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1. 时间离散梯度流的均匀化
Homogenization of Time-Discrete Gradient Flows
作者: Yunhong Deng, Yuan Gao, Li Wang
类别: math.AP, math-ph, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25206v1
摘要翻译: 本文研究平坦环面上具有快速振荡扩散系数的 Fokker–Planck 方程在时间离散水平上的均匀化极限。我们使用极小化移动格式对时间离散化，并比较两种度量选择：运输型度量和加权 L2 度量。我们证明这种选择至关重要：当尺度参数趋于零时，这两种格式导致不同的均匀化极限，因而在时间步长趋于零时导致不同的发展方程。特别地，只有基于加权 L2 度量的格式才能恢复正确的有效动力学。
Comment: 无
链接: http://arxiv.org/abs/2607.25206v1
1. 研究对象和背景
本文研究的是在平坦环面 $\mathbb{T}^d$ 上带有快速振荡扩散系数的 Fokker-Planck 方程的均匀化问题, 并特别关注其时间离散梯度流格式的渐近行为. 所考虑的 $\varepsilon$-Fokker-Planck 方程形如
$$ \frac{\partial \mu_\varepsilon}{\partial t} - \nabla \cdot \left[ A\left(\frac{x}{\varepsilon}\right)(\nabla \mu_\varepsilon + \mu_\varepsilon \nabla V_\varepsilon(x)) \right] = 0, \quad (x,t) \in \mathbb{T}^d \times (0,\infty), $$其中 $0 < \varepsilon \ll 1$ 表示微观与宏观尺度之比, $A: \mathbb{R}^d \to \mathbb{R}^{d\times d}$ 是 1-周期的光滑矩阵场, 满足一致椭圆性 $\Lambda_0 I \preceq A(x) \preceq \Lambda_1 I$, 约束势 $V_\varepsilon$ 及其梯度一致有界且 $V_\varepsilon$ 一致收敛到极限势 $V$. 标准的均匀化理论指出, 当 $\varepsilon \to 0$ 时, 方程的解收敛到有效 Fokker-Planck 方程
$$ \frac{\partial \mu}{\partial t} - \nabla \cdot \left[ A^* (\nabla \mu + \mu \nabla V(x)) \right] = 0, $$其中有效扩散矩阵 $A^*$ 由
$$ (A^*)_{jk} = \int_{\mathbb{T}^d} \left( A_{jk}(x) + \sum_{\ell=1}^d A_{j\ell}(x) \frac{\partial \chi_k(x)}{\partial x_\ell} \right) dx $$给出, $\chi$ 为校正子, 满足 $ -\nabla\cdot(A\nabla\chi) = \nabla\cdot A$. 从变分观点看, 方程也可写为自由能 $\mathcal{H}_\varepsilon(\mu) = \int_{\mathbb{T}^d} \log \mu\, d\mu + \int_{\mathbb{T}^d} V_\varepsilon\, d\mu$ 关于 $\varepsilon$-Wasserstein 距离的梯度流. Gao 与 Yip [20] 从该梯度流结构出发, 证明了均匀化极限是 $\mathcal{H}$ 关于有效 Wasserstein 距离 $W_*$ 的梯度流, 且该极限与上述有效 Fokker-Planck 方程相容. 然而, 他们同时指出, 若仅仅均匀化距离 $W_\varepsilon$ 本身, 会得到不同的度量 $W_{\rm GH}$, 这表明距离与动力学的均匀化并不可交换.
本文试图填补时间离散水平的空白: 自然的问题是, 若直接对连续梯度流施行时间离散 (即 JKO 格式), 其均匀化极限是否仍保持上述相容性? 作者系统地比较了两种时间离散方案: 一种基于运输度量 ( $\varepsilon$-JKO 格式), 另一种基于加权 $L^2$ 度量. 通过详尽的分析, 发现 JKO 格式并非渐近保持 (asymptotic-preserving): 其均匀化极限偏离了正确的有效动力学, 而是收敛到相应于 $W_{\rm GH}$ 的梯度流. 相反, 加权 $L^2$ 格式在时间离散后仍能恢复正确的有效极限. 这一发现对多尺度随机过程的数值模拟具有重要意义.
2. 主要定理或主要结果
本文的主要结果由如下定理精确给出:
定理 1.4 (加权 $L^2$ 格式的均匀化) 设 $\mu_{\varepsilon,\tau}$ 为加权 $L^2$ 格式 (定义 1.2) 的解, 初值 $\mu_0 \in H^1(\mathbb{T}^d)$, 则当 $\varepsilon \to 0$ 时, 对任意 $t > 0$ 有 $\mu_{\varepsilon,\tau}(t) \to \mu_\tau(t)$ 在 $L^2(\mathbb{T}^d)$ 意义下成立. 这里的均匀化解 $\mu_\tau$ 由 $\mu_\tau(t) = \mu^n$ 对 $(n-1)\tau < t \le n\tau$ 给出, 其中 $\mu^n$ 迭代定义为
$$ \mu^n = \mathop{\arg\min}_{\mu \in H^1(\mathbb{T}^d)} \left\{ Q(\mu) + \frac{1}{2\tau} K(\mu, \mu^{n-1})^2 \right\}, $$且
$$ Q(\mu) = \frac12 \int_{\mathbb{T}^d} \langle \pi_*(x) A^* \nabla\varrho,\, \nabla\varrho \rangle\, dx, \qquad K(\mu,\nu) = \left( \int_{\mathbb{T}^d} \frac{(\mu-\nu)^2}{\pi_*} dx \right)^{1/2}, $$其中 $\varrho = \mu/\pi_*$, $\pi_* = e^{-V}$, $A^*$ 为 (1.3) 定义的有效扩散常数.
