生成时间: 2026年05月25日 06:57:41
重点关注论文数量: 0
了解领域论文数量: 4
不相关论文数量: 14
了解领域论文（摘要翻译）
1. 直接ALE ADER-DG方法中通过四维时空孔洞状单元处理三维多面体动网格拓扑变化的研究
On the treatment of topology changes on 3D polyhedral moving meshes via 4D space-time hole-like elements in direct ALE ADER-DG methods
作者: Elena Gaburro, Matej Klima, Mauro Bonafini, Maurizio Tavelli
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23506v1
链接: http://arxiv.org/abs/2605.23506v1
摘要翻译
中文标题: 直接ALE ADER-DG方法中通过四维时空孔洞状单元处理三维多面体动网格拓扑变化的研究
摘要翻译: 本文研究一种利用ADER间断伽辽金格式在直接任意拉格朗日-欧拉（ALE）框架下，于具有拓扑变化的三维动多面体网格上对双曲型偏微分方程进行高阶演化求解的新方法。我们的直接ALE方法基于在四维（三维空间+时间）时空控制体上对偏微分方程进行积分，这些控制体连接着前后两个网格剖分的单元，从而以一种有效且高阶的方式同时在时间和两个不同网格之间演化解。通过这种方式，我们还避免了间接ALE格式中常见的复杂且昂贵的投影重构技术和网格求交操作。关键步骤在于构造时空控制体的策略，这些控制体还须连接因拓扑变化而具有不同形状和邻域的单元。事实上，仅通过坍缩或拉伸已有单元的边来连接它们，会在时空域中留下一个“孔洞”。为了填补该孔洞，我们引入了额外的退化单元，称之为孔洞状单元。这些单元是在时间步始末均具有零三维体积、但具有严格非零四维时空体积的四维对象。鉴于这种三维空间中时空方法的独特性以及刻画此类单元几何的必要性，本文的主要目标是给出该方法的严格几何与数值描述，并呈现新的直观可视化策略。特别地，我们详细刻画了在底层Delaunay四面体剖分上发生2-3、3-2和4-4翻转时对应出现的孔洞状单元。最后，数值实验表明，即使在频繁发生拓扑变化的情况下，该方法仍然是完全守恒的、满足几何守恒律（GCL）并能保持正确的收敛阶。
2. 多面体方法收敛分析技术的关键挑战与桥梁
Key challenges and bridges among convergence analysis techniques for polytopal methods
作者: Lourenço Beirão da Veiga, Daniele Antonio Di Pietro, Jérôme Droniou
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23405v1
链接: http://arxiv.org/abs/2605.23405v1
摘要翻译
中文标题: 多面体方法收敛分析技术的关键挑战与桥梁
摘要翻译: 多面体方法为一般网格上偏微分方程的数值逼近提供了一个灵活的框架。由于其固有的非协调性质，以及在许多情况下解的完全离散性质，其收敛分析提出了特定的挑战。考虑了两种主要技术：虚拟函数法（例如在虚拟元方法中使用）和完全离散法（例如离散de Rham方法的基础）。我们在此引入一个基于协调提升概念的新框架，即从离散空间到连续空间的有界且一致的映射。该方法在虚拟观点和完全离散观点之间架起了桥梁，澄清了虚拟函数的范数等价性的作用，并导致了一个可用于多面体方法的一致性误差分解。这三种方法在一个模型问题上进行了演示，这为讨论相关技术要点提供了机会。还与离散微分复形的收敛性质建立了联系。
3. 二维声学超材料中亚波长共振的傅里叶–伽辽金方法
Fourier–Galerkin Methods for Subwavelength Resonances in 2D Acoustic Metamaterials
作者: Jinghao Cao
类别: math.NA, math.AP
发布日期: 2026-05-22
arXiv ID: 2605.23251v1
链接: http://arxiv.org/abs/2605.23251v1
摘要翻译
中文标题: 二维声学超材料中亚波长共振的傅里叶–伽辽金方法
摘要翻译: 我们提出一种傅里叶–伽辽金框架，用于分析并计算有限域中二维散射问题的亚波长共振。从边界积分公式出发，我们将算子投影到傅里叶模式上，并导出一个显式的有限维有效矩阵，其奇异性刻画了共振频率。在亚波长区域，我们得到了该矩阵关于 $ω$ 和材料对比度的渐近展开，识别出了主阶算子及其核结构。这一约化将共振问题转化为一个低维非线性特征值问题，避免了大规模离散和全局寻根过程。有效矩阵的元可以显式计算，并允许使用基于FFT的求积进行快速求值。所得方法为一般光滑几何形状中的共振提供了一个高效且鲁棒的计算框架。
4. 基于隐式惩罚的高阶守恒间断伽辽金方法：广义 Korteweg-de Vries 方程和 Hirota-Satsuma KdV 系统
High-order Conservative Discontinuous Galerkin Methods via Implicit Penalization for the Generalized Korteweg-de Vries Equation and the Hirota-Satsuma KdV System
作者: M. Shan Tariq, Yanlai Chen, Bo Dong
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23181v1
链接: http://arxiv.org/abs/2605.23181v1
摘要翻译
中文标题: 基于隐式惩罚的高阶守恒间断伽辽金方法：广义 Korteweg-de Vries 方程和 Hirota-Satsuma KdV 系统
摘要翻译: 我们针对非线性波动问题发展了新的守恒型间断伽辽金（DG）方法，重点研究广义 Korteweg-de Vries（gKdV）方程和耦合的 Hirota-Satsuma KdV（HS-KdV）系统。所提出的方法通过数值迹的单值结构保持质量守恒，而能量和哈密顿量守恒则通过辅助守恒约束隐式确定数值迹中的惩罚参数来实现。在我们之前的工作 [11] 中，我们为 gKdV 方程发展了一种守恒型 DG 方法；然而，该构造涉及近似解跳跃的时间导数，这给超越二阶时间精度带来了困难。我们新的构造通过引入重新设计的迹配置消除了导数项，从而克服了这一限制。这一新增强功能无缝地为更高阶的时间离散铺平了道路，并且与之前的方法相比，每个时间步需要求解更少的非线性系统。对于耦合的 HS-KdV 系统，我们提出了第一个守恒型 DG 方法，该方法保持精确解的所有三个不变量。数值结果展示了所提出方法的精度和预期的收敛行为，以及对 gKdV 方程和 HS-KdV 系统的长时间稳定性和强守恒性质。
不相关论文（基本信息）
1. 免训练的循环Transformer
