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1. 基于优化的时空超材料波动方程有效系数的识别

Optimization-Based Identification of Effective Coefficients for Wave Equations in Spatio-Temporal Metamaterials

作者: Christian Döding, Vishnu Raveendran, Barbara Verfürth

类别: math.AP, math.NA, math.OC

发布日期: 2026-08-06

arXiv ID: 2608.05779v1

摘要翻译: 我们研究非均匀介质中波动方程有效系数的识别问题。这类方程出现在时空超材料的建模中，其底层材料性质在空间和时间上均呈现变化。均匀化方法在微观尺度趋于零的渐近情形下提供了有效模型，但从波的传播观测中确定宏观参数仍具有挑战性。我们引入一种基于优化的方法，通过极小化一个代价泛函来识别一个常值有效系数。该方法适用于底层时空依赖系数未知，而解可通过测量在空间和时间上获取的情形。我们将基于优化的系数识别技术从椭圆多尺度问题拓展到波动方程，并证明在均匀化极限存在的前提下，当微观尺度趋于零时，所识别的系数收敛到均匀化系数。此外，我们建立了相应有效解向非均匀解的收敛性，包括 $L^2$ 中的强收敛以及一阶空间和时间导数的弱收敛。数值实验进一步展示了该方法对于非周期时空非均匀介质的性能，其中包含均匀化极限未知的情形。

Comment: 无

链接: http://arxiv.org/abs/2608.05779v1

1. 研究对象和背景

本文研究时空调制波动方程中有效系数的识别问题. 研究对象是定义在 d-维环面 T^d 和无界时间区间 (0,T) 上的波动方程

$$ \partial_{tt} u_\varepsilon - \operatorname{div}\big(\mu_\varepsilon(x,t) \nabla u_\varepsilon\big) = 0, \qquad (x,t) \in T^d \times (0,T), $$

并配备初始条件 $u_\varepsilon|_{t=0} = v_0$, $\partial_t u_\varepsilon|_{t=0} = v_1$, 其中 $v_0 \in H^1(T^d)$, $v_1 \in L^2(T^d)$. 系数 $\mu_\varepsilon \in C^1([0,T], L^\infty(T^d, \mathbb{R}^{d\times d}_{\rm sym}))$ 是一致正定的, 即存在常数 $\lambda>0$ 使得对任意 $\xi\in\mathbb{R}^d$ 和几乎所有 $(x,t)$ 有 $\mu_\varepsilon(x,t)\xi\cdot\xi \ge \lambda |\xi|^2$. 该系数描述了时空超材料中的材料性质, 可能在空间和时间上表现出尺度为 $0<\varepsilon\ll 1$ 的快速振荡. 这种多尺度结构使得直接数值模拟成本高昂, 因而推导有效模型十分必要.

均质化理论为高度振荡介质提供了在 $\varepsilon\to0$ 时求取有效系数的严格数学框架. 对于纯空间调制的波动方程, 已有较为成熟的结果; 对于时空调制的情形, 由于时间振荡可能导致解的不稳定性, 相应研究较少. 已有工作包括基于形式均质化的渐近分析 [9,23] 和对弱时间调制的严格均质化 [5]. 然而, 均质化理论本质上是渐近的: 只有当微观尺度趋于零时, 均质化系数才准确描述异质介质. 在实际应用中, 微观尺度 $\varepsilon$ 固定为正, 因此均质化模型通常只会引入 $O(\varepsilon)$ 的误差.

在许多物理问题中, 异质系数 $\mu_\varepsilon$ 的具体时空依赖关系往往是未知的, 而仅能通过测量得到波传播的解 $u_\varepsilon$. 这自然地引出一个逆问题: 从观测数据中确定有效的宏观材料参数. 直接恢复 $\mu_\varepsilon$ 通常是病态的, 但识别一个常系数有效矩阵是可行的. 对于椭圆型多尺度问题, 已有多种识别方法, 包括参数估计、分离振荡技术、神经网络方法, 以及由 Le Bris, Legoll 和 Lemaire [14,15] 提出并后续发展的优化框架. 然而, 据作者所知, 尚无针对时空依赖波动方程的类似方法. 本文旨在填补这一空白, 将优化识别框架推广至波动方程, 并提出一种不依赖具体微观结构的有效系数识别方案, 同时从理论上证明当均质化极限存在时, 该方法具有一致性.

2. 主要定理或主要结果

本文在以下两个基本假设下获得主要结果:

假设 1 (初始数据的丰富性): 初始位移 $v_0$ 的傅里叶级数 $v_0(x) = \sum_{k\in\mathbb{Z}^d} c_k e^{2\pi i k\cdot x}$ 中存在一组波向量 $\{k_j\}_{j=1}^{d^*}$ ($d^* = d(d+1)/2$) 使得 $c_{k_j}\neq 0$ 且
$$\operatorname{span}\{ k_j k_j^\top : j=1,\dots,d^* \} = \mathbb{R}^{d\times d}_{\rm sym}.$$
该假设保证了解包含足够多的模态, 以确保优化问题的适定性.
假设 2 (均质化极限的存在性): 存在一族系数 $\mu_\varepsilon$ 和对应解 $u_\varepsilon$, 使得当 $\varepsilon\to0$ 时,
$$\nabla u_\varepsilon \rightharpoonup \nabla u_{\mathrm{hom}}, \quad \partial_t u_\varepsilon \rightharpoonup \partial_t u_{\mathrm{hom}} \quad \text{在 } L^2(0,T;L^2(T^d)) \text{ 中弱收敛},$$
且极限 $u_{\mathrm{hom}}$ 满足常系数均质化波动方程
$$\partial_{tt} u_{\mathrm{hom}} - \operatorname{div}(\mu_{\mathrm{hom}} \nabla u_{\mathrm{hom}}) = 0, \quad u_{\mathrm{hom}}|_{t=0}=v_0,\; \partial_t u_{\mathrm{hom}}|_{t=0}=v_1,$$
其中 $\mu_{\mathrm{hom}}$ 是一个正定对称矩阵. 例如, 对于形如 $\mu_\varepsilon(x,t) = \mu_0(x/\varepsilon) + \varepsilon \mu_1(t/\varepsilon)$ 的弱时空振荡系数, 该假设成立 (见 [5]).

