生成时间: 2026年06月17日 07:40:13
重点关注论文数量: 1
了解领域论文数量: 4
不相关论文数量: 21
重点关注论文（完整分析）
1. INI-VPINN：一种用于具有几何奇性的多材料区域的隐式 Neumann 与界面处理的变分物理信息神经网络
INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
作者: Shayan Dodge, Alessandro Formisano, Sami Barmada
类别: math.NA, cs.LG, physics.comp-ph
发布日期: 2026-06-16
arXiv ID: 2606.18032v1
摘要翻译: 我们提出一种新的弱形式物理信息神经网络方法（名为 INI-VPINN）。INI-VPINN 自然地将 Neumann 边界条件和界面条件纳入变分公式，从而消除了对额外损失项或多个子域网络的需求。该框架采用紧支撑权函数和分部积分来隐式施加通量和连续性约束，由此隐式地确保了跨材料边界的物理一致性。在具有尖锐界面和复杂几何的 Poisson 和 Laplace 问题上对所提方法进行了测试。结果表明，与其它几种基于物理信息神经网络的公式相比，INI-VPINN 一致地实现了更高的精度、更平滑和更快的收敛。所提出的框架提供了一种使用神经网络求解具有复杂几何和混合 Neumann-Dirichlet 边界条件的多材料问题的通用方法。实现已公开在 GitHub 仓库中。
Comment: Preprint version. Under peer review. Code available at: https://github.com/ShayanDodge/INI-VPINN
链接: http://arxiv.org/abs/2606.18032v1
1. 研究对象和背景
物理信息神经网络 (PINN) 将偏微分方程 (PDE) 嵌入神经网络训练，已成为求解正、反问题的重要工具。然而，强形式 PINN 面临三大挑战：(i) 在多材料问题中难以满足界面条件，需要子域分解与额外损失项 (如 cPINN、I-PINN、X-PINN)；(ii) 损失函数中 PDE 残差、边界条件、初始条件、界面条件等多项之间存在量级差异，导致训练不稳定和收敛困难；(iii) 在几何奇异域 (如含凹角的 L 形域) 中，点采样和边界条件施加难度大。
为缓解上述问题，变分 PINN (VPINN) 通过弱形式代替强形式残差，降低了正则性要求并改善了数值稳定性。早期工作中，Kharazmi 等 (2019) 提出基于全局测试函数的 VPINN，Khodayi-Mehr 等 (2020) 的 VarNet 采用局部紧支撑函数，Zang 等 (2020) 的 WAN 利用对抗训练，Kharazmi 等 (2021) 的 hp-VPINN 引入区域分解与 hp 细化。然而，这些弱形式方法仍存在两个关键空白：(1) 诺伊曼边界条件的有效嵌入；(2) 多材料界面条件的本征满足。例如，hp-VPINN 无法隐式处理诺伊曼条件，仍需专门的边界损失项；cPINN 需要每个子域独立网络并通过惩罚项耦合界面。
本文研究的核心问题是：
$$ \begin{cases} -\nabla\cdot(\kappa(\mathbf{x})\nabla u(\mathbf{x})) = f(\mathbf{x}), & \forall \mathbf{x}\in\Omega\subset\mathbb{R}^2,\\ u = g_D, & \text{on }\Gamma_D,\\ \partial_n u = g_N, & \text{on }\Gamma_N, \end{cases} $$其中 $\kappa(\mathbf{x})>0$ 是空间变化的材料系数，允许跨界面 $\Gamma_I$ 间断。在多材料域中，界面条件为
$$ [u]_{\Gamma_I}=0,\quad [\kappa\partial_n u]_{\Gamma_I}=0. $$物理上，这类问题描述稳态热传导、静电势或地下渗流，广泛存在于复合材料和跨尺度问题中。数值求解的关键在于正确处理诺伊曼边界和材料间断，传统有限元方法需通过特定测试函数空间或 Nitsche 方法实现，而 PINN 类方法则需额外设计损失项或网络结构。
本文试图填补的空白是：在变分 PINN 框架下，仅通过测试函数的选择与弱形式的局部积分，隐式地施加诺伊曼边界条件和界面条件，无需专门的损失项或多网络结构。该框架称为 INI-VPINN，其核心思想是通过分部积分将诺伊曼条件自然融入变分残差，并利用间断 $\kappa$ 在弱形式中的正则性自动满足界面跳跃条件。与现有方法相比，它能用单一网络求解整个多材料域，且损失函数中仅包含弱形式残差和狄利克雷边界惩罚项，极大地简化了训练流程并增强了对几何奇异点和材料界面的适应性。
2. 主要定理或主要结果
本文未给出严格的数学定理，主要贡献为数值实验验证。在如下假设下展示了方法性能：(1) 物理问题为二维 Poisson 或 Laplace 方程，材料系数分片常数或光滑过渡；(2) 域边界为分段直线，界面为直线或圆弧；(3) 测试函数取自奇数余弦、奇数正弦和雅可比（勒让德多项式差）族，元素为矩形剖分。
主要数值结论包括：
- 在 L 形域 Laplace 问题（具有凹角奇异性）上，INI-VPINN 的相对 $L^2$ 误差比经典 PINN 低约一个数量级，比 VPINN（显式诺伊曼强制）和 cPINN 低约 50% 左右（具体数值文中表格给出）。诺伊曼边界处通量误差显著减小，且误差分布更均匀。
- 在 T 形域混合边界条件 Poisson 问题中，INI-VPINN 不需调整损失权重即可稳定收敛，而 cPINN 和标准 VPINN 需仔细调整界面损失权重和采样密度，否则出现发散或精度下降。
- 多材料域（嵌套方形和圆形夹杂）测试中，INI-VPINN 自动满足界面通量连续性，即使材料系数比 $\kappa_2/\kappa_1=10^3$，解在界面上无明显数值反常，而 cPINN 在界面附近出现虚假振荡，除非加入大量界面配置点并调试损失权重。
- 收敛性实验表明，随着测试函数阶数 $p$ 增大或元素数 $h$ 增加，INI-VPINN 的弱形式残差与误差均呈代数或指数下降，显示出 $hp$-细化能力。训练损失曲线更平滑，波动幅度远小于对比方法。
- 在一类带有紧支撑源的 Poisson 问题中，INI-VPINN 能在相对粗糙的求积规则下保持较高精度，而强形式 PINN 由于对奇性源项敏感，需要极高采样密度。
3. 研究方法、关键技术和核心工具