Training-Free Looped Transformers
作者: Lizhang Chen, Jonathan Li, Chen Liang, Ni Lao, Qiang Liu
类别: cs.LG, math.NA, stat.ML
发布日期: 2026-05-22
arXiv ID: 2605.23872v1
链接: http://arxiv.org/abs/2605.23872v1
摘要: We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.
2. 大型常微分方程组自适应龙格-库塔方法的混合精度
Mixed-Precision in adaptive Runge-Kutta method for large ODE systems
作者: Mouhamad Al-Sayed, Samuel Bernard, Arsène Marzorati, Jonathan Rouzaud-Cornabas
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23727v1
链接: http://arxiv.org/abs/2605.23727v1
摘要: Mixed-precision methods combine low and high precision arithmetics to exploit low precision computational speed and high precision accuracy. Large ODE systems that contain many heterogeneous interactions lead to a high computational cost that could be tackled with mixed-precision solvers. We tested mixedprecision versions of the Bogacki-Shampine 3(2) Runge-Kutta pair over three benchmark systems: coupled linear oscillators, the Kuramoto model and a circadian clock model. Our study is performed in a way that can be adapted to any finite-precision format, software architecture and numerical scheme. We found that mixed-precision solvers can preserve most of the high-precision solver accuracy under a wide range of solver tolerances. Moreover, mixed-precision solver accuracy improves with system size, reaching levels equivalent to high-precision solvers in small system size. We also observed that mixed-precision arithmetic does not impact the number of evaluation in a way that balances the benefit of fast operations in low precision. Taken together, these results show that mixed-precision methods can offer significant computational speed-up at little or no loss of accuracy in large coupled ODE systems.
3. 基于随机预处理的变分资料同化集合加速
Accelerating an ensemble of variational data assimilations with randomized preconditioning
作者: I. Daužickaitė, S. Gürol, M. Destouches, L. Berre, A. T. Weaver
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23571v1
链接: http://arxiv.org/abs/2605.23571v1
摘要: Ensembles of variational data assimilations (EDA) require solving systems of linear equations with iterative methods. The solution process can be accelerated using a limited memory preconditioner constructed with approximations of the leading eigenpairs of the Hessian matrix. Randomized methods for low-rank matrix approximations provide a feasible approach for computing these eigenpairs. These methods use a random sketching matrix to obtain a low-rank representation of the Hessian matrix, which is then used for computing the eigendecomposition. The sketching matrix highly influences the quality of the approximation. In this paper, we show how the structure of the EDA can be exploited to construct a suitable sketching matrix, i.e., using the differences of the right-hand sides of the linear systems of equations. Idealised numerical experiments with the Lorenz-96 model show that the resulting preconditioner is able to accelerate the EDA solution process for all ensemble members, even if constructed from the control member only.