主要理论结果:

极小化子的存在性 (引理 3.1): 在假设 1 下, 定义在可容许集 $\mathcal{A} = \{\mu\in\mathbb{R}^{d\times d}_{\rm sym} : \mu\xi\cdot\xi \ge 0\}$ 上的二次代价泛函
$$\Phi_\varepsilon(\mu) = \|\square^{-1}\big(\partial_{tt}u_\varepsilon - \operatorname{div}(\mu\nabla u_\varepsilon)\big)\|_{L^2(0,T;L^2(T^d))}^2$$
连续且满足强制估计 $\Phi_\varepsilon(\mu) \ge \alpha |\mu|^2 - C$, 从而至少存在一个极小化子 $\mu_\varepsilon = \arg\min_{\mu\in\mathcal{A}} \Phi_\varepsilon(\mu)$. 在一维情形下, $\mu_\varepsilon$ 唯一并具有显式表达式 (3.12); 在高维下, 若矩阵 $A$ 正定则极小化子唯一并由 $\mu_\varepsilon = A^{-1} b$ 给出.
渐近一致性 (定理 4.1 及后续推论): 在假设 2 下, 有
$$\inf_{\mu\in\mathcal{A}} \Phi_\varepsilon(\mu) \to 0 \quad \text{当 } \varepsilon\to0.$$
进而可以证明, 所识别的系数收敛到均质化系数:
$$\lim_{\varepsilon\to0} \mu_\varepsilon = \mu_{\mathrm{hom}}.$$
此外, 对应的有效解 $u(\mu_\varepsilon)$ 满足
$$\|u(\mu_\varepsilon) - u_{\mathrm{hom}}\|_{L^2(0,T;L^2(T^d))} \to 0,$$
以及弱收敛
$$\nabla u(\mu_\varepsilon) \rightharpoonup \nabla u_{\mathrm{hom}}, \quad \partial_t u(\mu_\varepsilon) \rightharpoonup \partial_t u_{\mathrm{hom}} \quad \text{在 } L^2(0,T;L^2(T^d)) \text{ 中}.$$
数值验证: 数值实验在周期与非周期介质中进行, 均显示所识别系数能逼近均质化极限或提供有意义的有效近似, 即使均质化极限的存在性未获理论保证.

3. 研究方法、关键技术和核心工具

本文的主要创新在于将椭圆多尺度问题的优化识别框架系统性地推广到时空调制波动方程. 其技术要点可归纳如下:

1. 二次代价泛函的构造

核心难点在于双曲方程的正则性低于椭圆方程, 直接对残差 $\partial_{tt}u_\varepsilon - \operatorname{div}(\mu\nabla u_\varepsilon)$ 在 $L^2$ 空间中测量会导致对 $\partial_{tt}u_\varepsilon$ 和 $\mu\nabla u_\varepsilon$ 的过高正则性要求. 作者的解决方案是引入一个逆波算子 $\square^{-1}$: 它是齐次初值条件下具有单位速度的波动方程的解算子, 即 $v = \square^{-1}f$ 满足

$$\partial_{tt} v - \Delta v = f, \quad v|_{t=0}=0,\; \partial_t v|_{t=0}=0.$$

该算子将 $L^2(0,T;H^1(T^d)^*)$ 映射回 $L^2(0,T;H^1(T^d))$, 从而将原方程的残差正则化. 由此定义的代价泛函

$$\Phi_\varepsilon(\mu) = \|\square^{-1}(\partial_{tt}u_\varepsilon - \operatorname{div}(\mu\nabla u_\varepsilon))\|_{L^2(0,T;L^2(T^d))}^2$$

不仅是 $\mu$ 的二次型 (见 (3.10)–(3.11)), 便于数值优化, 还保留了 $\mu$ 的显式依赖关系, 为理论分析提供了方便.

2. 极小化子的存在性: 丰富初始条件的作用

证明 $\Phi_\varepsilon$ 强制 (coercive) 的关键在于线性算子 $\mathcal{Q}_\varepsilon(\mu) = \square^{-1}(\operatorname{div}(\mu\nabla u_\varepsilon))$ 的单射性. 若 $\mathcal{Q}_\varepsilon(\mu_0)=0$, 则 $\operatorname{div}(\mu_0\nabla u_\varepsilon)=0$; 特别地, 取 $t=0$ 得 $\operatorname{div}(\mu_0\nabla v_0)=0$. 代入傅里叶展开并利用假设 1 中波向量张成全空间的条件 (引理 A.1), 推出 $\mu_0=0$. 由于 $\mu$ 所在空间维数有限, 单射性等价于下界估计 $|\mu| \le C \|\mathcal{Q}_\varepsilon(\mu)\|_{L^2}$, 由此得到 $\Phi_\varepsilon$ 的二次下界, 加上连续性便得到极小化子的存在. 这一处理将原本无穷维的反问题转化为有限维约束下的二次优化, 为理论保证和高效计算奠定了基础.

3. 渐近一致性: 能量估计与特殊测试函数的选取

证明 $\inf \Phi_\varepsilon(\mu) \to 0$ 是本文最精巧的部分. 主要步骤是证明 $\Phi_\varepsilon(\mu_{\mathrm{hom}})\to 0$. 记 $v_\varepsilon = \square^{-1}(\partial_{tt}u_\varepsilon - \operatorname{div}(\mu_{\mathrm{hom}}\nabla u_\varepsilon))$, 它满足

$$\partial_{tt}v_\varepsilon - \Delta v_\varepsilon = \operatorname{div}((\mu_{\mathrm{hom}}-\mu_\varepsilon)\nabla u_\varepsilon), \quad v_\varepsilon|_{t=0}=0,\; \partial_t v_\varepsilon|_{t=0}=0.$$

选取巧妙的时间依赖测试函数 $\varphi(t) = -\Delta^{-1}(\partial_t v_\varepsilon(t))$ (即泊松方程的解, 且均值为零), 则 $\nabla\varphi$ 与 $\partial_t v_\varepsilon$ 满足对偶关系, 代入弱形式后得到能量恒等式:

$$\frac{1}{2}\frac{d}{dt}\big(\|\nabla\varphi(t)\|_{L^2}^2 + \|v_\varepsilon(t)\|_{L^2}^2\big) = -\langle (\mu_\varepsilon - \mu_{\mathrm{hom}})\nabla u_\varepsilon, \nabla\varphi \rangle.$$

利用系数有界性和 Cauchy-Schwarz 不等式, 并记 $p_\varepsilon(t) = \|\nabla\varphi\|_{L^2}^2 + \|v_\varepsilon\|_{L^2}^2$, 可得微分不等式

$$\frac{d}{dt}p_\varepsilon(t) \le C \|\nabla u_\varepsilon(t)\|_{L^2} \sqrt{p_\varepsilon(t)}.$$

由于均质化假设给出 $\nabla u_\varepsilon$ 在 $L^2(L^2)$ 中有界, 应用 Gronwall 型不等式可证 $p_\varepsilon(t) \le C \varepsilon$ (或至少 $p_\varepsilon \to 0$). 从而 $\|v_\varepsilon\|_{L^2(L^2)} \to 0$, 即 $\Phi_\varepsilon(\mu_{\mathrm{hom}})\to 0$. 再结合 $\Phi_\varepsilon$ 的强制性、极小化子性质以及常规的弱收敛论证, 即可推出 $\mu_\varepsilon \to \mu_{\mathrm{hom}}$ 和有效解的收敛. 该方法成功避免了依赖具体均质化展开, 仅依靠抽象收敛假设便完成了误差估计.

4. 非周期与不适定情形下的适用性

尽管理论分析建立在均质化极限存在的前提上, 数值实验表明该方法即使在未知均质化极限存在与否的情况下仍能稳定地提取出有效的常系数, 展现出良好的鲁棒性和实用潜力.