本工作的核心技术创新在于弱形式构造与测试函数的几何自适应选择，具体体现在以下几个方面：
1. 隐式诺伊曼条件嵌入
从强形式出发，乘以测试函数 $v$ 并在整个域 $\Omega$ 上积分，应用格林第一公式得到
若选取测试函数 $v$ 在 $\Gamma_D$ 上为零，则狄利克雷边界积分消失；而不强制 $v$ 在 $\Gamma_N$ 上为零，则诺伊曼条件 $\partial_n u = g_N$ 可代入边界积分，得到
$$ \int_{\Omega} \kappa\nabla u\cdot\nabla v\,d\Omega - \int_{\Omega} f v\,d\Omega = \int_{\Gamma_N} \kappa g_N v\,d\Sigma, \quad \forall v\in H_0^1_{\Gamma_D}(\Omega). $$该形式将 $\partial_n u$ 的计算替换为已知函数 $g_N$ 的积分，消除了对解函数的法向导数需求，且无需在损失函数中添加诺伊曼惩罚项。传统 VPINN 若不特殊处理，通常需要显式计算该项并加入损失。
2. 紧支撑测试函数与单元剖分
为获得局部适应性和高效积分，采用局部紧支撑测试函数，并对域进行矩形单元划分。每个单元 $V_k$ 上定义一组张量积测试函数：
其中一维测试函数族来自以下三种：
- 奇数余弦模式：$v^{(\cos)}_n(x) = \cos\left(\frac{(2n-1)\pi}{2}\frac{x-a}{b-a}\right)$，在左端点值为 $+1$，右端点为 $0$；
- 奇数正弦模式：$v^{(\sin)}_n(x) = (-1)^n\sin\left(\frac{(2n-1)\pi}{2}\frac{x-a}{b-a}\right)$，在左端点为 $0$，右端点为 $+1$；
- 雅可比模式（勒让德多项式差）：$v^{(\text{Jac})}_n(x) = P_{n+1}(x)-P_{n-1}(x)$，在两端点均为 $0$。
测试函数的选择规则直接基于单元边界类型：
- 内部单元（不接触外边界）：使用雅可比函数，因其在界面上为零，自动保证跨单元通量连续性的弱形式可加性；
- 接触诺伊曼边界的单元：在该边界方向选用在其上非零的奇数正弦或余弦（取决于边界位于哪一侧），使得诺伊曼条件自然嵌入；
- 狄利克雷边界方向：可选用雅可比函数或零函数，配合后续惩罚项。
在二维中，对每个单元独立选取 $x$ 和 $y$ 方向测试函数族，组合成张量积形式。图 3 展示了一个具有四边混合边界的示例：角部单元为 $C\times J$ 或 $S\times J$，内部为 $J\times J$。这种选择机制是 INI-VPINN 区别于其他变分 PINN 的关键。
3. 界面条件的隐式满足
对于材料间断，$\kappa$ 呈阶梯函数形式。强形式中 $\nabla\kappa$ 产生 Dirac 奇性，但弱形式不涉及 $\kappa$ 的导数，仅需计算 $\kappa$ 与 $\nabla u\cdot\nabla v$ 的乘积积分，因此即使 $\kappa$ 间断，积分仍良定。从物理角度看，弱形式自动保证了界面通量连续性 $[\kappa\partial_n u] = 0$（自然边界条件），无需显式施加。为实现自动微分和求积的稳定性，实际训练中采用 $\tanh$ 光滑化 $\kappa$：
其中小参数 $t$ 控制界面厚度。当 $t\to 0$ 时，光滑化模型收敛到尖锐界面问题。这使得自动微分梯度有界，且高斯-洛巴托-雅可比 (GLJ) 求积规则能精确捕获过渡层，避免了锯齿采样误差。
4. 损失函数构造与训练
每个单元 $k$ 对一组测试函数 $(n,m)$ 计算弱残差：
残差为 $R^{(k)}_{n,m} = U^{(k)}_{n,m} - F^{(k)}_{n,m}$。总弱形式损失为所有单元所有模式残差的 MSE：
$$ L_V = \frac{1}{N_x N_y N}\sum_{k=1}^{N}\sum_{n=1}^{N_x}\sum_{m=1}^{N_y} \big(R^{(k)}_{n,m}\big)^2. $$狄利克雷边界条件通过惩罚项 $L_D = \frac{1}{N_{D}}\sum_{\Gamma_D}(u-g_D)^2$ 施加。总损失为 $L = \lambda_D L_D + \lambda_V L_V$，其中权重 $\lambda_D,\lambda_V$ 预先设定（文中多取为 1）。
积分采用 GLJ 求积规则，阶数 $N_{\text{quad}}$ 可调，因其包含端点，利于捕捉边界贡献。通过以上设计，训练只需一个神经网络，无需子域分解或额外网络，且损失项数目固定，摆脱了繁琐的权重调整。
4. 与之前工作的比较
本文与多种弱形式及强形式 PINN 方法进行了系统对比，推进主要体现在以下维度：
1. 与 VPINN 及 hp-VPINN 的比较 (Kharazmi et al., 2019; 2021)
传统 VPINN 使用全局测试函数（如多项式或三角函数），适应性差，难以捕获局部奇异性。hp-VPINN 通过区域分解和局部测试函数改善了这一点，但二者均未能隐式处理诺伊曼边界：其弱形式仍包含诺伊曼边界积分项，需通过显式计算 $\partial_n u$ 并加入损失予以惩罚，等同于在损失函数中增加一项边界残差。INI-VPINN 通过精心选取测试函数，使诺伊曼条件直接转化为包含已知 $g_N$ 的边界积分，损失中不含未知函数的法向导数项，从而避免了额外的损失分量与权重调节。此外，hp-VPINN 在界面处一般需要显式约束或惩罚，而 INI-VPINN 依赖弱形式的自然连续性，仅通过单一网络即可保证界面条件近似满足。
2. 与 cPINN 的比较 (Jagtap et al., 2020)
cPINN 是处理多材料问题的代表性方法，其思路是将域按材料分区，为每个子域分配独立网络，并在界面处加入通量连续性损失和函数连续性损失。这导致参数数量随材料数线性增加，界面损失需精细调整权重和采样密度，且多个网络的训练难以同步收敛。INI-VPINN 使用单一网络覆盖全域，参数规模与单材料问题相当，无需界面损失项，因此在计算效率和训练稳定性上优势明显。数值结果显示，cPINN 在界面附近易产生数值振荡，除非增加大量界面配置点并精心调整损失权重；INI-VPINN 即使采用少量界面求积点也能获得光滑解。
3. 与 DEM 的比较 (Samaniego et al., 2020)
DEM 直接最小化能量泛函，自动满足自然边界条件，但要求问题具有能量形式且材料性质满足一定对称性，对于非线性问题或强制边界难以处理。INI-VPINN 不依赖能量形式的存在，适用于任何可写为弱形式的线性或非线性椭圆问题，且同样能自然施加诺伊曼条件。