定理 1.5 ( $\varepsilon$-JKO 格式的均匀化) 设 $\mu_{\varepsilon,\tau}$ 为 $\varepsilon$-JKO 格式 (定义 1.3) 的解, 初值 $\mu_0 \in \mathcal{P}(\mathbb{T}^d)$ 且 $S(\mu_0) < \infty$, 则当 $\varepsilon \to 0$ 时, 对任意 $t > 0$ 有 $\mu_{\varepsilon,\tau}(t) \to \mu_\tau(t)$ 在 $W_2$ 距离下成立. 这里的均匀化解 $\mu_\tau$ 由 $\mu_\tau(t) = \mu^n$ 对 $(n-1)\tau < t \le n\tau$ 给出, 其中 $\mu^n$ 迭代定义为
$$ \mu^n = \mathop{\arg\min}_{\mu \in \mathcal{P}(\mathbb{T}^d)} \left\{ H(\mu) + \frac{1}{2\tau} W_{\rm GH}(\mu, \mu^{n-1})^2 \right\}, $$这里 $H(\mu) = S(\mu) + \int_{\mathbb{T}^d} V(x) d\mu$, $W_{\rm GH}$ 是基于均匀化成本 $c_{\rm hom}$ 的 Wasserstein 距离, $c_{\rm hom}$ 由
$$ c_{\rm hom}(x,y) = \min_{z} \left\{ \int_0^1 A_{\rm hom}(\dot z) dt : z(0)=x, z(1)=y, z\in H^1([0,1];\mathbb{T}^d) \right\} $$定义, 其中 $A_{\rm hom}(\xi)$ 是度量均匀化给出的极限行动函数, 满足 $\Lambda_1^{-1}|\xi|^2 \le A_{\rm hom}(\xi) \le \Lambda_0^{-1}|\xi|^2$.
定理 1.7 (加权 $L^2$ 均匀化格式的时间连续极限) 设 $\mu_\tau$ 由定理 1.4 给出, 则其解的迭代满足弱形式
$$ \frac{\mu^n - \mu^{n-1}}{\tau} - \nabla\cdot\left[ A^*(\nabla\mu^n + \mu^n \nabla V) \right] = 0, $$且当 $\tau \to 0$ 时, $\mu_\tau(t) \to \mu(t)$ 在 $L^2(\mathbb{T}^d)$ 中成立, $\mu(t)$ 为有效 Fokker-Planck 方程 (1.2) 的解.
定理 1.8 (均匀化 JKO 格式的时间连续极限, 需额外正则性假设) 若 $A_{\rm hom}$ 一致强凸, 设 $\mu_\tau$ 由定理 1.5 给出, 则存在 $\mu(t)$ 使得 $\mu_\tau(t) \to \mu(t)$ 在 $W_2$ 距离下成立. 进一步假设 $\mu_\tau(t) \to \mu(t)$ 在 $W^{1,1}(\mathbb{T}^d)$ 意义下成立, 则 $\mu(t)$ 满足拟线性抛物方程
$$ \frac{\partial \mu}{\partial t} - \nabla\cdot\left[ \left(\frac12 \nabla A_{\rm hom}\right)^{-1} \left( \nabla\mu + \mu\nabla V \right) \right] = 0. $$特别地, 在一维情形, 计算表明 $(\frac12 \nabla A_{\rm hom})^{-1}(\xi) = ( \int_0^1 A(x)^{-1/2} dx )^{-2} \xi$, 而有效扩散系数为 $A^* = ( \int_0^1 A(x)^{-1} dx )^{-1}$, 两者不同. 因此, 只有加权 $L^2$ 格式在时间连续极限下恢复了正确的有效动力学.
3. 研究方法、关键技术和核心工具
本文的核心分析工具是 $\Gamma$-收敛和最优传输中的度量均匀化理论, 并针对两种格式采取了不同的技术路线.
加权 $L^2$ 格式的均匀化
加权 $L^2$ 格式将每一步的最小化问题耦合在一个二次泛函中. 定义 $\varrho = \mu/\pi_\varepsilon$ 后, 原问题等价于最小化
其中振荡系数 $\pi_\varepsilon A(x/\varepsilon)$ 和 $\pi_\varepsilon$ 依赖于 $\varepsilon$. 处理该泛函的 $\Gamma$-极限时, 可使用标准椭圆均匀化的振荡测试函数方法: 构造校正子 $\chi$ 满足 $ -\nabla\cdot(A\nabla\chi) = \nabla\cdot A$, 并利用两尺度收敛技术, 将极限能量中的 $A$ 替换为均匀化矩阵 $A^*$, 同时 $\pi_\varepsilon$ 替换为 $\pi_* = e^{-V}$. 由于该格式的空间离散可以逐点处理, 无需复杂的耦合, 均匀化极限在每一步独立成立, 从而 $\Gamma$-极限的恢复序列和 liminf 不等式均可通过构造显式测试函数 (涉及 $\nabla\chi$) 得到. 这一部分的主要技术障碍在于同时处理扩散系数和势能中统一的修正, 但总体框架相对直接.