4. RA-DCA：一种用于极大值结构 DC 规划中方向驻点性的随机有效集 DCA
RA-DCA: A Randomized Active-Set DCA for Directional Stationarity in Max-Structured DC Programs
作者: Yi-Shuai Niu
类别: math.OC, cs.AI, math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23550v1
链接: http://arxiv.org/abs/2605.23550v1
摘要: We study nonsmooth difference-of-convex programs whose subtracted convex term is a finite maximum of smooth convex functions. In this setting, standard DCA iterations may converge to critical points that are not directionally stationary, whereas exact active-vertex screening can be expensive when active sets are large or combinatorial. We propose RA-DCA, a vertex-first randomized active-set DCA that projects active gradients onto sampled directions, checks a sampled vertex residual, and uses a small linear program only as a low-residual convex-combination fallback. The method preserves the descent structure of DCA and reduces the randomized screening layer to matrix multiplications. Under the stated regularity, numerical active-set consistency, and random-embedding assumptions, every accumulation point generated by the safeguarded method is directionally stationary with probability one. MATLAB experiments first test the theorem on degenerate max-affine, max-quadratic, and sparse support-function models, where the safeguard avoids nonstationary critical points and closely tracks a full active-vertex scan. Block top-k tests then show that the same screening idea remains useful when exact aggregate enumeration is combinatorial. Trimmed-regression, complementarity, and QUBO diagnostics separate cases where active-set selection helps from cases dominated by multistart search, the DC split, or other problem-specific features.
5. 基于GLT理论的雅可比加权直方插值矩阵的谱分布
Spectral distribution of Jacobi weighted histopolation matrices via GLT theory
作者: Allal Guessab, Federico Nudo, Stefano Serra-Capizzano
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23501v1
链接: http://arxiv.org/abs/2605.23501v1
摘要: In this paper we study a weighted histopolation problem on $[-1,1]$ associated with Jacobi weights. In the first part of the present work we prove results in approximation theory, while in the second we analyze the resulting matrices from an asymptotic linear algebra perspective. More in detail, in the first part, given weighted cell averages, we construct a reconstruction operator based on weighted primitives of Jacobi polynomials and investigate the resulting discretization matrices. At any fixed discretization level, we derive an exact factorization of the histopolation matrix through a backward-difference operator and a sampling operator of Jacobi weighted primitives. Combining a sharp integration by parts identity with the three-term recurrence of Jacobi polynomials, we further show that the primitive sampling operator admits an explicit decomposition involving a tridiagonal coupling matrix in the Jacobi spectral index. This yields a tridiagonal factor representation of the histopolation matrix. In the second part, under standard mesh-regularity assumptions, we show that all the various induced matrix sequences belong to the Generalized Locally Toeplitz (GLT) class, by describing in detail the related GLT symbols. As a consequence, we provide the corresponding spectral distributions and discuss their implications for numerical stability when solving the associated linear systems.
6. 一种用于线性四阶椭圆问题的高阶节点保界质量守恒方法及其在非线性抛物方程中的应用
A high-order nodally bound-preserving and mass-conservative method for linear fourth-order elliptic problems and its applications to nonlinear parabolic equations
作者: Jie Shen, Zuodong Wang
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23486v1
链接: http://arxiv.org/abs/2605.23486v1
摘要: We propose a high-order finite element method for linear fourth-order elliptic problems that is both nodally bound-preserving and mass-conservative, based on a variational inequality formulation. The method admits an equivalent strictly convex minimization structure, which ensures well-posedness and enables an optimal error estimate in the $H^1$-seminorm, under suitable regularity assumptions. This framework is further extended to nonlinear fourth-order parabolic problems through space–time high-order discretizations that combine variational inequalities, BDF schemes, and scalar auxiliary variable (SAV) techniques. The fully discrete schemes preserve nodal bounds and mass, and a modified energy stability result is established for the first-order temporal scheme. We also apply the same framework to nonlinear second-order parabolic problems by introducing a consistent fourth-order regularization, leading to space–time high-order schemes with the same bound-preserving and mass-conservative properties. Extensive numerical results, including challenging tests with singularities and low regularity, demonstrate the stability, efficiency, and high-order accuracy of the proposed methods.