4. 与 WAN 的比较 (Zang et al., 2020)
WAN 采用对抗训练，求解一个极小极大问题来寻找最优测试函数，计算代价高且收敛不稳定。INI-VPINN 采用固定、预设的测试函数族，免去对抗训练开销，并依据边界条件与界面位置进行确定性选择，更简单可解释。
5. 与 VarNet 及 RVPINN 的比较 (Khodayi-Mehr and Zavlanos, 2020; Rojas et al., 2024)
VarNet 也使用紧支撑 FE 型测试函数，但测试函数只考虑局部多项式基，未根据边界类型进行差异化选取，其精度依赖于支撑域大小，且缺乏系统化选择规则。RVPINN 通过最小残差原理优化测试函数，提高了鲁棒性，但计算代价更高（需计算函数评价泛函）。INI-VPINN 通过简单的几何驱动规则确定测试函数类型，在计算效率与精度之间取得平衡。
6. 通用性与局限性
INI-VPINN 的普遍性表现在：(a) 适用于任意混合边界条件的 Poisson/Laplace 问题，可推广至一般二阶椭圆方程；(b) 多材料界面可自由形状，只要适当定义光滑 $\kappa$ 并划分单元；(c) 具备 $hp$-细化能力，可通过增加单元数 (h) 或测试函数阶数 (p) 提高精度。
局限性：(1) 在 3D 问题中，张量积测试函数导致求积点数量为 $N_{\text{quad}}^{3}$，计算负担大幅增加，文中指出 3D 扩展仍需进一步研究；(2) 单元划分目前限于矩形（二维）或长方体（三维），对复杂非结构化域适应能力有限，可能需要与切割有限元等方法结合；(3) 测试函数选择依赖于对边界和界面位置的先验知识，当界面移动或材料分布未知时，需要自适应策略。
总体而言，INI-VPINN 在保持弱形式 PINN 优点的同时，通过巧妙设计测试函数选择规则，显著减少了损失项数量与人工调参需求，为弱形式 PINN 在多材料与复杂边界问题中的应用提供了更为简洁高效的方案。
了解领域论文（摘要翻译）
1. 具有容许切向运动的几何梯度流的极小化运动框架
A minimizing-movement framework for geometric gradient flows with admissible tangential motion
作者: Xiaoxiao Liu, Quan Zhao
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18177v1
链接: http://arxiv.org/abs/2606.18177v1
摘要翻译
中文标题: 具有容许切向运动的几何梯度流的极小化运动框架
摘要翻译: 我们为具有容许切向运动的几何梯度流的参数有限元逼近发展了一个极小化运动框架。在每一时间步，离散变分问题将法向位移的度量耗散项与曲面Dirichlet能量相结合。该度量决定了法向几何演化：$L^2(Γ)$ 度量给出平均曲率流，而 $H^{-1}(Γ)$ 度量给出表面扩散流。切向速度通过对变形映射的弱约束独立选取。核心的结构条件是容许性，即恒等映射满足该约束条件。该条件使得恒等映射可作为比较函数，并导出自然的稳定性估计。该框架从无约束形式复原了经典的 Barrett–Garcke–Nürnberg (BGN) 格式，并从 MDR 约束复原了对偶的最小变形率 (MDR) 格式。我们进一步引入了两种新的容许变体：一种容许 BGN 格式和一种松弛 MDR 格式。对于得到的全离散格式，我们在自然的非退化假设下证明了存在唯一性，并建立了无条件能量稳定性。数值实验比较了容许格式与经典格式，并展示了它们的稳定性性质和网格质量表现。
2. 基于质量集中与变换光滑子的有限元外微积分多重网格预处理
Multigrid Preconditioning for FEEC using Mass-Lumping and Transforming Smoothers
作者: Radovan Dabetić
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17934v1
链接: http://arxiv.org/abs/2606.17934v1
摘要翻译
中文标题: 基于质量集中与变换光滑子的有限元外微积分多重网格预处理
摘要翻译: 对于自然置于de Rham复形中的偏微分方程，保结构的混合和鞍点有限元离散通常会生成不定线性系统。我们提出了一种多重网格预处理框架，将质量集中（可显式求逆）的FEEC质量矩阵与变换光滑子相结合，后者将算子映射为具有正定对角块的分块形式，从而可在变换后的系统上执行Gauss-Seidel型松弛。在温和的h一致范数等价假设下（且对于平凡拓扑），我们证明了质量集中系统的稳定性，进而得到质量集中与原始FEEC算子之间的谱等价性，这促使我们使用为质量集中算子设计的多重网格循环作为相容FEEC系统的预处理器。虽然我们的主要焦点在于算法设计而非形式收敛理论，但对Hodge-Dirac算子、混合Hodge-Laplacian以及二维和三维静磁学鞍点系统的大量数值实验证明了该方法的鲁棒性。
3. 用于双调和Poisson-Nernst-Planck Navier-Stokes系统的协调虚拟元方法
Conforming Virtual Element Method for Biharmonic Poisson-Nernst-Planck Navier-Stokes systems
作者: Ankur Ankur, Andrea Cangiani
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17918v1
链接: http://arxiv.org/abs/2606.17918v1
摘要翻译
中文标题: 用于双调和Poisson-Nernst-Planck Navier-Stokes系统的协调虚拟元方法
摘要翻译: 我们针对四阶Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) 系统提出并分析了一种协调虚拟元方法（VEM）。所提出的格式基于各组分的相容离散：对四阶静电势方程采用 (H^2)-协调 VEM，对 Nernst-Planck 方程采用 (H^1)-协调 VEM，对 Navier-Stokes 方程采用\textit{无散度}且\textit{压力鲁棒}的 VEM。时间积分采用向后 Euler 格式以确保稳定性。我们证明了连续问题在三维情形下的适定性，并通过不动点论证建立了全离散解的存在唯一性。此外，我们推导了先验误差估计，表明静电势、浓度和速度分别在 Bochner 范数 (L^\infty\bigl(0,T; H^2(Ω)\bigr))、(L^2\bigl(0,T; H^1(Ω)\bigr)) 和 (L^2\bigl(0,T; \bm{H}^1(Ω)\bigr)) 下具有最优收敛性。该分析需要精细的论证，以避免对强制性和连续性常数的任何限制性假设，并处理涉及三个不同变量的三线性形式。压力鲁棒的设计允许使用最低阶压力逼近，而不会影响其他变量的收敛速度，从而降低计算成本。数值实验证实了不同多项式阶数下的理论收敛速度，并展示了格式的鲁棒性，包括在低黏度情形下。