** $\varepsilon$-JKO 格式的均匀化**
对于 $\varepsilon$-JKO 格式, 难度更大. 每一步的泛函为
为了得到 $\Gamma$-极限, 需要分别处理能量 $\mathcal{H}_\varepsilon$ 和距离 $W_\varepsilon$ 的渐近行为. $\mathcal{H}_\varepsilon$ 的 $\Gamma$-极限是 $H$, 因为势能 $V_\varepsilon$ 一致收敛, 而 Boltzmann 熵 $S$ 本身是下半连续的. 核心障碍在于 $W_\varepsilon(\cdot, \nu)$ 的 $\Gamma$-极限, 其中 $\nu$ 固定. $W_\varepsilon$ 不是点态收敛的; 事实上 Gao 和 Yip 已指出 $\lim_{\varepsilon\to0} W_\varepsilon(\mu,\nu) \neq W_*(\mu,\nu)$. 作者借助 Riemannian 度量的均匀化理论 [9, 41], 证明 $W_\varepsilon(\cdot, \nu)^2$ 的 $\Gamma$-极限是 $W_{\rm GH}(\cdot, \nu)^2$. 其证明策略如下: 首先, 对于给定的两个测度, 可以通过求解 Monge 问题将 $W_\varepsilon$ 表示为基于振荡成本 $c_\varepsilon$ 的运输距离. 利用度量均匀化的紧性结果, 成本函数 $c_\varepsilon(x,y)$ 依某种意义收敛到 $c_{\rm hom}(x,y)$. 接着, 需要在 $\Gamma$-收敛框架下处理最优运输泛函. 关键引理可能是借助最优运输映射的稳定性和度量均匀化中极限行动函数 $A_{\rm hom}$ 的性质 (如一致凸性, 引理 2.5–2.6) 来建立 liminf 不等式和恢复序列. 整个 $\Gamma$-极限的建立依赖连续 $\Gamma$-收敛的概念: 由于 $\mathcal{H}_\varepsilon$ 和 $W_\varepsilon^2$ 都是连续泛函 (在 $W_2$ 拓扑下), 可以利用 $\Gamma$-收敛的稳定性 (引理 2.13) 将两者的极限相加.
一旦得到 $F_\varepsilon$ 的 $\Gamma$-极限为 $F(\mu) = H(\mu) + \frac{1}{2\tau} W_{\rm GH}(\mu, \mu^{n-1})^2$, 结合紧性 (由环面紧致性保证) 和 $\Gamma$-收敛基本定理 (定理 2.16), 便得到极小元列的收敛. 由于熵函数的严格凸性, 极小元唯一, 故全序列收敛无需抽取子列.
时间连续极限
对于加权 $L^2$ 格式的均匀化极限, 直接写出欧拉-拉格朗日方程, 得到关于 $\mu^n$ 的线性椭圆型方程, 然后对时间差分取极限, 当 $\tau\to0$ 就恢复有效 Fokker-Planck 方程. 对于 JKO 情形, 均匀化极限的欧拉-拉格朗日方程是非线性的, 利用最优传输映射表示, 可推导出离散时间演化方程. 在假设 $A_{\rm hom}$ 一致强凸下, 可保证运输映射的正规性质, 但为了导出极限 PDE (1.23), 需要更强的紧性: $\mu_\tau$ 在 $W^{1,1}$ 中收敛. 这一紧性在标准 (非均匀化) JKO 格式中有时可通过熵耗散估计建立 [26,36,14], 但本文未给出证明, 而是将其列为假设, 并指出这是技术性开放问题.
4. 与之前工作的比较
本文与已有工作的关联和推进主要体现在以下三个维度:
与时间连续梯度流均匀化的对比
Gao 和 Yip [20] 从 Wasserstein 梯度流框架出发研究了连续 Fokker-Planck 方程的均匀化, 得到了与经典均匀化一致的结果, 并发现了度量均匀化与动力学均匀化的不一致性 ($W_*$ vs $W_{\rm GH}$). 本文则将视野扩展到时间离散层面, 系统探讨了 JKO 格式的均匀化极限. 主要推进在于: (i) 揭示出即使连续梯度流的均匀化正确, 其自然的时间离散化 (JKO) 却并非渐近保持, 会收敛到错误的极限方程; (ii) 找到了一种替代格式 (加权 $L^2$), 该格式在时间离散后仍保持正确的均匀化极限, 为数值计算提供了可行的方案. 这表明在涉及多尺度问题时, 变分结构的相容性至关重要.与最优传输中度量均匀化理论的结合
最优传输中的度量均匀化结果 (如 [9] 定理 2.6, [41]) 给出了振荡成本函数的极限 $c_{\rm hom}$ 和行动函数 $A_{\rm hom}$ 的性质. 本文将这些工具创造性地应用于优化问题 $W_\varepsilon(\mu,\nu)$ 的 $\Gamma$-收敛分析中, 这是此前未曾尝试的. 与纯粹的度量均匀化不同, 这里需要处理带有熵和势能的变分问题, 且要求 $\Gamma$-极限在测度空间 $\mathcal{P}(\mathbb{T}^d)$ 中成立, 因此需要结合 Wasserstein 距离的拓扑和最优运输映射的紧性. 作者通过连续 $\Gamma$-收敛和恢复序列的构造成功解决了这一问题.与 JKO 格式收敛性工作的关系
对于经典 Fokker-Planck 方程, JKO 格式的收敛性 ( $\tau\to0$ ) 已被广泛研究, 特别是在紧空间上可建立强收敛结果 [26, 36, 14]. 本文在均匀化 JKO 格式的时间连续极限部分依赖于类似的想法, 但极限 PDE 变为拟线性的, 且需要 $W^{1,1}$ 紧性假设. 从现有内容看, 作者并未证明这一紧性, 而是注明其为下一步工作. 这暗示均匀化 JKO 格式在 $\tau\to0$ 极限下的严格收敛性仍是一个开放问题.代价与局限
本文的结果清楚展示了加权 $L^2$ 格式的优势, 但该格式需要将方程转化为热方程形式, 并设计相应的加权 Dirichlet 能量和加权 $L^2$ 距离, 这可能限制其对一般扩散过程的推广. 而 $\varepsilon$-JKO 格式虽然几何意义自然, 但其均匀化极限的 $\tau\to0$ 收敛需额外正则性, 且最终方程与物理有效方程不符. 此外, 本文的理论是在平坦环面 $\mathbb{T}^d$ 上建立的, 并假设系数光滑和势能一致有界, 向更一般区域、粗糙系数或非线性扩散的推广尚未涉及. 从文章结构来看, 各定理的详细证明并未在提供的摘录中完全展开, 只在第 3、4 节给出分析, 因而某些技术细节 (如恢复序列的具体构造) 需参阅原文.