4. 与之前工作的比较

本文与现有文献的比较主要体现在以下几个方面:

1. 与椭圆多尺度问题优化识别方法的比较

Le Bris, Legoll 和 Lemaire [14,15] 率先提出了一种针对椭圆多尺度方程的有效系数识别方法, 其核心是通过解一个加权最小二乘问题来拟合观测解和有效解. 后续工作如 [2,20] 进一步拓展了该框架. 本文首次成功将该思想移植到波动方程. 移植过程中需克服的本质困难有:

波动方程的解只具有时间方向上的有界能量, 而非椭圆方程 $H^1$ 空间中的强紧性, 因此代价泛函的强制性的获取必须依赖于初始数据的丰富性假设.
波动方程的均质化涉及时间导数, 因而在渐近分析中需要处理时间振荡可能带来的不稳定, 如文献 [6] 所指出的; 本文通过引入逆波算子 $\square^{-1}$ 并结合能量估计巧妙地化解了该困难.

2. 与均质化理论的比较

经典均质化理论 (G-收敛或 H-收敛 [22]) 在算子意义下确定有效系数, 一般需要求解辅助的细胞问题, 且系数通常不具备点态收敛性. 本文通过优化识别直接给出一个常数矩阵 $\mu_\varepsilon$, 并证明了它在 $\varepsilon\to0$ 时点态收敛到 $\mu_{\mathrm{hom}}$ (当后者存在时). 这种点态收敛性比 G-收敛更强, 为实际计算提供了更直接的指导.

此外, 传统均质化通常以系数的周期性或特定结构为前提. 本文的方法仅需满足抽象的“均质化极限存在”假设, 因此可涵盖周期性介质 (如 (4.18) 中的弱调制情形) 以及其他可能无可直接利用的结构 (例如数值实验中考虑的非周期或随机介质), 适用范围更广.

3. 与时空波动方程严格均质化工作的比较

对于时空调制波动方程, 已知的严格均质化结果依赖于小微尺度且弱时间调制的特殊形式 (例如 [5] 中 $\mu_\varepsilon(x,t)=\mu_0(x/\varepsilon)+\varepsilon \mu_1(t/\varepsilon)$). 本文并没有提出新的均质化定理, 而是发展了一个系统性的优化识别框架, 其优势在于:

不需要预先知道 $\mu_\varepsilon$ 的具体解析形式或结构, 仅利用解的测量数据.
即使对于均质化不成立或难以分析的介质, 方法仍能产生有实际意义的有效系数.
理论分析为均质化系数提供了另一种一致性保证, 为工程应用中的参数识别提供了数学基础.

4. 与其他有效系数识别方法的比较

在椭圆问题中, 已有参数估计 [17]、分离振荡 [12]、神经网络 [18] 等方法. 这些方法要么依赖于多次测量或先验信息, 要么理论保证不足. 相比之下, 本文方法具有以下特点:

在理论上, 在合理的假设下严格证明了识别系数收敛到均质化系数以及有效解收敛.
在数值上, 优化问题归结为求解一个低维线性系统 (矩阵 $A$ 和向量 $b$), 计算代价低, 适合大规模三维问题.
通过选取适当的初始条件 (满足假设 1), 仅需一次或少数几次测量即可完成识别, 避免了对大量测量数据的需求.

文中并未与其他识别方法进行量化比较, 但理论上的进步和数值实验的展示充分证明了该方法的竞争力.

2. 非均匀结构动态非线性湿热-热-力耦合问题的高效高阶多尺度方法及其收敛性估计

Efficient higher-order multi-scale method and its convergence estimate for dynamic nonlinear hygro-thermo-mechanical coupling problems of heterogeneous structures

作者: Yifei Ding, Hao Dong, Jiale Linghu, Yangshuai Wang

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.05580v1

摘要翻译: 本文提出一种新颖的高阶多尺度（HOMS）计算框架，用于非均匀结构中非线性湿热-热-力（H-T-M）耦合问题的高效、高精度和低成本模拟。所研究模型的内在非线性主要来源于温度或湿度相关的材料性质，并且该模型还考虑了由放热、耗湿化学反应（如水化反应）引起的依赖于温度的内部热源项和湿度汇项。本文的主要贡献如下。首先，利用多尺度渐近方法结合泰勒级数展开，针对具有强空间非均匀性的非均匀结构中的非线性H-T-M耦合问题，构造了一个包含高阶校正项的高精度多尺度渐近模型。其次，对多尺度渐近解在点态和积分意义下给出了严格的误差估计，从理论上证明了所提出的HOMS方法的必要性和优越性。第三，基于有限差分法和有限元法，开发了一个包含离线阶段和在线阶段的高效两阶段数值算法，并严格证明了其收敛性。最后，进行了二维和三维数值实验以评估所提出的HOMS方法的计算性能，结果表明该方法具有优异的数值精度和鲁棒性，且计算开销低。

Comment: 无

链接: http://arxiv.org/abs/2608.05580v1

1. 研究对象和背景

本文研究一类描述非均匀结构在多物理场耦合作用下的非线性偏微分方程组。设宏观区域 $\Omega \subset \mathbb{R}^n (n=2,3)$ 具有周期性微观结构，其单胞 $Y=[0,1]^n$，小参数 $\varepsilon$ 为微观尺度与宏观尺度之比。温度场 $T^\varepsilon(\mathbf{x},t)$，湿度场 $\omega^\varepsilon(\mathbf{x},t)$ 和位移场 $\mathbf{u}^\varepsilon(\mathbf{x},t)$ 满足以下非线性的湿热-力耦合问题:

$$ \begin{cases} \rho^\varepsilon(\mathbf{x},T^\varepsilon)c^\varepsilon(\mathbf{x},T^\varepsilon)\frac{\partial T^\varepsilon}{\partial t} - \frac{\partial}{\partial x_i}\left(k_{ij}^\varepsilon(\mathbf{x},T^\varepsilon)\frac{\partial T^\varepsilon}{\partial x_j}\right) = h(\mathbf{x},t) + Q^\varepsilon_{\text{hyd}}(\mathbf{x},T^\varepsilon), \\ \frac{\partial \omega^\varepsilon}{\partial t} - \frac{\partial}{\partial x_i}\left(g_{ij}^\varepsilon(\mathbf{x},\omega^\varepsilon)\frac{\partial \omega^\varepsilon}{\partial x_j}\right) = m(\mathbf{x},t) - S^\varepsilon_{\text{hyd}}(\mathbf{x},T^\varepsilon), \\ -\frac{\partial}{\partial x_j}\left(C_{ijkl}^\varepsilon(\mathbf{x},T^\varepsilon)\frac{\partial u_k^\varepsilon}{\partial x_l} - \alpha_{ij}^\varepsilon(\mathbf{x},T^\varepsilon)(T^\varepsilon-\tilde T) - \beta_{ij}^\varepsilon(\mathbf{x},T^\varepsilon)(\omega^\varepsilon-\tilde \omega) \right) = f_i(\mathbf{x},t), \end{cases} $$