4. 一种用于四旋度问题的IP DG方法的收敛性
Convergence of an IP DG Method for the Quad-Curl Problem
作者: Xianhao Zeng
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17428v1
链接: http://arxiv.org/abs/2606.17428v1
摘要翻译
中文标题: 一种用于四旋度问题的IP DG方法的收敛性
摘要翻译: 本文分析并修正了 [Chen, G., Qiu, W., & Xu, L. (2021). Analysis of an interior penalty DG method for the quad-curl problem. IMA Journal of Numerical Analysis, 41(4), 2990-3023.] 中针对四旋度问题所施加的内罚(IP)间断伽辽金(DG)方法，在非凸多面体区域中引入了一个分片常数系数矩阵。我们得到两个主要结果：在最小正则性假设下，我们证明数值解在 $ H(\text{curl}) \times H^1(Ω)$ 范数下强收敛到真解。在稍高的正则性下，我们建立了依赖于解的正则性的收敛速度的最优估计。这两个结果作为对现有文献的补充，完整回答了所关注的IP DG方法在低正则性四旋度问题上的表现。
不相关论文（基本信息）
1. LGNO: 一种用于双曲守恒律的局部-全局神经算子
LGNO: A Local-Global Neural Operator for Hyperbolic Conservation Laws
作者: Hao Wang, Chi-Wang Shu, Qi Tang
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18221v1
链接: http://arxiv.org/abs/2606.18221v1
摘要: Solutions of hyperbolic conservation laws exhibit both smooth structures across large scales and sharp localized features such as shocks and contact discontinuities, making them difficult to approximate accurately with existing neural operators. The Fourier Neural Operator (FNO) captures long-range interactions well but tends to smear localized structures through excessive numerical dissipation. To address this, we propose a Local-Global Neural Operator (LGNO) that learns a one-step discrete flow map by combining a global FNO branch for representing smooth dynamics at large scales with a local multiresolution branch for enhancing localized discontinuities and nonsmooth features. The model is trained with a one-step loss that combines a physical space prediction term and a spectral penalty on high frequencies to suppress spurious oscillations near steep fronts. On a large collection of benchmarks in one and two dimensions, LGNO consistently outperforms FNO baselines with matched parameter counts, reducing one-step errors by factors of 2-5 and remaining significantly more accurate over long autoregressive rollouts. Most strikingly, although it is trained only on short-time data from a high-order WENO-Z scheme, the long-time rollout of LGNO on a coarse $256^2$ grid exhibits lower numerical dissipation than the same WENO-Z scheme run on a finer $512^2$ grid, while being orders of magnitude cheaper to evaluate. These results suggest that, with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps. They further suggest that such learned operators have the potential to control long-time numerical dissipation better than the conventional shock-capturing schemes that generate the training data.