了解领域论文（摘要翻译）
1. 面向干热气候下低成本墙体材料热性能排序的物理信息神经网络算子
A Physics-Informed Neural Operator for Thermal Ranking of Low-Cost Wall Materials in Hot-Dry Climates
作者: Muhammad Akbar Khan, Fahim Raees, Ubaida Fatima
类别: cs.LG, math.NA, physics.comp-ph
发布日期: 2026-07-28
arXiv ID: 2607.25668v1
链接: http://arxiv.org/abs/2607.25668v1
摘要翻译
中文标题: 面向干热气候下低成本墙体材料热性能排序的物理信息神经网络算子
摘要翻译: 识别能够最小化墙壁热渗透的、具有成本效益的本土建筑材料，对于夏季温度经常超过45°C的干热气候下低收入农村住宅的室内热舒适至关重要。我们提出了一个用于五种低成本本土墙体材料热性能排序的两阶段计算框架，这五种材料是：泥砖、粘土-秸秆土坯、石灰稳定竹板、烧结粘土砖和石灰-泥复合材料。首先，经过验证的Crank-Nicolson有限差分方法（FDM）在日间太阳辐射和室外空气温度的强迫下，求解具有Robin边界条件的一维瞬态热方程，通过拉丁超立方采样在九维参数空间中生成了1500个周期性日解。其次，一个以傅里叶神经网络算子（FNO）为主干的物理信息神经网络算子（PINO）学习参数到解的算子 mu -> T(x,t)，同时保证数据保真度和PDE一致性。训练后的PINO达到了5.14e-4的相对L2场误差和0.201 K的内表面峰值温度平均绝对误差，并精确保持了FDM的材料排序；在150个FDM样本上训练的PINO与在双倍数量上训练的纯数据FNO性能相当，因此当数据稀缺时，物理损失最有价值。周期性日公式还给出了ISO 13786的时间延迟和衰减因子，再现误差分别在0.99 h和0.010以内。在标称干热夏季条件下，粘土-秸秆土坯在广泛可获得的材料中取得了最佳的成本-性能指标。通过FDM抽查确认的气候扫描揭示了一个工况边界：在室外温度低于环境温度的条件下，材料排序颠倒为导热的烧结粘土砖，从而划分了隔热和排热两种工况。该框架为干热地区洪水后重建中基于证据的材料选择提供支持。
2. 基于表示论的离散外微积分在弯曲与三维网格上的并行化
Parallelisation of Discrete Exterior Calculus via Representation Theory on Curved and Three-Dimensional Meshes
作者: Leon D. da Silva, Marcelo P. Santos, José D. da Silva, Gilson Ferreira
类别: math.NA, math.DG, physics.comp-ph
发布日期: 2026-07-28
arXiv ID: 2607.25192v1
链接: http://arxiv.org/abs/2607.25192v1
摘要翻译
中文标题: 基于表示论的离散外微积分在弯曲与三维网格上的并行化
摘要翻译: 我们针对对称网格上的离散外微积分（DEC）算子建立了一个通用的块对角化框架，从而使得可证明FLOP减少的易并行求解器成为可能。我们证明了两个基本的 DEC 算子，即离散外微分 $d$ 和 Hodge 星算子 $\star$，在单纯复形上的等距有限群作用下是等变的。该证明利用了群作用在上链空间上诱导的置换表示。因此，任何由 $d$ 和 $\star$ 组合而成的算子（包括 Hodge Laplace 算子、余微分算子、Maxwell 型算子和弹性算子）都在单一的对称适应基下继承块对角结构，而该基对于每个网格仅需计算一次。与仅限于平坦柏拉图区域的谱方法不同，该框架天然适用于弯曲流形，并且原则上可应用于对称区域上的计算电磁学和几何流体模拟。在测地球面（$I_h$ 对称）和六边形环面（$D_{6h}$ 对称）上的数值实验，相对于稠密直接分解，分别获得了高达 $62$ 倍和 $182$ 倍的基于 FLOP 的并行加速比。在具有 $T_d$ 对称的平坦三维环面 $T^3$ 的体心立方（BCC）镶嵌上的进一步实验，在三种网格分辨率下，对于形式次数 $k=0,1,2$，以机器精度确认了外微分、Hodge 星算子和 Hodge Laplace 算子的等变性。基于 FLOP 的顺序加速比接近其理论渐近值 $\approx 9.07$ 倍，而标准的 Schur 重数约化又进一步加深了约 $|G|$ 量级的因子。这些结果表明，单一的对称适应基降低了在弯曲和三维网格上保结构 DEC 计算的线性求解成本。
3. 一种针对大尺度界面演化问题的自适应相场框架：基于强形式梯度光滑方法
An adaptive phase field framework for large-scale interface evolution problems using a strong-form gradient smoothing approach
作者: Zirui Mao, Alice Xu, Shenyang Hu, Panos Stinis, Yuyan Shao, Ang Li, Yulan Li
类别: math.NA, math-ph
发布日期: 2026-07-27
arXiv ID: 2607.25142v1
链接: http://arxiv.org/abs/2607.25142v1
摘要翻译
中文标题: 一种针对大尺度界面演化问题的自适应相场框架：基于强形式梯度光滑方法
摘要翻译: 具有演化界面的多尺度问题在科学与工程中普遍存在。相场模型是计算力学与材料建模中模拟界面主导现象的有力工具，但其在大尺度问题中的应用常受到在整个区域内解析薄扩散界面所需高计算代价的制约。本文提出一种高效的强形式相场求解器，将梯度光滑方法（GSM）与分层自适应移动结构化网格相耦合，能够自动在狭窄的界面区域内局部化求解分辨率，同时在体域中保持粗离散化。引入了一种分层细化设计，以保持跨界面局部均匀的分辨率，使得尽管远离界面处网格非均匀性很强，GSM 离散仍能维持整体二阶精度。尽管 GSM 每个自由度上的计算代价高于标准有限差分格式，但自适应框架大幅减少了总自由度数，从而相比均匀网格方法的二次计算规模，实现了近似线性的计算规模。基于 Allen-Cahn 和 Cahn-Hilliard 方程的数值算例表明，所提出的自适应 GSM 求解器在界面演化中达到了期望精度，同时获得了比现有弱形式和强形式求解器更优的计算复杂度 $O(N)$，对于具有薄界面或界面面积占整个区域比例较小的大尺度问题，效率显著提高。
4. 基于评分的含时问题稳定化
Score-Based Stabilization for Time-Dependent Problems
作者: Eshed Gal, Eldad Haber, Uri Ascher
类别: cs.LG, math.NA
发布日期: 2026-07-27
arXiv ID: 2607.25119v1
链接: http://arxiv.org/abs/2607.25119v1
摘要翻译
中文标题: 基于评分的含时问题稳定化
