeneous, having different axial diffusion coefficients. This work is motivated from the consideration of composite material which can exhibit different material properties along, and perpendicular to, internal planar structures. Careful attention is paid to accurately capture the boundary behavior of the solution. A spectral approximation scheme is used for the spatial discretization and a backward Euler approximation used for the temporal discretization. Following an error analysis for the approximation scheme, numerical experiments are given to demonstrate the effects of the nonhomogeneous domain and to support the theoretical analysis.
32. CATO：面向神经PDE算子的图表注意力
CATO: Charted Attention for Neural PDE Operators
作者: Chun-Wun Cheng, Sifan Wang, Carola-Bibiane Schönlieb, Angelica I. Aviles-Rivero
类别: cs.AI, cs.LG, math.NA
发布日期: 2026-05-09
arXiv ID: 2605.09016v1
链接: http://arxiv.org/abs/2605.09016v1
摘要: Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76% over the strongest competing baselines while reducing the number of parameters by 81.98%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.
33. Karhunen-Loève展开入门
A Primer on the Karhunen-Loève Expansion
作者: Alen Alexanderian
类别: math.NA, math.FA, math.PR
发布日期: 2026-05-09
arXiv ID: 2605.08959v1
链接: http://arxiv.org/abs/2605.08959v1
摘要: This article provides a primer on the spectral representation of random fields via the Karhunen-Loève Expansion (KLE). The goal is to bridge the gap between the theoretical foundations of the KLE and its application in computational modeling under uncertainty. We detail how tools from operator theory and probability are combined to analyze the convergence and optimality of the KLE. We also emphasize the associated computational and mathematical modeling considerations.
34. 扩散恢复：实时马尔可夫链蒙特卡洛光照传输
Diffusion Restore: Real-Time Markov Chain Monte Carlo Light Transport
作者: Sascha Holl, Gurprit Singh, Hans-Peter Seidel
类别: cs.CE, math.NA, math.PR
发布日期: 2026-05-09
arXiv ID: 2605.08916v1
链接: http://arxiv.org/abs/2605.08916v1
摘要: We present Diffusion Restore, a real-time framework for diffusion-based MCMC light transport. MCMC methods are highly suitable for sampling from complex high-dimensional distributions and for approximating integrals over them. In practice, they are often the only viable solution when direct sampling is not possible and alternative methods are either inefficient or cannot be applied due to the structure of the target distribution. However, controlling the exploration of the target distribution in MCMC methods remains challenging. Efficient exploration requires a balance between local exploration and global discovery, and local dynamics must rapidly explore individual modes without getting stuck or exhibiting excessive backtracking. The problem of global discovery has recently been addressed by the introduction of the Restore framework. In this work, we build on this framework and focus on improving local exploration. We show how to choose diffusion-based local dynamics within the Restore framework while completely avoiding Metropolis-adjustment, which is known to slow down convergence. Furthermore, we model these dynamics as nonreversible, introducing momentum in the drift and thereby enabling more directed exploration of the target distribution compared to reversible, random-walk-like dynamics. We provide a theoretical justification for the validity of our choice of local dynamics. Empirically, we demonstrate across diverse scenes that Diffusion Restore outperforms all existing MCMC light transport methods and establishes a new state of the art. In addition, we present a GPU implementation in ray tracing and compute shaders and achieve real-time frame rates. This demonstrates that Diffusion Restore is not only superior in offline rendering, but also outperforms traditional Path Tracing methods in real-time rendering settings, such as interactive applications and games.
35. 神经变分方法中解的非唯一性
Non-Uniqueness of Solutions in Neural Variational Methods
作者: Andreas Langer
类别: math.NA
发布日期: 2026-05-09
arXiv ID: 2605.08877v1
链接: http://arxiv.org/abs/2605.08877v1
摘要: Recent work has shown that strong-form physics-informed neural networks (PINNs) based on pointwise enforcement of differential operators can be ill-posed due to the combination of sufficiently expressive neural network trial spaces with finitely many measurements. In this work, we develop an abstract analytical framework that isolates this finite-information mechanism and extends its applicability beyond strong-form formulations. We apply the framework to three representative variational neural discretizations: the Deep Ritz method, neural network discretizations of variational regularization functionals, and weak PINNs. Despite their differing formulations, these methods constrain the neural trial function only through finitely many linear measurements, such as quadrature evaluations or finite-dimensional test spaces. We show that this structural feature leads to ill-posed discrete optimization problems, manifested by non-uniqueness or degeneracy of minimizers, independently of the well-posedness of the underlying continuous variational problem.