2. 一种用于审计学习的偏微分方程模拟器的诊断软件套件
A Diagnostic Software Suite for Auditing Learned PDE Simulators
作者: Lennon J. Shikhman
类别: cs.MS, math.NA, physics.comp-ph
发布日期: 2026-06-16
arXiv ID: 2606.18200v1
链接: http://arxiv.org/abs/2606.18200v1
摘要: Learned PDE simulators are increasingly used as low-cost replacements for expensive numerical solvers, but standard relative $L^2$ error does not determine whether a learned model behaves as a coherent numerical time propagator. This paper presents a diagnostic software suite for auditing learned PDE simulators as approximate evolution operators. The suite provides architecture-independent, post hoc diagnostics for relative state error, semigroup consistency, finite-difference generator discrepancy, energy behavior, integral balance, admissibility constraints, perturbation response, and scaling-law consistency. The software is designed around a minimal contract: reference trajectories, a learned propagator or saved predictions, equation metadata, and a diagnostic configuration specifying which structures are meaningful for the problem under study. We validate the suite on five benchmark PDE tasks: two-dimensional incompressible Navier-Stokes, shallow-water dynamics, active matter, three-dimensional compressible Navier-Stokes, and three-dimensional magnetohydrodynamics, using FNO, DeepONet, U-Net, and ResNet-style surrogate models together with controlled underfit and oversmoothed variants. The validation study shows that relative $L^2$ error can remain moderate, or even improve, while structural diagnostics deteriorate substantially. The package therefore supports software-level auditing of learned PDE simulators by reporting an interpretable diagnostic panel rather than collapsing model behavior into a single state-error score.
3. 一种用于识别矩阵若尔当结构的编码器-变换器架构
An Encoder-Transformer Architecture for Recognition of the Jordan Structure of a Matrix
作者: Michał Trojanowski, Michał Wojtylak
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18185v1
链接: http://arxiv.org/abs/2606.18185v1
摘要: We propose a machine-learning framework for detecting whether a given matrix is a perturbation of a matrix with a large Jordan block. The proposed model achieves high classification accuracy on synthetically generated, robustly perturbed data and outperforms a classical numerical baseline. Moreover, we demonstrate that the learned model generalizes to several classes of matrices not seen during training. These results suggest that the architecture captures structural properties associated with matrix defectiveness.
4. 一种用物理信息神经网络求解非线性偏微分方程的凸拟线性化方法
A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks
作者: Gbenga T. Awojinrin, Abdul-Akeem Olawoyin, Rami M. Younis
类别: math.NA, cs.LG, physics.comp-ph
发布日期: 2026-06-16
arXiv ID: 2606.18175v1
链接: http://arxiv.org/abs/2606.18175v1
摘要: We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.
5. 一种在Loewner序下精确计算最小上界的算法
An algorithm to exactly compute minimal upper bounds in the Loewner order
作者: Adam Humeniuk, Gabriel Jarry-Bolduc, Patrick Pascua, Nejaunie Williams
类别: math.FA, math.NA, math.OC
发布日期: 2026-06-16
arXiv ID: 2606.18173v1
链接: http://arxiv.org/abs/2606.18173v1
摘要: The Loewner order on Hermitian matrices is a partial order that compares matrices in terms of positive semidefiniteness. The Loewner order plays a key role in many fields such as optimization, numerical linear algebra, control theory, operator theory, and quantum information. A fundamental difficulty is that two or more Hermitian matrices do not necessarily have a unique minimal upper bound (or maximal lower bound). In this paper, we propose an iterative method to exactly compute a minimal upper bound for any finite collection of $n\times n$ Hermitian matrices. It is shown that the algorithm terminates in at most $n$ iterations. The exactitude of the algorithm is proved using standard results from finite-dimensional linear algebra. A self-contained proof of an algebraic characterization of minimality originally explored by Stott is provided. We illustrate the algorithm in examples and also provide an implementation of the algorithm in Python.
6. Chebfun编程。案例研究：Richards方程
Programming with Chebfun. Case study: Richards equation
作者: Nicolae Suciu
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18118v1
链接: http://arxiv.org/abs/2606.18118v1
摘要: The Chebfun software system is a Matlab extension essentially based on representations of (piece-wise) smooth one-variable functions by expansions in Chebyshev polynomials. One of Chebfun’s attractive features is the ability to provide solutions to nonlinear boundary value problems (BVP) with accuracy close to the machine precision. This is done by the chebop class which provides automatic solutions by performing linearizations with a Newton method in function spaces of the nonlinear BVP, automatic differentiation, and using Fast Fourier Transform computations for the coefficients of the Chebyshev polynomials. A drawback of chebop automatic approach is the possible lack of convergence of the Newton method if the initial guess is not close enough to the exact solution. An explicit functional linearization done for each particular shape of the differential operator (i.e. without automatic differentiation) proves to be more robust than the chebop class and allows an enlargement of the range of convergence. Another alternative is the implicit L-scheme (quasi-Newton approach with derivatives replaced by suitable positive constants L), with a much simpler implementation and globally convergent. While chebop is the easiest way to solve the BVP, provided that it converges, the last two approaches largely overcome the convergence issues, yielding accurate solutions to a wide class of steady-state one-dimensional problems governed by Richards’ equation. Chebfun2 and Chebfun3, which at the current stage cannot solve BVPs, provide efficient tools for accuracy and convergence assessments of the non-steady solutions in one or two spatial dimensions obtained by classical discretization schemes.