摘要翻译: 我们提出一种用于偏微分方程数值模拟的基于评分的稳定化框架，其中一个学习到的评分模型定义了一个作用于临时数值更新的稳定化算子。该算子通过将迭代驱向容许状态流形的修正来强化结构一致性与物理一致性，从而增强标准时间步进格式。我们证明该稳定化算子充当朝向该流形的收缩映射，产生一个具有吸引域条件稳定性的修正机制。在平流方程、Korteweg-de Vries (KdV) 方程、非线性薛定谔 (NLS) 方程和 Burgers 方程上的数值实验表明，该方法提高了鲁棒性，抑制了非物理不稳定性，并保持了定性动力学特性。
5. 构造满足熵条件的Active Flux\yolo{-type}方法
Construction of entropy satisfying Active Flux-type methods
作者: Remi Abgrall, Yongle Liu
类别: math.NA
发布日期: 2026-07-27
arXiv ID: 2607.25111v1
链接: http://arxiv.org/abs/2607.25111v1
摘要翻译
中文标题: 构造满足熵条件的Active Flux\yolo{-type}方法
摘要翻译: 本文分析Active Flux\yolo{-type}格式对于配备一个熵不等式的双曲系统的熵稳定性性质。这种格式演化两组自由度：在覆盖计算域的元素边界上选择的点值，以及这些元素内解的平均值。我们证明唯一需要做的是得到平均值的熵不等式，点值自由度不起任何作用。我们构造一个整体格式，该格式是保界的\cite{BP_Pampa_VEM}，遵循\cite{PampaDG}的无振荡，并且熵递减。熵条件在Tadmor框架\cite{TadmorEntropy}中实现，即仅对半离散格式。该格式在Kurganov-Popov-Petrova测试案例\cite{KPP}上进行测试，该案例已知对满足熵不等式\yolo{be}敏感。我们表明我们的熵修正是有效的：如果不激活保界条件或无振荡条件，我们会得到带有一些虚假振荡的正确解，正如预期。尽管该格式的开发、实现和测试是用三角形版本进行的，相同的方法也可以用于多边形网格，遵循\cite{BP_Pampa_VEM}。
6. 多物种趋化中亚临界与超临界动力学的混合物理信息神经网络框架
A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
作者: Hamid El Bahja, Jan C. Riedel, Peter Jung
类别: math.NA, math.AP
发布日期: 2026-07-27
arXiv ID: 2607.24949v1
链接: http://arxiv.org/abs/2607.24949v1
摘要翻译
中文标题: 多物种趋化中亚临界与超临界动力学的混合物理信息神经网络框架
摘要翻译: 我们研究一种两物种趋化系统，该系统呈现两种定性不同的状态：亚临界动力学（其中解保持光滑）和超临界动力学（其中强聚集可能导致有限时间爆破）。在亚临界状态下，我们使用标准连续时间物理信息神经网络（PINN）对耦合场进行交替训练，并证明它提供了准确且高效的近似。然而，在超临界状态下，该格式不够稳健，无法解析与爆破相关的高度局部化结构和陡峭梯度。为解决这一局限性，我们引入一种基于向后欧拉时间步进的离散时间PINN，并结合对数变换以稳定大解值，以及一种残差自适应配置策略，将训练点集中在高活动区域附近。该混合框架使我们能够在解光滑时使用简单且计算成本低的PINN，而当奇异动力学出现时切换到更稳健的离散格式。数值实验证实，连续时间格式准确解析了亚临界动力学，而离散时间格式可靠地捕捉了超临界状态下的奇异聚集和有限时间爆破。
不相关论文（基本信息）
1. 拟SVD：学习一种受李群约束的矩阵分解用于实时成像
Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging
作者: Christopher Hahne
类别: cs.CV, cs.LG, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25967v1
链接: http://arxiv.org/abs/2607.25967v1
摘要: Singular Value Decomposition (SVD) underlies matrix factorisation tasks across computational imaging, with medical applications increasingly demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in clinical pipelines. This study introduces Quasi-SVD, a differentiable, fully parallelized matrix factorization framework for GPUs. Rather than enforcing orthogonality on both factors, it guarantees exact orthogonality for a single Lie-parameterized factor while recovering the remaining components through soft constraints, enabling efficient parallel decomposition without iterative singular-vector optimization. This asymmetric design, provably sufficient for valid factorisation, achieves reconstruction fidelity of SSIM = 0.89-0.94 and accelerates computation by 3-20x relative to cuSOLVER and randomised SVD, enabling throughput above 25 FPS. Performance is evaluated on two medical imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, and (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices. Across both regimes and multiple imaging instruments, the proposed framework demonstrates robust domain transfer and throughput exceeding 25 FPS at clinical matrix scales, a rate sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, Quasi-SVD makes structured matrix factorisation practical for real-time imaging.