配备适当的边界条件和初始条件。其中，系数 $\{\rho^\varepsilon, c^\varepsilon, k_{ij}^\varepsilon, g_{ij}^\varepsilon, C_{ijkl}^\varepsilon, \alpha_{ij}^\varepsilon, \beta_{ij}^\varepsilon\}$ 为材料参数, 是微观变量 $\mathbf{y} = \mathbf{x}/\varepsilon$ 的快速振荡周期函数, 且显式依赖于温度 $T^\varepsilon$ 或湿度 $\omega^\varepsilon$, 因此整个系统是强非线性的。此外, $Q^\varepsilon_{\text{hyd}}$ 和 $S^\varepsilon_{\text{hyd}}$ 分别表示由于温度依赖的化学反应 (如水泥水化或热固性树脂固化) 产生的内部热源和湿汇项, 也具有温度依赖性。这类问题广泛存在于电子封装可靠性分析、复合材料湿热老化、混凝土早期开裂等工程领域, 具有重要的实际意义。

传统数值方法 (有限差分、有限元、有限体积法) 直接求解时, 需极细网格捕捉微观振荡, 导致计算量巨大甚至不可行, 且非线性、振荡性和多物理耦合对方法的精度和收敛性带来严峻挑战。因此, 发展高效且具有理论保证的多尺度方法成为该领域的前沿课题。

目前已有多种多尺度方法, 如渐近均匀化方法 (AHM) (Bensoussan et al., 2011)、异质多尺度方法 (HMM) (E et al., 2005)、变分多尺度法 (VMS) (Hughes et al., 1998)、多尺度有限元 (MsFEM) (Hou et al., 1999) 等。特别是, Cui 团队在过去三十年建立了一套高阶多尺度方法 (HOMS) 用于复合材料的热、力学及多物理行为的精确模拟 (Wang et al., 2015; Cao et al., 2002; Dong et al., 2019a; Yang et al., 2016; Dong et al., 2019b), 但仅限于线性问题。对于非线性问题, Andrianov et al. (2017), Chung et al. (2001) 等用 AHM 处理了温度相关热传导, Efendiev 和 Hou (2009) 提出了 MsFEM 用于非线性椭圆和抛物方程; Dong et al. (2023, 2025b) 将 HOMS 推广到非线性热力耦合问题并给出严谨的收敛分析。然而, 据本文作者所知, 针对同时包含热、湿、力耦合且具有材料非线性的多尺度问题, 现有研究甚少。Gholami et al. (2022) 采用一阶计算均匀化 (FE$^2$) 进行湿热条件下的失效分析, 但缺乏系统的高阶理论框架和误差估计; Dong et al. (2025a) 构造了准周期结构湿热力耦合的高阶多尺度方法, 但未考虑材料和源项对温度、湿度的依赖性。因此, 本研究试图填补这一空白, 为具有高度振荡和强非线性系数的结构湿热力耦合问题建立一种高效且严格的高阶多尺度计算框架。

2. 主要定理或主要结果

在本文构造的高阶多尺度渐近框架下, 未知场 $T^\varepsilon$, $\omega^\varepsilon$, $\mathbf{u}^\varepsilon$ 展开为 (2.3): 零阶项 $T^{(0)}(\mathbf{x},t)$, $\omega^{(0)}(\mathbf{x},t)$, $\mathbf{u}^{(0)}(\mathbf{x},t)$ 不依赖微观变量, 满足宏观均匀化方程 (2.23), 其中的均匀化系数 $\hat{k}_{ij}$, $\hat{g}_{ij}$, $\hat{C}_{ijkl}$, $\hat{\alpha}_{ij}$, $\hat{\beta}_{ij}$, 以及源项 $\hat{Q}_{\text{hyd}}$, $\hat{S}_{\text{hyd}}$ 由 (2.24) 定义, 它们同时经由材料参数的零阶项和一阶细胞函数依赖于宏观量 $T^{(0)}$ 或 $\omega^{(0)}$。一阶修正项 $T^{(1)}$, $\omega^{(1)}$, $u_i^{(1)}$ 由 (2.17) 给出, 通过一阶细胞函数 $H_{\alpha_1}(\mathbf{y},T^{(0)})$, $J_{\alpha_1}(\mathbf{y},\omega^{(0)})$, $X_{ih}^{\alpha_1}(\mathbf{y},T^{(0)})$, $M_i(\mathbf{y},T^{(0)})$, $N_i(\mathbf{y},T^{(0)})$ 与宏观解梯度耦合, 这些细胞函数满足单胞上的椭圆边值问题 (2.18)-(2.22), 且由于系数依赖宏观量而呈现出准周期性。

为了更高保真度地捕捉微观振荡, 作者进一步构造了二阶修正项 $T^{(2)}$, $\omega^{(2)}$, $u_i^{(2)}$, 其控制方程源自 $O(\varepsilon^0)$ 阶关系式并消去宏观均匀化部分, 如 (2.25), 这些方程同样定义在单胞上, 右端项含宏观解的高阶导数和细胞函数及其导数。

本文主要的严格结果是:

通过对多尺度渐近解与真解误差的逐点比较和积分估计, 理论证明了高阶多尺度 (HOMS) 解相较于传统低阶多尺度 (LOMS) 解 (仅含一阶项) 在捕捉微观振荡信息上的必要性和优越性;
在某些简化和正则性假设下, 导出了 HOMS 解的显式误差收敛阶, 即对温度、湿度和位移, 在适当范数下, 误差满足 $\| T^\varepsilon - T_{\text{HOMS}} \| \le C \varepsilon^2$ (类似对 $\omega$ 和 $\mathbf{u}$);
证明了所设计的两阶段算法 (离线求解细胞问题 + 在线求解宏观均匀化问题并重构多尺度解) 的收敛性, 即数值误差可控制且与渐近截断误差协调;
二维和三维数值实验显示该方法在极低计算开销下具有优异的精度和鲁棒性。

(注: 具体定理的精确表述和详细假设条件在提供的摘录中未给出, 但文中明确提及了严格误差估计和收敛性证明。)

3. 研究方法、关键技术和核心工具

本文的技术核心在于处理非线性材料/源项以及建立高阶多尺度模型的系统方法。主要障碍是: 方程组中所有材料系数和源项均显式依赖于未知函数 $T^\varepsilon$ 或 $\omega^\varepsilon$, 这使得直接运用线性周期均匀化理论失效。作者的创新在于将泰勒展开与经典的多尺度渐近分析相结合。

具体做法如下:

假设 $T^\varepsilon$, $\omega^\varepsilon$, $\mathbf{u}^\varepsilon$ 可表为形如 (2.3) 的 $\varepsilon$ 幂级数。对于任一依赖于 $T^\varepsilon$ 的系数, 例如弹性张量 $C_{ijkl}^\varepsilon(\mathbf{x},T^\varepsilon) = C_{ijkl}(\mathbf{y},T^\varepsilon)$, 在 $T^{(0)}$ 处作泰勒展开 (2.4):
$$C_{ijkl}(\mathbf{y},T^\varepsilon) = C_{ijkl}(\mathbf{y},T^{(0)}) + \varepsilon T^{(1)} D_{(0,1)} C_{ijkl}(\mathbf{y},T^{(0)}) + \varepsilon^2 \left[ T^{(2)} D_{(0,1)} C_{ijkl} + \frac12 (T^{(1)})^2 D_{(0,2)} C_{ijkl} \right] + O(\varepsilon^3).$$
从而将非线性的振荡系数分解为不同阶次的量: $C_{ijkl}^{(0)}(\mathbf{y},T^{(0)})$, $\varepsilon C_{ijkl}^{(1)}(\mathbf{x},\mathbf{y},T^{(0)})$, $\varepsilon^2 C_{ijkl}^{(2)}(\mathbf{x},\mathbf{y},T^{(0)})$, 类似处理其他参数和源项 (2.5)-(2.12)。这样, 原非线性问题转化为一系列具有已知系数的线性扰动问题。
将展开代入控制方程, 并利用链锁法则 (2.2), 比较 $\varepsilon$ 同次幂系数, 得到各阶方程。
- $O(\varepsilon^{-2})$: 得出零阶项与 $\mathbf{y}$ 无关, 即 $T^{(0)} = T^{(0)}(\mathbf{x},t)$ 等。
- $O(\varepsilon^{-1})$: 分离变量得到一阶修正与宏观梯度的关系, 引入一阶细胞函数 $H_{\alpha_1}, J_{\alpha_1}, X_{ih}^{\alpha_1}, M_i, N_i$, 它们满足形如 (2.18)-(2.22) 的单胞问题, 系数仅为 $\mathbf{y}$ 和宏观量 $T^{(0)}$ (或 $\omega^{(0)}$) 的函数, 因而在宏观求解时需视 $T^{(0)}$ 为参数。这些问题的弱形式是典型的椭圆方程, 可用有限元法在单胞上求解。
- $O(\varepsilon^0)$: 在单胞上积分并利用周期边界条件导出宏观均匀化方程 (2.23), 其系数通过细胞函数的微观平均确定 (2.24)。同时, 从 $O(\varepsilon^0)$ 关系中减去宏观平均部分, 得到 $T^{(2)}$ 等的控制方程 (2.25), 其右端包含 $T^{(0)}$ 的高阶导数及细胞函数的贡献, 可视为驱动二阶修正的源。这样, 无需直接求解全球的细尺度问题, 仅通过求解一系列单胞问题及一个宏观问题即可重构高精度的多尺度近似解。
误差分析方面: 文中采用了两种视角。首先是逐点比较, 通过分析低阶微尺度解 (LOMS) 与高阶解 (HOMS) 在各尺度上的差异, 展示高阶项在恢复微观振荡方面不可或缺。其次是基于能量的积分估计, 作者借助均匀化理论中的余项估计思想, 将误差表示为边界层效应和余项的组合, 利用椭圆正则性、Sobolev 嵌入以及 Gronwall 不等式等技术, 导出了显式的 $\varepsilon$ 阶收敛界。核心困难在于处理非线性项造成的线性化误差, 作者通过泰勒展开的余项进行控制, 并在足够光滑的假设下获得最优阶。此外, 文中还证明了两阶段数值算法 (细胞问题用 FEM, 宏观问题用 FDM 或 FEM) 的收敛性, 借助 Strang 引理一类的抽象框架, 将离散误差与渐近误差结合。
计算实现: 设计了两阶段离线-在线算法。离线阶段: 对于给定的宏观量 $T^{(0)}$ 和 $\omega^{(0)}$ 的可能范围, 预求解所有一阶和二阶细胞问题, 存储细胞函数及其梯度。在线阶段: 求解宏观均匀化方程得到 $T^{(0)}$, $\omega^{(0)}$, $\mathbf{u}^{(0)}$, 再利用存储的细胞函数通过组合 (如 (2.17) 及二阶修正的类似表达式) 直接合成多尺度解。这种方式避免了在每个时间步或宏观迭代步反复求解微观问题, 极大降低了计算成本, 同时保留了高阶精度。

4. 与之前工作的比较

与线性 HOMS 方法对比: 以往高阶多尺度工作 (如 Wang et al. 2015; Dong et al. 2019a; Yang et al. 2016) 仅针对线性材料, 即系数不依赖于温度、湿度等未知场。本文通过泰勒展开将非线性问题分解为一系列线性细胞问题, 但均匀化系数仍宏观相关 (依赖 $T^{(0)}$ 或 $\omega^{(0)}$), 同时引入了高阶修正项以恢复微观细节。这在理论和算法上都是一次重要的非线性推广。与另一类基于宏微观耦合迭代的非线性均匀化 (如增量式均匀化) 相比, 本方法不需求解非线性微观问题, 只需求解参数化的线性细胞问题, 计算效率更高。
与现有非线性多场多尺度方法对比: 对于热力耦合, Dong et al. (2023, 2025b) 已建立含收敛分析的高阶多尺度框架, 但未涉及湿扩散。本文耦合了湿度方程, 且考虑了温度依赖的内部热源和湿汇, 使模型更接近实际工程环境 (如水泥水化、电子封装中的吸湿效应)。Gholami et al. (2022) 采用了 FE$^2$ 方法, 缺乏高阶修正和收敛理论, 且每个宏观积分点需求解一个非线性微观问题, 代价高昂。本文方法通过离线-在线分离, 只需在线求解几个宏观 PDE, 计算量远低于 FE$^2$。此外, 本文的理论误差估计是 FE$^2$ 所不具备的。
与 Dong et al. (2025a) 的对比: Dong et al. (2025a) 首次将 HOMS 应用于湿热力耦合结构, 并考虑了准周期性, 然而该工作假设材料性质与温度、湿度无关 (线性化)。本文的关键推进在于: 使用泰勒展开处理温度/湿度依赖的材料和源项, 将其纳入渐近框架, 从而使细胞函数成为准周期的 (依赖于 $T^{(0)}$ 或 $\omega^{(0)}$), 并且宏观均匀化系数也显式依赖于这些宏观场。这既增强了物理适用性, 也带来了理论分析的新挑战, 如一致性误差估计中需额外控制泰勒余项。
与经典渐近均匀化及 HMM 的比较: 传统 AHM 对非线性问题一般仅给出零阶均匀化解, 缺少高阶修正和严格误差分析。HMM 是一种通用数值框架, 也可处理非线性, 但其误差分析通常基于解的正则性假设, 较少给出显式收敛阶。本工作继承了 AHM 展开方法的解析深度, 能够提供点态和积分范数下的显式误差阶, 且计算上利用周期性可预先计算基础函数, 效率更高。不过, 本方法依赖于周期假设和泰勒展开, 对于强非线性或塑性等路径相关问题, 需要进一步拓展。
局限性与展望: 文中假设了材料系数对温度和湿度的光滑性, 以便进行泰勒展开; 结构需为严格周期排列; 求解细胞问题时需要预知宏观量的变化范围以便离散参数空间, 这可能带来一定的内存开销。对于更一般的多尺度问题 (如随机介质、损伤演化), 本方法尚不能直接应用。尽管如此, 该工作为非线性多物理场耦合的多尺度计算提供了坚实的理论基础和实用算法。