36. 签名核与施温格-戴森核方程作为双参数粗糙微分方程
Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations
作者: Thomas Cass, Dan Crisan, Andrea Iannucci, William F. Turner
类别: math.PR, math.NA
发布日期: 2026-05-09
arXiv ID: 2605.08844v1
链接: http://arxiv.org/abs/2605.08844v1
摘要: We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger–Dyson kernel equations. The theory is formulated in spaces of jointly controlled rough paths and is based on a robust two-parameter rough integration framework. In particular, we introduce a notion of rough integration over two-dimensional simplices at low regularity extending previous results in the literature. Within this setting, we show that the signature kernel equation arises naturally as a two-parameter rough differential equation and establish well-posedness and stability. We also extend the Schwinger–Dyson kernel equation, previously formulated for bounded-variation paths, to rough driving signals, proving existence and uniqueness in appropriate controlled rough path spaces. In the smooth rough path regime, we relate the resulting equations to PDE and integro-differential formulations. Finally, we derive and analyse a numerical scheme for the rough Schwinger–Dyson equation, including runtime and memory complexity estimates, and illustrate its performance with numerical experiments.
37. 核-晕分解：大规模不动点问题的去中心化
Core-Halo Decomposition: Decentralizing Large-Scale Fixed-Point Problems
作者: Haixiang, Yang Xu, Jiefu Zhang, Xudong Wu, Zihan Zhou, Jun He, Jiayu Chen
类别: stat.ML, cs.AI, cs.LG, math.NA
发布日期: 2026-05-09
arXiv ID: 2605.08681v1
链接: http://arxiv.org/abs/2605.08681v1
摘要: We study solving large-scale fixed-point equation (x^\star=\bar F(x^\star)) with decomposition. Standard strict decomposition assigns each agent a disjoint block and evaluates updates using only owned coordinates. For most operators, however, a block update may depend on variables outside the block. Truncating these dependencies by strict decomposition changes the mean operator and creates structural bias that cannot be removed by more samples, smaller stepsizes, or additional consensus. We therefore propose Core-Halo decomposition, which separates write ownership from read-only evaluation context: each agent updates its own core and reads from an overlapping halo. By aligning the Core-Halo decomposition with the block-dependence structure of $\bar F$, the original fixed-point problem can be implemented faithfully in a decentralized multi-agent system. We further characterize the fundamental obstruction faced by strict decomposition through a Bellman closure condition and a blockwise bias lower bound, showing that local-only updates can alter the original fixed-point operator. Finally, we conduct extensive experiments across a range of application settings, and demonstrate that Core-Halo achieves near-centralized performance while retaining the parallelism benefits of decentralization.
38. 贝叶斯物理信息神经网络求解椭圆型偏微分方程的后验集中性
Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs
作者: Yuxuan Zhao, Yulong Lu
类别: math.ST, math.NA, stat.ML
发布日期: 2026-05-09
arXiv ID: 2605.08672v1
链接: http://arxiv.org/abs/2605.08672v1
摘要: We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a strong solution in a Hölder space and using with a suitably constructed prior on the neural network weights, we prove that the posterior distribution concentrates around the exact solution at a near-minimax rate. Furthermore, the chosen prior is rate-adaptive: the posterior contracts at an (almost) optimal rate without prior knowledge of the smoothness level of the exact solution. Our results provide statistical guarantees for uncertainty quantification of PDEs via Bayesian PINNs.
39. Hilbert空间上凸Lipschitz泛函的有限样本保结构重构
Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples
作者: Anastasis Kratsios
类别: math.FA, cs.LG, cs.NE, math.NA, math.OC
发布日期: 2026-05-08
arXiv ID: 2605.08559v1
链接: http://arxiv.org/abs/2605.08559v1
摘要: Convex functionals are ubiquitous in applied analysis, appearing as value functions, risk measures, super-hedging prices, and loss functionals in machine learning. In many applications, however, the functional is only observed through finitely many exact pointwise evaluations. We ask whether a convex functional on a separable Hilbert space $H$ can be reconstructed, up to arbitrary uniform accuracy, by an explicit formula which preserves convexity and Lipschitz regularity and is finitely computable. We answer this affirmatively. For every compact convex $C\subseteq H$, every $L$-Lipschitz convex functional $ρ:C\to\mathbb{R}$, and every $\varepsilon>0$, we construct an explicit finite-sample reconstruction which is convex, $L$-Lipschitz, and uniformly $\varepsilon$-accurate on $C$. The construction uses only finitely many linear measurements $\langle b,\cdot\rangle_H$, with $b$ lying in a finite-dimensional subspace of $H$, and is exactly implementable by a $\operatorname{ReLU}$-MLP. Building on this, we introduce convex neural functionals (CNFs), a structured trainable architecture class containing our reconstruction, whose every admissible parameter configuration is automatically convex and Lipschitz, providing a principled foundation for learning convex functionals from finite data.
40. 一个基于偏微分方程的具有接触约束的材料参数识别问题
On a PDE-based material parameter identification problem with contact constraints
作者: Simon Hubmer, Stefan Kindermann, Ekaterina Sherina
类别: math.NA
发布日期: 2026-05-08
arXiv ID: 2605.08514v1
链接: http://arxiv.org/abs/2605.08514v1
摘要: We consider the identification of a scalar coefficient in a PDE-based parameter estimation problem with contact constraints. The considered problem can be used as an idealized model of a membrane under forces, constrained by a barrier or indenter. More generally, it serves as a benchmark for the analysis of more complex contact problems and the development of corresponding reconstruction algorithms. In this paper, we discuss both the forward and inverse parameter estimation problems, as well as uniqueness and non-uniqueness issues caused by the contact constraints. Furthermore, we consider the design and implementation of reconstruction approaches which we test on numerical examples, illustrating both uniqueness and non-uniqueness as well as parameter identifyability.