2. 针对 Trixi$.$jl 与 deal$.$II 的并行自适应多物理场模拟的多求解器耦合
Multi-Solver Coupling for Parallel Adaptive Multi-Physics Simulations with Trixi$.$jl and deal$.$II
作者: Vivienne Ehlert, Gregor Gassner, Martin Kronbichler, Hendrik Ranocha, Michael Schlottke-Lakemper
类别: math.NA, cs.MS
发布日期: 2026-07-28
arXiv ID: 2607.25871v1
链接: http://arxiv.org/abs/2607.25871v1
摘要: Many standalone frameworks for numerical solvers have been developed to tackle the simulation of specific single- or multi-physics problems. For solving coupled problems, common approaches are to extend existing solvers, to develop an entirely new solver or to couple two existing solvers by using the interface provided by a coupling library or framework. However, to the best of our knowledge, there is not yet a framework that allows the easy and direct development of coupled solvers. In this work, we prototype a portable reproducible cross-language framework for the development of coupled parallel adaptive solvers using Trixi$.$jl and deal$.$II for the numerical simulation of coupled multi-physics problems. Currently, this is tightly entangled with an example coupled solver. We show its usability by developing a partitioned strongly-coupled multi-physics solver for the dynamics of Newtonian self-gravitational gases. For the coupled solver, we validate the expected order of convergence, physical sensibility of the results and mesh adaptivity. Finally, we investigate its parallel scaling. A publicly accessible reproducibility repository for the numerical results and code is available.
3. 使用Julia的大规模并行数值模拟
Massively parallel numerical simulations with Julia
作者: Simon Candelaresi, Benedict Geihe, Marco Artiano, Lars Christmann, Valentin Churavy, Andrés Rueda-Ramírez, Hendrik Ranocha, Gregor J. Gassner, Michael Schlottke-Lakemper
类别: math.NA, cs.DC, cs.MS, cs.PF
发布日期: 2026-07-28
arXiv ID: 2607.25866v1
链接: http://arxiv.org/abs/2607.25866v1
摘要: The Julia programming language aims to provide a modern approach to develop high-performance computing (HPC) applications. It tries to achieve this by combining a high-level, dynamic interface with just-in-time compilation to native machine code, thereby facilitating high developer productivity and native code performance at the same time. While this approach has already been shown to work well for serial applications, it is not clear if it readily translates to traditional, massively parallel HPC work loads. In this paper, we fill this gap by analyzing the parallel performance of the numerical computational fluid dynamics simulation code Trixi$.$jl, written in Julia, and compare it to the Fortran code FLUXO. We show some of the challenges of using Julia at scale and discuss possible solutions, specifically with respect to code loading and compilation at startup. Finally, we demonstrate the parallel scaling of our Julia code on up to 61440 CPU cores.
4. 关于松弛投影的无穷乘积的有界性：扰动恢复性与动态弦平均
On the boundedness of infinite products of relaxed projections: perturbations resilience and dynamic string-averaging
作者: Daniel Reem, Yair Censor
类别: math.OC, math.FA, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25797v1
链接: http://arxiv.org/abs/2607.25797v1
摘要: Very recently (2026), Bauschke and Tung extended from finite- to infinite-dimensional Hilbert spaces a result published by Meshulam in 1996 (following an earlier result of Aharoni-Duchet-Wajnryb from 1984) regarding the boundedness of infinite products of relaxed projections onto a finite family of closed affine subspaces. In the present note we extend in various ways the result of Bauschke and Tung by allowing certain perturbations and proving perturbation resilience, by considering a mixture of closed half-spaces and closed hyperplanes, and by using dynamic weighted sums of dynamic strings (of dynamic lengths) of relaxed projections in the iterative process. We also discuss the limitation to generalize the Bauschke-Tung result to arbitrary closed and convex sets by presenting a large family of counterexamples in which the associated control is not cyclic and not even almost cyclic. Along the way we establish a general theorem of independent interest regarding the uniform boundedness and uniform unboundedness of infinite products of nonexpansive mappings in a normed space setting.
5. 铁磁磁流体动力学模型的线性全离散有限元方法的误差估计
Error Estimates for a Linear Fully-Discrete Finite Element Method for the Ferromagnetic Magnetohydrodynamical Model
作者: Noah Vinod, Thanh Tran
类别: math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25523v1
链接: http://arxiv.org/abs/2607.25523v1
摘要: The ferromagnetic magnetohydrodynamic equations are a non-linear PDE system that models the flow of an electrically conducting fluid that also has intrinsic magnetisation. In this paper, we propose a fully-discrete linear finite element scheme based on Euler’s method to approximate the solutions to these equations and perform an error analysis for the scheme. Numerical experiments have been included to corroborate our theoretical results.