摘要翻译: 我们提出并分析了一族针对单位球面上切向向量场的无散核插值方法。从欧氏空间中的标量径向核出发，我们构造了切空间值、曲面无散的矩阵核，且无需反复应用曲面微分算子。该构造将无散约束的几何实施与标量生成元的选择分离开来，并允许得到较经典势方法正则性要求更低的低阶变体。借助向量球谐函数，我们推导出显式的核表示并刻画了其 Fourier 乘子。我们还引入一种逆 Laplace–Beltrami 构造，它保留了底层标量球带核的乘子。对于散乱节点插值，我们建立了插值矩阵最小特征值的下界，并导出了逐点和 Sobolev 误差估计，其中对光滑性高于原生空间的目标函数还包含超收敛结果。数值实验验证了理论收敛性和稳定性结果，而在非球面曲面上的额外算例则说明了该核公式在球面以外的适用性。

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arXiv ID: 2608.06107v1

链接: http://arxiv.org/abs/2608.06107v1

摘要: Machine learning offers a promising avenue to accelerate physical simulations by replacing computationally expensive traditional Partial Differential Equation (PDE) solvers with fast, differentiable surrogate models. However, standard auto-regressive ML emulators often suffer from error accumulation over long horizons and struggle to capture the stochasticity of complex physical systems. In this paper, we propose Kastor, a comprehensive methodology to adapt a deterministic physics foundation model into a highly efficient and accurate generative surrogate. First, we introduce a two-stage inference scheme that combines a large-stride causal auto-regressive model with a non-causal temporal super-resolution network, significantly reducing error accumulation while minimizing computational cost. Second, we present Mean prediction regularization (MPR), a novel training objective that constrains the generative model to predict the deterministic distribution mean under null noise conditioning. This regularization dramatically improves the performance and stability of both Functional Generative Networks (FGN) and diffusion-based emulators. Finally, we demonstrate that incorporating spatial gradient matching improves the accuracy and physical fidelity of the simulations as measured by power spectrum density. Extensive evaluations on diverse simulation datasets of the benchmark The Well show that with these components, our model outperforms competing methods in forecasting accuracy, spectral consistency, and computational efficiency. Our model achieves a 42.9% average reduction in forecasting compared to our reference based on the Walrus finetuning methodology, and outperforms Walrus for 8 out of 10 datasets on variance-normalized RMSE (VRMSE).

7. 肿瘤生长的热力学一致Cahn-Hilliard-Navier-Stokes模型

A Thermodynamically Consistent Cahn-Hilliard-Navier-Stokes Model for Tumor Growth

作者: Chenyang Li, Ping Lin, Hui Yu, Haibiao Zheng

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.06099v1

链接: http://arxiv.org/abs/2608.06099v1

摘要: This work develops a thermodynamically consistent phase-field model for tumor growth based on the energetic variational framework. The model couples the Cahn-Hilliard equations for tumor evolution and nutrient transport with the incompressible Navier-Stokes equations. A first-order time discretization scheme based on the Multiple Scalar Auxiliary Variables (MSAV) approach together with a pressure-correction strategy is proposed to efficiently handle the nonlinear and coupled structure of the system. The proposed scheme is rigorously proved to be unconditionally energy stable and mass conservative. Furthermore, optimal first-order temporal error estimates are established for the tumor phase-field variable, the nutrient concentration, and the fluid velocity. Finally, numerical experiments demonstrate the effectiveness and robustness of the proposed method and verify the theoretical convergence rates.

8. 电机热瞬态模拟的约化基方法

Reduced Basis Method for Simulating Thermal Transients in Electric Machines

作者: Herbert Egger, Eva-Maria Haslhofer

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.05904v1

链接: http://arxiv.org/abs/2608.05904v1

摘要: Reduced Basis (RB) methods provide low-dimensional approximations of parametrized partial differential equations with controllable accuracy. We discuss the construction of RB approximations for transient thermal simulation of an induction motor and compare their performance to Finite Element (FE) models and Lumped Parameter Thermal Networks (LPTNs) after calibration to measurements. Numerical results demonstrate that RB models can achieve FE accuracy, allow for a fully automatic construction, and offer computational efficiency comparable to traditional LPTNs.

9. Rothe 时间离散与截断 Westervelt 系统的弱解

Rothe time discretization and weak solutions for a cutoff Westervelt system

作者: Marvin Fritz

类别: math.NA, math.AP

发布日期: 2026-08-06

arXiv ID: 2608.05865v1

链接: http://arxiv.org/abs/2608.05865v1

摘要: We study a fully implicit Rothe time discretization for a cutoff first-order formulation of the Westervelt equation. The key ingredients are the enthalpy variable and the primitive mobility variable, which turn each nonlinear time step into a uniformly monotone elliptic problem and avoid higher-order energy estimates and inverse inequalities. For every time step, the discrete problem reduces to a monotone elliptic equation, which yields well-posedness of the Rothe scheme by the Browder-Minty theorem. We derive a discrete energy inequality, establish compactness for the transformed variable by an Alt-Luckhaus type argument, and pass to the limit to obtain existence of weak solutions for the cutoff first-order system. We prove a weak-strong uniqueness principle for the cutoff problem and formulate a conditional a posteriori criterion under which the cutoff is inactive. For sufficiently regular solutions of the forced cutoff problem, we establish consistency estimates which identify the temporal residuals produced by the Rothe approximation in the natural discrete-in-time weak norms. These estimates provide a rigorous consistency basis for the observed temporal behavior. Finally, numerical experiments illustrate the stability of the scheme, its observed near first-order temporal behavior, and inactivity of the cutoff along the computed discrete trajectories in the tested regimes.

10. 动理学 Fokker-Planck 方程的核公式

A Kernel Formula for Kinetic Fokker-Planck Equations

作者: Fuqun Han, Wuchen Li

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.05770v1

链接: http://arxiv.org/abs/2608.05770v1

摘要: We derive an explicit one-step kernel formula for the kinetic Fokker-Planck equation. The construction begins with a reference dynamics admitting an explicit transition kernel and uses a carefully chosen auxiliary weight to encode the drift perturbation. The resulting kernel formula provides an explicit approximation of the density evolution. In a weighted phase-space setting, we establish local weak consistency and first-order finite-time weak convergence under suitable regularity and moment assumptions. We also give a formal variational interpretation of the formula in terms of a regularized Wasserstein-type proximal operator and present analogous constructions for a broader class of parabolic equations. Numerical experiments illustrate the accuracy of the kernel approximation and its application to deterministic particle schemes for kinetic sampling.