7. 一种用于大规模正则化的草图广义Krylov子空间方法
A Sketched Generalized Krylov Subspace Method for Large-Scale Regularization
作者: Davide Palitta, Mirjeta Pasha
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18073v1
链接: http://arxiv.org/abs/2606.18073v1
摘要: The generalized Krylov subspace (GKS) method is an effective projection technique for large-scale Tikhonov regularization with a general regularization matrix. As the subspace expands, however, two computational bottlenecks limit scalability: the thin QR factorizations of the tall projected matrices formed by the forward operator and the regularization matrix applied to the basis, and the full reorthogonalization of each new basis vector against all previous columns. We propose a sketched variant, named sGKS, that addresses both bottlenecks. The QR factorizations are performed on compressed matrices of much smaller row dimension, maintained incrementally via rank-one updates. Moreover, we observe that explicit reorthogonalization can be skipped entirely without compromising the quality of the approximation subspace, since no step of GKS relies intrinsically on the orthogonality of the basis. The resulting algorithm is independent of the choice of sketching operator and preserves the approximation quality of the original method: we show that, in the absence of sketching in the projected solve, sGKS produces iterates identical to those of standard GKS, and that the sketched projected solve delivers quasi-optimal residual norms controlled by the embedding quality. For more challenging problems where the loss of basis orthogonality becomes significant, we show that incorporating a small number of iterative refinement steps in the projected solve restores the spectral properties of the basis and recovers the full accuracy of the unsketched method. Numerical experiments on image deblurring, X-ray computerized tomography, seismic travel-time tomography, and dynamic computerized tomography demonstrate that sGKS matches the reconstruction quality of standard GKS while significantly reducing per-iteration costs and overall wall-clock time.
8. 可压缩流动的四阶守恒自适应多分辨率小波迎风格式
A Fourth-order Conservative Adaptive Multiresolution Wavelet Upwind Scheme for Compressible Flows
作者: Bing Yang, Xiaojing Liu, Youhe Zhou, Feng Xiao, Jizeng Wang
类别: math.NA, physics.comp-ph
发布日期: 2026-06-16
arXiv ID: 2606.18026v1
链接: http://arxiv.org/abs/2606.18026v1
摘要: A fourth-order conservative adaptive multiresolution average-interpolating wavelet upwind scheme is proposed for compressible flows governed by hyperbolic conservation laws. A family of asymmetric average-interpolating wavelets with upwind properties is constructed for conservative finite volume discretization, while symmetric average-interpolating wavelets are employed for multiresolution decomposition and reconstruction of physical variables in the adaptive procedure. Since both the conservative discretization and the adaptive multiresolution representation are constructed from cell-average quantities, the proposed scheme preserves strict conservation during both numerical evolution and adaptive cell redistribution. Unlike hybrid adaptive wavelet methods that use wavelets mainly for data compression and mesh adaptation, the present adaptive wavelet upwind scheme utilizes average-interpolating wavelet multiresolution approximation to reconstruct the interface values directly for numerical flux evaluation, thereby avoiding additional ghost-cell marking and reconstruction near coarse–fine mesh interfaces. The boundary variation diminishing reconstruction is incorporated at the finest resolution level to achieve non-oscillatory shock-capturing capability. Numerical tests demonstrate that the proposed scheme achieves the expected fourth-order accuracy, maintains conservation errors close to machine precision, and controls numerical errors around the prescribed threshold. The proposed method also sharply captures shock waves and contact discontinuities without spurious oscillations and resolves multiscale smooth structures through a sparse adaptive representation. These results indicate that the proposed scheme provides an efficient, conservative, and reliable approach for high-resolution simulations of compressible flows.
9. 反高斯拉格朗日插值：克里斯托费尔-达布形式、重心表示与正交展开
Anti-Gauss Lagrange interpolation: Christoffel-Darboux form, barycentric representation, and orthogonal expansion
作者: Patricia Díaz de Alba, Luisa Fermo, Valerio Loi
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.18018v1
链接: http://arxiv.org/abs/2606.18018v1
摘要: The paper deals with new formulations of a Lagrange interpolant polynomial based on the nodes of the well-known anti-Gauss rule. A first representation is given in terms of the classical Christoffel-Darboux kernel appropriately modified. The second one closely follows the barycentric form of the classical Lagrange polynomial, while the third formulation represents the interpolant as a combination of an orthonormal family of polynomials with respect to the discrete anti-Gauss inner product. A numerical test shows the performance of the explored forms.
10. 用于两层薄膜模型的基于黎曼不变量的替代 WENO 格式
Riemann invariant-based alternative WENO scheme for a two-layer thin film model
作者: Biswarup Biswas, Rahul Barthwal, Rakesh Kumar
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17862v1
链接: http://arxiv.org/abs/2606.17862v1
摘要: In this article, we develop a multi-dimensional two-layer thin film model extending the thin film model proposed in \cite{barthwal2025hyperbolic}. The model considered in \cite{barthwal2025hyperbolic} considered a very specific Marangoni scale by choosing Marangoni numbers in both layers to be $1$. We relax this condition here and prove that the obtained system possesses a full set of Riemann invariants. Based on these findings, we develop a Riemann Invariant-based Local Characteristic Decomposition WENO (RI-WENO) method for the two-layer thin film model in one and two dimensions. The method is built upon a specially designed variable transformation constructed from the derived Riemann invariants of the system. This transformation partially diagonalizes the governing equations and yields a sparse structure in the transformed eigenvector matrices. As a result, the proposed RI-WENO framework significantly reduces the computational cost of the standard Local Characteristic Decomposition WENO approach while retaining its strong capability to suppress spurious oscillations. Numerical experiments, including new benchmark test cases, demonstrate that the RI-WENO method achieves an effective balance between accuracy and computational efficiency, making it a promising and practical choice for solving the two-layer thin film model.