6. 深度学习算子的自动化数值稳定性分析
Automated Numerical Stability Analysis of Deep Learning Operators
作者: Xinye Chen
类别: math.NA, cs.AI
发布日期: 2026-07-28
arXiv ID: 2607.25494v1
链接: http://arxiv.org/abs/2607.25494v1
摘要: Finite-precision arithmetic unavoidably introduces numerical approximation errors. Numerical computations may use insufficient precision or an improper formulation, which leads to numerical instability. In this paper, we introduce the first unified software tool that integrates CESTAC for detecting the numerical stability of deep learning operators. Our developed software not only enables numerical validation with a single computation pass but also detects the sources of numerical instability and provides numerical stability monitoring during deep learning training and inference. We verified its effectiveness on the detection of polluted operators with injected numerical instabilities across various tasks. We believe that our developed method and tool provide valuable insights into developing numerically stable computing kernels, which are particularly critical for numerically stable and efficient deep learning training and inference.
7. 关于由最大角 $n$ 等分算法生成的三角剖分
On Triangulations Generated by the Largest-Angle $n$-Section Algorithm
作者: Jérôme Michaud, Sergey Korotov
类别: cs.CG, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25457v1
链接: http://arxiv.org/abs/2607.25457v1
摘要: We define a mesh refinement algorithm based on the rule of dividing the largest angles of triangular elements of planar partitions in focus into $n$ equal parts, and analyse the (geometric) properties of triangulations generated by this technique. This largest-angle $n$-section rule is compared with the classical longest-edge $n$-section rule, where it is the longest edges which are split into $n$ equal parts. The longest-edge bisection and trisection are known to produce nondegenerate triangulations (possibly with hanging nodes), but the longest-edge $n$-sections with $n\geq 4$ always produce (infinite) sequences of triangles with minimum angles tending to zero (moreover, their relevant maximum angles tend to $π$), thus breaking the minimum and maximum angle conditions. We show that this degeneration effect is not a consequence of $n$-section itself. For every $n\geq 2$, the largest-angle $n$-sections produce partitions satisfying the minimum angle condition (and, therefore, the maximum angle condition). More precisely, if the initial triangle has its smallest angle $γ_0>0$, then all descendant triangles have angles bounded below by $m_n=\min\left\{γ_0,\fracπ{3n}\right\},$ and, correspondingly, bounded above by $π-2m_n<π$. We also show that the recursive largest-angle $n$-section algorithm always produces a family of triangular partitions, i.e. the maximum diameter of level-$k$ descendants tends to zero as $k \to \infty$.
8. 经典多项式优化的量子估计
Quantum estimates for classical polynomial optimization
作者: Oleg Evnin
类别: math-ph, math.NA, quant-ph
发布日期: 2026-07-28
arXiv ID: 2607.25445v1
链接: http://arxiv.org/abs/2607.25445v1
摘要: The problem of finding lower and upper bounds on multivariate homogeneous polynomials is both difficult and important given its applications to questions ranging from dynamical stability in complex potential landscapes to data analysis. From the standpoint of tensor eigenvalue theory, the question is equivalent to finding the smallest and the largest eigenvalues of the coefficient tensor corresponding to the given polynomial. Standard approaches outlined in the literature amount to running nonlinear iterations in search for the optimal rays along which the growth of the polynomial is fastest or slowest. Unlike the case of matrices (or their corresponding multivariate quadratic forms) convergence of such algorithms for higher-rank tensors is capricious due to the complex topography of polynomial objective functions. In this essay, a very different strategy, inspired by quantum-mechanical variational methods, is introduced for finding bounds on polynomials. The original polynomial is replaced by an operator acting in a suitably chosen (large) space of states, such that in an appropriate “classical” limit this operator approaches the original polynomial expression made of commutative variables. As a result, approximating the smallest and largest eigenvalues of the coefficient tensor, and thus finding bounds on polynomials, amounts to diagonalizing the resulting quantum operator, represented as a large matrix, and then inspecting the smallest and largest eigenvalues of this matrix. This approach is then successfully applied to standard test examples from tensor eigenvalue literature and other problems of interest in mathematical physics including Strichartz-type inequalities.
9. 一种非马尔可夫量子态扩散的通用一阶和二阶数值求解器
A General First- and Second-Order Numerical Solver for Non-Markovian Quantum State Diffusion
作者: Zhenning Cai, Quanhui Zhu
类别: quant-ph, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25427v1
链接: http://arxiv.org/abs/2607.25427v1
摘要: The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.
10. 线性统计模型的顺序预处理共轭梯度法
Sequential Preconditioned Conjugate Gradient Method for Linear Statistical Models
作者: Guan-Yu Chen, Dong-Yue Xie, Xi Yang, Zun-Hao Zheng
类别: math.NA, stat.ML
发布日期: 2026-07-28
arXiv ID: 2607.25272v1
链接: http://arxiv.org/abs/2607.25272v1
摘要: We propose a randomized iterative method for the ordinary least-squares estimation problem in large-scale linear statistical models, namely the Sequential Preconditioned Conjugate Gradient Method (SPCG). SPCG constructs a sequence of sketched least-squares subproblems with increasing sketch sizes, applies PCG as the inner solver, and warm-starts each subproblem from the previous solution. A final refinement stage is then performed on the full-scale problem. Since most iterations are carried out on smaller subproblems, the overall computational cost is significantly reduced. We establish the convergence theory, prove that SPCG attains OLS prediction accuracy, and derive per-subproblem iteration bounds and complexity estimates. Numerical experiments show that SPCG reaches the target prediction accuracy with fewer iterations and less CPU time than full-data PCG and Iterative Double Sketching (IDS).