11. 势匹配最优传输：用于精确 $p$-Wasserstein 动力学的连续正规化流

Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact $p$-Wasserstein Dynamics

作者: Lishuo Zhang, Ruizhi Huang, Yang Yu, Lei Li

类别: cs.LG, math.NA

发布日期: 2026-08-06

arXiv ID: 2608.05666v1

链接: http://arxiv.org/abs/2608.05666v1

摘要: We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou–Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model’s own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou–Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

12. 动态子结构中基于高阶Craig-Bampton方法的降阶模型先验误差估计器

A priori error estimator for reduced-order models based on the higher-order Craig-Bampton method in dynamic substructuring

作者: Jaemin Kim, Seung-Hwan Boo

类别: math.NA, math-ph

发布日期: 2026-08-06

arXiv ID: 2608.05625v1

链接: http://arxiv.org/abs/2608.05625v1

摘要: The Craig-Bampton (CB) method is a widely used dynamic substructuring technique based on component mode synthesis (CMS). The higher-order Craig-Bampton (HCB) method augments the CB basis with residual modes from a Neumann series expansion of the residual flexibility matrix, where HCB-n retains terms up to the n-th order and is reduced back to the CB size via the System Equivalent Reduction Expansion Process (SEREP), achieving improved accuracy at the same model dimension. However, assessing the accuracy of a reduced model without solving the full-order problem remains a fundamental challenge: if the full-order solution is required to evaluate the error, the purpose of model reduction is defeated. Despite the demonstrated superiority of the HCB method, no a priori error estimator (one that predicts eigenvalue errors without solving the full-order eigenvalue problem) has been proposed for it. The present work addresses this gap with a hierarchical estimation framework that exploits the nested Ritz subspace structure of the HCB method, where each higher-order solution serves as a reference for estimating the error of the preceding order. The framework provides (i) a generalized CB error estimator derived from a Rayleigh quotient perturbation analysis, and (ii) a novel HCB-1 error estimator using the HCB-2 eigensolution as a reference. Numerical examples across models of varying geometric complexity validate both estimators.

13. 中等长波方程与扰动Benjamin–Ono模型的分裂方法

Splitting methods for Intermediate Long Wave and perturbed Benjamin–Ono models

作者: Yvonne Alama Bronsard, Clémentine Courtès, Benjamin Melinand

类别: math.NA, math.AP

发布日期: 2026-08-06

arXiv ID: 2608.05540v1

链接: http://arxiv.org/abs/2608.05540v1

摘要: Recently, novel integrability techniques, and most notably Birkhoff coordinates followed by an explicit formula, have been introduced to solve the Benjamin–Ono (BO) equation, leading to major theoretical and computational advances. We propose to perturb this explicit BO formula via a novel splitting method for numerically solving a class of PDEs, which includes the Intermediate Long Wave (ILW) equation, as well as other quasilinear equations that are not necessarily integrable. Using the Birkhoff coordinates of BO we establish first-order convergence in the $H^s$-norm ($s \ge 0$) while requiring only one additional derivative on the initial data $u_0 \in H^{s+1}_{0}$, significantly improving upon the regularity requirements of typical splitting methods for nonlinear dispersive equations. Furthermore, we prove that in the deep-water limit, the scheme for the ILW equation converges to the BO solution. Computational advantages of these schemes are shown in simulations: unlike classical splitting methods, our approach does not require a restrictive quadratic time step condition to nearly preserve energy over long time scales, thus rendering efficient and accurate long-time simulations feasible. As an application, we numerically explore the soliton resolution conjecture for both the ILW and a KdV–BO equation.

14. 一种使用总压力的耦合Stokes–Biot模型的二阶整体格式

A Second-Order Monolithic Scheme for the Coupled Stokes–Biot Model Using Total Pressure

作者: Talal Alshehri, Mingchao Cai, Xiaoqin Shen

类别: math.NA

发布日期: 2026-08-06

arXiv ID: 2608.05501v1

链接: http://arxiv.org/abs/2608.05501v1

摘要: We develop and analyze a second-order, fully implicit, monolithic time-discretization for the coupled time-dependent Stokes and quasi-static Biot system. The Biot subsystem is formulated in a three-field total-pressure formulation, in which the total pressure is introduced as an additional unknown together with the solid displacement and pore pressure. This reformulation improves robustness in nearly incompressible regimes and suppresses volumetric locking. All variables are discretized in time using the BDF2 scheme, and the coupled problem is solved monolithically without operator splitting. Discrete energy stability is established using the BDF2 $G$-stability identity. The resulting estimates exhibit robustness with respect to the Lamé parameter, Biot–Willis coefficient, and storage coefficient. A consistency–stability argument is then used to derive second-order convergence in time together with optimal-order convergence in space. Numerical experiments confirm the theoretical results and demonstrate second-order temporal accuracy in the corresponding energy norms.

15. 基于基与流形先验的侧向切片采样结构化张量近似

Structured Tensor Approximation from Lateral Slice Sampling via Basis and Manifold Priors

作者: Jeongmin Chae, Usama Saleem, Selin Bac, Shaama Mallikarjun Sharada, Urbashi Mitra

类别: math.NA

发布日期: 2026-08-05

arXiv ID: 2608.05463v1

链接: http://arxiv.org/abs/2608.05463v1

摘要: In this work, we consider a structured tensor approximation problem, where only a limited number of lateral slices are observed. The proposed algorithm , called Basis and Manifold prior Tensor Approximation (BMTA), exploits both global and local structures of the evolution of a global tensor. Specifically, BMTA integrates two signal models: (i) a quasi-basis model that captures smooth global variations along a physical trajectory, and (ii) a manifold-guided interpolation model that characterizes local relationships among tensor slices. A low-rank Tucker reconstruction framework is incorporated to efficiently capture the priors, resulting in coefficients for basis function estimation and a tensor optimization. In addition, we provide a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch. Numerical experiments are performed on both synthetic and real-world datasets, including quantum chemistry and spatiotemporal sensing applications.

16. 离散能量作为有限元代理模型的精确无标签训练目标

Discrete energy as an exact label-free training objective for finite-element surrogates

作者: Ruifeng Cao, Xidan Song

类别: cs.CE, cs.LG, math.NA

发布日期: 2026-08-05

arXiv ID: 2608.05437v1

链接: http://arxiv.org/abs/2608.05437v1

摘要: Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.

17. 参数线性控制系统的围道积分方法与模型降阶

Contour integral methods and model order reduction for parametric linear control systems

作者: Serkan Gugercin, Mattia Manucci

类别: math.OC, math.NA

发布日期: 2026-08-05

arXiv ID: 2608.05363v1

链接: http://arxiv.org/abs/2608.05363v1

摘要: This paper introduces a contour integral method (CIM) for efficiently computing outputs of parametric linear systems in control form over specified time intervals and to a user-prescribed accuracy. The CIM approximates the inverse Laplace transform via a quadrature rule applied along a modified integration contour. For parametric systems, we show how CIM integrates effectively with projection-based model order reduction (MOR) where a greedy algorithm builds the projection spaces following an error estimate we derive for this setting. We additionally demonstrate that the developed projection framework naturally enforces Hermite interpolation conditions. This combination substantially lowers the cost of evaluating the input-output relations across the parameter domain, for a wide range of input functions, and for initial conditions well captured by a low-dimensional subspace.. We demonstrate the accuracy and efficiency of the approach on benchmark non-parametric and parametric control systems, comparing against state-of-the-art projection-based MOR methods.