11. 有限差分实现的高阶时空耦合紧致气体动理学格式
Finite Difference Implementation of a High-order Space-Time Coupled Compact Gas-Kinetic Scheme
作者: Fengxiang Zhao, Yibing Chen, Kun Xu
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17772v1
链接: http://arxiv.org/abs/2606.17772v1
摘要: This study presents a high-order compact finite difference gas-kinetic scheme (FD-CGKS) that introduces a novel spatial discretization strategy for the efficient implementation of space-time coupled high-order schemes on structured grids. A conservative nonlinear compact discretization is achieved by formulating numerical fluxes from physical fluxes at both nodal and interfacial locations. To simplify the multidimensional spatial reconstruction required for the GKS flux evaluation, we propose a dual-grid approach that updates conservative variables on both a primary grid and an identical dual grid, offset by half the mesh spacing. By leveraging the time-accurate interface solutions from the gas-kinetic evolution model, the scheme explicitly updates averaged spatial derivatives between virtual interfaces, naturally enabling compact high-order reconstruction. Furthermore, a nonlinear GENO method is incorporated to capture flow discontinuities with high resolution and robustness, effectively suppressing spurious oscillations. The proposed framework, which also offers new perspectives for designing schemes based on space-time decoupled Riemann solvers, is systematically validated. Comprehensive benchmark computations of inviscid and viscous flows demonstrate the scheme’s high accuracy in resolving a wide spectrum of flow features, from smooth multiscale structures to strong shock discontinuities.
12. 时间积分作为滤波：一种时空离散感知的LES格式
Time integration as filtering: a space-time discretization-aware LES formulation
作者: Syver Døving Agdestein
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17759v1
链接: http://arxiv.org/abs/2606.17759v1
摘要: Discretization-aware LES yields an exact expression for the discrete target flux in finite-volume LES by recognizing that a coarse finite difference is a top-hat-filtered exact derivative (the “filter-swap” property). That argument is purely spatial; here we observe that the forward-Euler time difference is itself a (one-sided) top-hat-filtered exact time derivative, and repeat the construction in space-time. The resulting exact discrete flux decomposition extends the spatial one with a single temporal term: a flux-quadrature error that shrinks with the quadrature order of the time integrator. In a Burgers experiment this term grows with the CFL number while the spatial terms do not, and a Smagorinsky closure augmented with its leading order - a Lax-Wendroff-type diffusion - stays accurate at coarse time steps where space-only closures degrade.
13. 基于Lyapunov框架的振动弦点阻尼器最优布置与调谐
Optimal placement and tuning of pointwise dampers for vibrating strings via a Lyapunov framework
作者: Josip Tambača, Ninoslav Truhar
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17748v1
链接: http://arxiv.org/abs/2606.17748v1
摘要: We study the optimal placement and tuning of a small number of pointwise viscous dampers for a vibrating string. Starting from a finite element discretization of the damped wave equation, the system is transformed into a first-order phase-space formulation, which enables a unified Lyapunov trace framework. Three optimization criteria are considered: average total energy, average total displacement, and energy for a fixed initial state. For all criteria, explicit gradient formulas with respect to damper positions and viscosities are derived, requiring only one primal and one dual Lyapunov solve. Due to the strong non-convexity of the problem, a simple heuristic based on an explicit single-damper formula is proposed to generate effective initial guesses. Numerical examples illustrate the influence of spectral selection and discretization on the optimal damping configuration.
14. 关于大规模非自治微分Riccati方程求解的数值研究
On the Solution of Large-scale Non-autonomous Differential Riccati Equations: a Numerical Study
作者: Eugenio Mancinelli, Davide Palitta, Jens Saak
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17744v1
链接: http://arxiv.org/abs/2606.17744v1
摘要: We explore the numerical solution of large-scale non-autonomous Differential Riccati Equations (DREs). While we assume to discretize the differential operator using a Backward Differentiation Formula (BDF) of order s, we solve the generalized Algebraic Riccati Equation (gARE) resulting at each time step by different state-of-the-art methods. In particular, we compare the performance of the inexact Newton- Kleinman method with line search and the low-rank RADI iteration, considering for both methods two different initialization strategies: zero initialization and warm-start. A comprehensive panel of numerical results illustrate the potential and limitations of these methods when employed within a numerical pipeline for the solution of DREs, rather than for the isolated solution of a single gARE, as commonly considered in the existing literature.
15. 当粗糙数据有益时：积分算子核恢复中收敛速度的相变
When Rough Data Helps: A Phase Transition in Convergence Rates for Kernel Recovery in Integral Operators
作者: Jihong Wang, Fei Lu, Yue Yu
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17672v1
链接: http://arxiv.org/abs/2606.17672v1
摘要: Learning kernels in operators from data is a fundamental task that arises in nonlocal continuum mechanics, operator learning, and interacting particle systems. A central question is how the roughness of input data impacts the accuracy of kernel recovery. We quantify the roughness of the input data via its spectral decay exponent and analyze how it determines the degree of ill-posedness of the inverse problem and, consequently, the convergence rates of the Tikhonov-regularized estimator in the small-noise limit. Within this framework, we identify a phase transition between an under-rough regime, in which rougher data improves recovery, and an over-rough regime, in which further roughening leads to slower rates. These theoretical findings are supported by numerical experiments ranging from idealized settings to more realistic configurations, with quantitative agreement in the former and broad consistency of the main trends in the latter.