11. 希尔伯特空间中变分包含问题的双惯性步投影收缩方法
Projection and contraction methods with double inertial steps for variational inclusion problems on Hilbert spaces
作者: Moin Uddin, Mohammed Alshahrani, Qamrul Hasan Ansari
类别: math.OC, math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25203v1
链接: http://arxiv.org/abs/2607.25203v1
摘要: In this paper, we propose three projection–contraction algorithms for solving variational inclusion problems in the setting of Hilbert spaces, each incorporating a double inertial technique into the projection–contraction framework. The first algorithm \algoO achieves weak convergence under monotonicity and Lipschitz continuity of the single-valued operator, with an adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. A modified variant \algoT of \algoO attains $R$-linear convergence under the strong monotonicity assumption. The third algorithm \algoDIM obtains strong convergence to the minimum-norm solution without requiring strong monotonicity. We illustrate the proposed methods on an abstract variational inclusion problem and apply them to the split feasibility problem and the elastic net regularization problem, comparing them with existing methods in the literature. The $R$-linear convergence of \algoT is also verified through numerical simulations.
12. 相场晶体方程的凸分裂谱方法：能量稳定性、计算稳定性图与三维GPU模拟
A Convex Splitting Spectral Method for the Phase Field Crystal Equation: Energy Stability, Computational Stability Maps, and Three-Dimensional GPU Simulations
作者: Saulo Orizaga, Peimeng Yin, Deep Choudhuri
类别: math.NA
发布日期: 2026-07-28
arXiv ID: 2607.25177v1
链接: http://arxiv.org/abs/2607.25177v1
摘要: We present an efficient Fourier spectral method based on the convex splitting framework of Eyre~\cite{eyre1998unconditionally} for the phase field crystal (PFC) equation. The proposed first-order scheme is unconditionally energy stable and conserves mass to machine precision. An energy stability theorem is established using a truncated potential argument, and a semi-analytical neutral stability curve is derived from a dominant-mode energy balance, providing a closed-form characterization of the practical stability boundary. The classical sufficient condition $a \geq 2$ is shown to be conservative: a computational stability map obtained from 40,000 GPU-accelerated PFC simulations reveals that energy-stable solutions persist for values of $a$ significantly below this threshold. Crucially, accuracy analysis demonstrates that smaller values of $a$ within the stable region consistently yield lower $L^2$ errors. This high-fidelity regime is rigorously verified through an extended asymptotic stress test consisting of a long-time three-dimensional simulation on a $256^3$ grid up to $T_f = 10,000$, successfully executing $10^6$ continuous temporal increments deep within the relaxed stability regime, well below the classical convex splitting limit ($a < 2$), while preserving strict monotonic energy dissipation and machine-precision mass conservation. Finally, two-dimensional and three-dimensional simulations at resolutions up to $512^3$ are performed on a single consumer GPU, demonstrating the scalability of the proposed framework for resolving complex phase-field dynamics without requiring HPC infrastructure. The code is made publicly available on GitHub.
13. 主动子空间的黎曼观点
A Riemannian View on Active Subspaces
作者: Zachary Grey
类别: math.NA, cs.LG, math.DG
发布日期: 2026-07-28
arXiv ID: 2607.25163v1
链接: http://arxiv.org/abs/2607.25163v1
摘要: Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.
14. 巴龙空间中的椭圆正则性理论及其在深度里兹方法中的应用
Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Method
作者: Stephan Wojtowytsch
类别: math.AP, math.NA, stat.ML
发布日期: 2026-07-27
arXiv ID: 2607.25100v1
链接: http://arxiv.org/abs/2607.25100v1
摘要: We prove that harmonic functions with Dirichlet boundary data in Barron space, a function class tailored to wide ReLU networks with a single hidden layer and suitably bounded weights, are generally neither Lipschitz continuous nor in the Sobolev class $H^2$. A fortiori, they are not in any function class in which the norm controls the Lipschitz constant, which rules out not only Barron space regularity, but also regularity in function classes for deeper ReLU networks with bounded coefficients. They can, however, be approximated to accuracy $\sim \varepsilon$ by Barron functions of low norm $\sim |\log\varepsilon|$ in various Lebesgue and Sobolev norms (with at most two derivatives). The positive result holds on very simple domains: Half-spaces in arbitrary dimension and rectangular domains in two dimensions. As an application of this regularity theory, we obtain a priori error estimates for Deep Ritz neural PDE solvers.
15. 关于 Koopman 算子特征值问题的扩展动态模式分解解的残差界与向后遮蔽稳定性
On residual bounds, backward shadowing stability of the Extended Dynamic Mode Decomposition solution to the eigenvalue problem for the Koopman operator
作者: Zlatko Drmač
类别: math.NA, cs.MS, math.DS
发布日期: 2026-07-27
arXiv ID: 2607.25086v1
链接: http://arxiv.org/abs/2607.25086v1
摘要: This paper introduces shadowing theory based backward stability assessment of data driven computational analysis of discrete dynamical systems in the framework of the Koopman composition operator and the Extended Dynamic Mode Decomposition. Data driven spectral analysis of the dynamics using the approximate (computed) eigenpairs of the Koopman operator is cast in terms of the backward error analysis. The individual residuals of the computed eigenpairs are aggregated in a backward perturbation of the Koopman operator, so that the computed eigenpairs are exact for a perturbed operator that is not a composition operator. Then, the impact of the perturbation in the operator is carried over to the map and the initial condition of the original system, showing that the computed approximations correspond exactly to a pseudo–trajectory of the system. In the final step, the pseudo–trajectory is shadowed by an exact trajectory of the system. This interpretation of errors is in particular suitable in data driven scenarios where the data is contaminated by noise. New insights into the numerical shadowing are provided.