18. 空间离散薛定谔方程的透明边界条件：一维晶格中的无反射量子输运

Transparent boundary conditions for the spatially discrete Schrödinger equation: Reflectionless quantum transport in 1D lattices

作者: Mashrab E. Akramov, Jambul R. Yusupov, Matthias Ehrhardt, Davron U. Matrasulov

类别: math-ph, math.NA, quant-ph

发布日期: 2026-08-05

arXiv ID: 2608.05338v1

链接: http://arxiv.org/abs/2608.05338v1

摘要: We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain

19. 用于移动边界达西流中贝叶斯反演的变分框架

A variational framework for Bayesian inversion in moving boundary Darcy flow

作者: Michael E. Causon, Marco A. Iglesias, Michael V. Tretyakov

类别: math.NA

发布日期: 2026-08-05

arXiv ID: 2608.05317v1

链接: http://arxiv.org/abs/2608.05317v1

摘要: We develop an infinite-dimensional Bayesian framework for recovering spatially varying log-permeability from pressure observations in single-phase Darcy flow with a moving boundary, motivated by resin transfer moulding. The main contribution is an observation-wise representer formulation linking sensitivities of the parameter-to-observable map to the moving-boundary state and adjoint systems. This yields a reduced Levenberg–Marquardt method for maximum a posteriori (MAP) estimation on the Cameron–Martin space of a Gaussian prior and an explicit finite-rank covariance for the linearised MAP (LMAP) approximation. Once the representers are computed, the LM update and covariance assembly reduce to linear algebra in observation space, independently of the discretisation dimension. In 1D, the explicit moving-boundary solution is used to establish posterior well-posedness and existence of MAP estimators, and to derive closed-form Fréchet derivatives and representers. In 2D, we introduce a very weak mixed formulation coupling pressure, interface velocity and domain evolution. A formal one-sided directional shape linearisation yields the corresponding linearised state and adjoint systems, with observation-wise representers computed through independent, parallel adjoint problems. Numerical experiments show that the 1D LMAP approximation closely matches a reference MCMC posterior while requiring (\mathcal{O}(10^1)) rather than (\mathcal{O}(10^5\text{–}10^6)) forward simulations. In 2D, the method produces reconstructions and uncertainty estimates comparable to ensemble Kalman inversion with large ensembles, at substantially lower computational cost. The workflow is demonstrated across several geometries and experimental configurations without reformulating the inversion methodology.

重点关注论文（完整分析）#

1. 基于优化的时空超材料波动方程有效系数的识别#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

2. 非均匀结构动态非线性湿热-热-力耦合问题的高效高阶多尺度方法及其收敛性估计#

1. 研究对象和背景#

2. 主要定理或主要结果#

3. 研究方法、关键技术和核心工具#

4. 与之前工作的比较#

了解领域论文（摘要翻译）#

1. 具有傅里叶增强特征的物理信息神经网络的交替Levenberg-Marquardt训练#

摘要翻译#

2. 从神经算子视角看逆成像问题中的U-Net#

摘要翻译#

3. 基于最小化的多项式修正用于固定和移动区域上的高阶曲线边界：在有限体积和不连续伽辽金格式上的评估#

摘要翻译#

4. 准静态电孔弹性的总压力形式迭代解耦方法#

摘要翻译#

5. 基于矩阵值核的切向量场无散插值#

摘要翻译#

不相关论文（基本信息）#

1. 逆生成元问题的一个反例及相关问题#

2. 点云上的梯度下降及其在学习型算子校正中的应用#

3. Muon 在 Stiefel 流形上具有精确的闭式更新#

4. 条件期望算子与条件均值嵌入的可验证正则性准则及其在非参数回归、贝叶斯逆问题与库普曼算子中的应用#

5. ChromOps.jl: 色谱模型的高阶模拟与离散正向灵敏度分析#

6. Kastor: 一种用于偏微分方程模拟生成式仿真的高效微调策略#

7. 肿瘤生长的热力学一致Cahn-Hilliard-Navier-Stokes模型#

8. 电机热瞬态模拟的约化基方法#

9. Rothe 时间离散与截断 Westervelt 系统的弱解#

10. 动理学 Fokker-Planck 方程的核公式#

11. 势匹配最优传输：用于精确 $p$-Wasserstein 动力学的连续正规化流#

12. 动态子结构中基于高阶Craig-Bampton方法的降阶模型先验误差估计器#

13. 中等长波方程与扰动Benjamin–Ono模型的分裂方法#

14. 一种使用总压力的耦合Stokes–Biot模型的二阶整体格式#

15. 基于基与流形先验的侧向切片采样结构化张量近似#

16. 离散能量作为有限元代理模型的精确无标签训练目标#

17. 参数线性控制系统的围道积分方法与模型降阶#

18. 空间离散薛定谔方程的透明边界条件：一维晶格中的无反射量子输运#

19. 用于移动边界达西流中贝叶斯反演的变分框架#

重点关注论文（完整分析）

1. 基于优化的时空超材料波动方程有效系数的识别

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

2. 非均匀结构动态非线性湿热-热-力耦合问题的高效高阶多尺度方法及其收敛性估计

1. 研究对象和背景

2. 主要定理或主要结果

3. 研究方法、关键技术和核心工具

4. 与之前工作的比较

了解领域论文（摘要翻译）

1. 具有傅里叶增强特征的物理信息神经网络的交替Levenberg-Marquardt训练

摘要翻译

2. 从神经算子视角看逆成像问题中的U-Net

摘要翻译

3. 基于最小化的多项式修正用于固定和移动区域上的高阶曲线边界：在有限体积和不连续伽辽金格式上的评估

摘要翻译

4. 准静态电孔弹性的总压力形式迭代解耦方法

摘要翻译

5. 基于矩阵值核的切向量场无散插值

摘要翻译

不相关论文（基本信息）

1. 逆生成元问题的一个反例及相关问题

2. 点云上的梯度下降及其在学习型算子校正中的应用

3. Muon 在 Stiefel 流形上具有精确的闭式更新

4. 条件期望算子与条件均值嵌入的可验证正则性准则及其在非参数回归、贝叶斯逆问题与库普曼算子中的应用

5. ChromOps.jl: 色谱模型的高阶模拟与离散正向灵敏度分析

6. Kastor: 一种用于偏微分方程模拟生成式仿真的高效微调策略

7. 肿瘤生长的热力学一致Cahn-Hilliard-Navier-Stokes模型

8. 电机热瞬态模拟的约化基方法

9. Rothe 时间离散与截断 Westervelt 系统的弱解

10. 动理学 Fokker-Planck 方程的核公式

11. 势匹配最优传输：用于精确 $p$-Wasserstein 动力学的连续正规化流

12. 动态子结构中基于高阶Craig-Bampton方法的降阶模型先验误差估计器

13. 中等长波方程与扰动Benjamin–Ono模型的分裂方法

14. 一种使用总压力的耦合Stokes–Biot模型的二阶整体格式

15. 基于基与流形先验的侧向切片采样结构化张量近似

16. 离散能量作为有限元代理模型的精确无标签训练目标

17. 参数线性控制系统的围道积分方法与模型降阶

18. 空间离散薛定谔方程的透明边界条件：一维晶格中的无反射量子输运

19. 用于移动边界达西流中贝叶斯反演的变分框架