16. 用于求解线性双曲守恒律的稳定且可推广的数据驱动离散格式的设计原则
Design principles for stable and generalizable data-driven discretizations for solving linear hyperbolic conservation laws
作者: Antoine-Alexis Nasser, Alistair Adcroft
类别: math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17497v1
链接: http://arxiv.org/abs/2606.17497v1
摘要: We investigate data-driven finite-volume discretizations of the linear advection equation in one dimension. Neural networks for use as numerical advection schemes are constructed adhering to first principles of numerical analysis, allowing us to examine how normalization, training data, and architectural choices influence stability, accuracy, and shape preservation. (i) We show that reconstruction based solely on cell averages leads to a multi-valued learning problem, explaining limited generalization when training data includes widely different curvature regimes. (ii) Numerical stability and good generalization can be achieved by enforcing semilinearity (Lin and Rood 1998) through local stencil-scale normalization, which ensures invariance under affine transformations of the inputs. (iii) A new data-driven flux limiter is introduced that outperforms the classical ‘OSTVD3’ (Arora and Roe, 1997) scheme in shape preservation by introducing mild antidiffusion in near-linear regimes, while higher-order reconstruction in non-monotonic regions provides limited benefit. (iv) We show that training on polynomial profiles yields stable, high-order accurate discretizations, with the polynomial degree controlling the formal order of accuracy. Together, these results illustrate how the representational, architectural, and training choices govern the stability and generalization of data-driven finite-volume schemes for linear advection.
17. 算子Boosting产生帕累托有效的PDE代理模型
Operator Boosting Produces Pareto-Efficient PDE Surrogates
作者: Lennon J. Shikhman
类别: cs.LG, math.NA, physics.comp-ph
发布日期: 2026-06-16
arXiv ID: 2606.17460v1
链接: http://arxiv.org/abs/2606.17460v1
摘要: Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.
18. Sobolev空间中多输入神经算子学习的泛化保证
Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces
作者: Yahong Yang, Zecheng Zhang, Wei Zhu, Wenjing Liao, Hao Liu
类别: cs.LG, math.NA
发布日期: 2026-06-16
arXiv ID: 2606.17419v1
链接: http://arxiv.org/abs/2606.17419v1
摘要: We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a (\log\log/\log)-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.
19. 流体-多孔弹性结构相互作用的Stokes-Biot模型的区域分解方法
Domain decomposition methods for the Stokes-Biot model of fluid-poroelastic structure interaction
作者: Manraj Ghumman, Manu Jayadharan, Ivan Yotov
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.17363v1
链接: http://arxiv.org/abs/2606.17363v1
摘要: We develop a non-overlapping domain decomposition method for the numerical solution of the Stokes-Biot model of fluid-poroelastic structure interaction in a mixed form. The model is based on a velocity-pressure formulation for the free fluid, a three-field stress-displacement-rotation formulation with weakly symmetric stress for the solid deformation, and a Darcy velocity-pressure formulation for the fluid in the poroelastic media. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman slip with friction condition are imposed on the interface. The interface conditions are incorporated through Lagrange multipliers modeling the traces of the displacement and the Darcy pressure. The system is discretized using stable mixed finite element spaces for Stokes flow, elasticity, and Darcy flow. The domain is decomposed into a union of subdomains of either Stokes or Biot type with three types of interfaces: Stokes-Stokes, Biot-Biot, and Stokes-Biot. On the Stokes-Stokes interfaces, a normal stress Lagrange multiplier is introduced to impose weakly velocity continuity, while the Biot-Biot and Stokes-Biot interfaces are equipped with displacement and pressure Lagrange multipliers to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced via Schur complement to an interface problem for the Lagrange multipliers, which is solved by GMRES. Each iteration requires the solution of local Stokes or Biot problems, which can be performed in parallel. We show that the resulting interface operator is positive definite and analyze the convergence of the GMRES iteration through fields-of-value analysis. Numerical experiments are presented to illustrate the performance of the method.
20. 贝叶斯泊松-随机伽马张量分解及其在国际贸易流中的应用
Bayesian Poisson-Randomized Gamma Tensor Factorization with Application to International Trade Flows
作者: Jie Jian, Aaron Schein
类别: stat.ME, econ.EM, math.NA, stat.AP, stat.ML
发布日期: 2026-06-15
arXiv ID: 2606.17267v1
链接: http://arxiv.org/abs/2606.17267v1
摘要: We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter–importer–product–year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational–Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.
21. 面向Krylov方法的混合数字-模拟近似逆预处理
Hybrid Digital-Analog Approximate Inverse Preconditioning for Krylov Methods
作者: Shikhar Shah, Rui Peng Li, Tayfun Gokmen, Vassilis Kalantzis, Lior Horesh, Yuanzhe Xi
类别: math.NA
发布日期: 2026-06-15
arXiv ID: 2606.17227v1
链接: http://arxiv.org/abs/2606.17227v1
摘要: Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid digital-analog setting. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while preconditioner applications are performed through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We compare various block Jacobi preconditioning schemes including exact block inverses, sparse approximate inverses, Monte Carlo approximate inverses (MCAI), damping, and nested block Jacobi schemes. Numerical experiments with realistic analog matrix-vector simulations show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.