41. 随机化拟蒙特卡洛球面行走法
Randomized quasi-Monte Carlo for walk on spheres
作者: Valerie N. P. Ho, Art B. Owen
类别: math.NA, stat.CO
发布日期: 2026-05-08
arXiv ID: 2605.08483v1
链接: http://arxiv.org/abs/2605.08483v1
摘要: We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the integrands of interest are periodic indicator functions over regions $Θ$ in the torus $\mathbb{T}^k$. We give conditions for $\partialΘ$ to have $k-1$ dimensional Minkowski content which allows us to use results of He and Wang (2015). The RQMC estimates involve multiple values of $k$. We see sampling variances decreasing with the number $n$ of sample points at slightly better than Monte Carlo rates. The median variance rate in $4$ RQMC methods over $5$ worked examples, including some with $d=3$ and some with nonzero source functions, was slightly better than $O(n^{-1.1})$. The variance reduction factors ranged from $1.8$ to $10.7$ at $n=2^{17}$. None of the four RQMC methods dominated the others.
42. Transformer 可以为上下文内高斯核回归实现预处理 Richardson 迭代
Transformers Can Implement Preconditioned Richardson Iteration for In-Context Gaussian Kernel Regression
作者: Mingsong Yan, Dongyang Li, Charles Kulick, Sui Tang
类别: cs.LG, cs.AI, math.NA, math.OC
发布日期: 2026-05-08
arXiv ID: 2605.08475v1
链接: http://arxiv.org/abs/2605.08475v1
摘要: Mechanistic accounts of in-context learning (ICL) have identified iterative algorithms for linear regression and related linear prediction tasks, often using linear or ReLU attention variants. For nonlinear ICL, prior work has related softmax and kernelized attention to functional-gradient-type dynamics, but it remains unclear whether a standard transformer with softmax attention can implement a convergent solver with an end-to-end prediction-error guarantee. In this paper, we study in-context kernel ridge regression (KRR) with Gaussian kernels and show that a standard softmax-attention transformer can approximate the KRR predictor during its forward pass by implementing preconditioned Richardson iteration on the associated kernel linear system. Under bounded-data assumptions, we construct a single-head transformer with $O(\log(1/ε))$ blocks and MLP width $O(\sqrt{N/ε})$ that achieves $ε$-accurate prediction for prompts of length $N$. Our construction reveals a functional decomposition within the transformer architecture: softmax attention produces a row-normalized Gaussian-kernel operator needed for cross-token interactions, while ReLU MLP layers act locally to approximate the intra-token scalar arithmetic required by the update. Empirically, we train GPT-2-style transformers on Gaussian-process regression tasks to further test the preconditioned Richardson interpretation. Through linear probing, we compare the transformer’s layer-wise predictions with the step-wise outputs of classical KRR solvers and find that its error profiles align most consistently with preconditioned Richardson iteration. Ablation studies further support this interpretation. Together, our theory and experiments identify preconditioned Richardson iteration as a concrete mechanism that softmax-attention transformers can realize for nonlinear in-context Gaussian-kernel regression.
43. 基于核的线性系统辨识：增广Krylov子空间方法
Kernel-based linear system identification using augmented Krylov subspaces
作者: Fabio Matti, Martin Skovgaard Andersen, Tianshi Chen, Daniel Kressner
类别: math.NA
发布日期: 2026-05-08
arXiv ID: 2605.08362v1
链接: http://arxiv.org/abs/2605.08362v1
摘要: We propose a novel Krylov subspace method for estimating the finite impulse response (FIR) of a one-dimensional linear time-invariant systems. The method approximates the system’s FIR using a kernel-based formulation combined with hyperparameter selection based on maximum likelihood estimation (MLE), which requires repeated evaluation of two terms: The data fit $\boldsymbol{y}^{\top} (λ\boldsymbol{I} + \boldsymbol{A})^{-1} \boldsymbol{y}$ and the model complexity $\log(\det (λ\boldsymbol{I} + \boldsymbol{A}))$, where $\boldsymbol{A}$ is a certain positive semidefinite matrix that admits fast matrix–vector products and $λ> 0$ is a regularization parameter. Instead of approximating these two quantities separately, we jointly approximate them using a single augmented Krylov subspace for $\boldsymbol{A}$. One major benefit of augmentation is that we obtain accelerated convergence when approximating the data fit quadratic form, through implicit preconditioning. Thanks to the shift invariance of Krylov subspaces, the extracted approximations can be used to evaluate the MLE objective for many values of $λ$ at little additional cost. We derive error bounds for the approximations, reflecting the benefits of augmentation demonstrated through multiple numerical experiments.