7. 随机广义采样
Stochastic Generalized Sampling
作者: Luca Finotti, Matteo Santacesaria
类别: math.FA, cs.IT, math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23421v1
链接: http://arxiv.org/abs/2605.23421v1
摘要: Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for recovering elements in arbitrary separable Hilbert spaces, deterministic approaches suffer from severe basis-dependent dimensionality constraints, often requiring a quadratic sample complexity $m \gtrsim n^2$ to avoid numerical instability. In this paper, we introduce a fully stochastic framework for GS that natively overcomes these deterministic barriers. By drawing measurements according to an optimal leverage-score probability distribution, we prove that stable recovery is guaranteed with high probability at a near-linear sample complexity of $m \gtrsim n\log n$. Crucially, this optimal rate is universal-independent of the specific choice of measurement and reconstruction bases-and holds even when the sensing system is a highly redundant frame. To establish these guarantees, we derive a novel matrix Bernstein inequality for random rectangular operators, allowing us to rigorously control the aliasing error governed by the empirical cross-term. Finally, we demonstrate the practical efficacy of our approach on the classical problem of recovering analytic functions from continuous Fourier measurements via Legendre polynomials, where our randomized method achieve near-exponential convergence rates.
8. 基于克罗内克预条件优化的多物理场物理信息神经网络中的耦合鲁棒精度
Coupling-Robust Accuracy in Multiphysics Physics Informed Neural Networks via Kronecker-Preconditioned Optimization
作者: Youngjae Park, Jaemin Kim, Junghwa Hong
类别: cs.LG, math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23391v1
链接: http://arxiv.org/abs/2605.23391v1
摘要: Physics-informed neural networks (PINNs) for coupled multiphysics systems suffer systematic accuracy degradation as inter-equation coupling strengthens. We provide a theoretical explanation for this phenomenon through neural tangent kernel (NTK) analysis: for linearly coupled systems, we prove that the standard NTK’s spectral radius grows as $Ω(γ^2)$ with coupling strength $γ$, shrinking the stable learning rate, while block-diagonal Gauss–Newton (GN) preconditioning yields a preconditioned NTK $K_P = J H^{+} J^\top$ (where $H$ is the block-diagonal GN Hessian) whose spectral radius is bounded by $S$ ($S$ = number of networks), independent of $γ$. We verify the $Ω(γ^2)$ growth numerically across symmetric, asymmetric, and nonlinear coupled PDE systems, and confirm $λ_{\max}(K_P) = S$ with equality in all cases. Combining the Kronecker-preconditioned optimizer SOAP with inverse-gradient-norm loss balancing (SOAP+GN) yields coupling-robust accuracy: across 234 experiments spanning three 1D systems of increasing nonlinearity and a 2D electroosmotic flow benchmark, SOAP+GN maintains final-epoch $L_2$ degradation $\leq 1.1\times$ (ratio of strong- to weak-coupling error) even as coupling parameters vary over one to two orders of magnitude, compared with $> 10^2\times$ for Adam+GN. SOAP+GN further scales to a 2D, 6-PDE electroosmotic flow system at EDL-resolved conditions – a regime that all prior PINN electrokinetics studies have avoided through simplified physics – where Adam+GN fails entirely ($L_2 > 0.9$).
9. Gross-Pitaevskii能量泛函的非协调有限元逼近与能量下界估计
Nonconforming Finite Element Approximation and Energy Lower Bound Estimation for the Gross–Pitaevskii Energy Functional
作者: Chen Zhang, Heyan Zhu, Wenbin Chen
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23334v1
链接: http://arxiv.org/abs/2605.23334v1
摘要: The ground state of Bose–Einstein condensates can be described as the minimizer of the Gross–Pitaevskii energy functional subject to a mass conservation constraint. In this paper, we study the corresponding discrete optimization problem in nonconforming finite element spaces and establish a priori error estimates for the discrete ground state energy, the discrete eigenvalue, and the discrete ground state. Specifically, we derive explicit convergence rates for the a priori error in the particular case of the $EQ_1^{\mathrm{rot}}$ finite element. Furthermore, we proof that within the $EQ_1^{\mathrm{rot}}$ finite element framework, the discrete ground state energy provides a lower bound estimation to the exact energy. Finally, numerical experiments are presented to validate the theoretical analysis.
10. 勒贝格函数的几何性质
Geometric properties of the Lebesgue function
作者: Leokadia Białas-Cież, Stefano De Marchi, Mateusz Suder
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23299v1
链接: http://arxiv.org/abs/2605.23299v1
摘要: We present a collection of observations concerning the peculiar behavior of the Lebesgue function in the setting of the interval $[-1,1]\subset \mathbb{R}$ and the square $[-1,1]^2\subset \mathbb{R}^2$. We provide numerical results and formulate several open problems related to the geometry of the Lebesgue function.
11. 面向具有随机水力传导张量的纳维-斯托克斯-达西流的高阶、长时间稳定且并行解耦的GBDF$k$ SAV集合格式
High-order, long-time stable and parallel decoupled GBDF$k$ SAV ensemble schemes for the Navier–Stokes–Darcy flow with random hydraulic conductivity tensors
作者: Wei-Wei Han, Fukeng Huang, Changxin Qiu
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23290v1
链接: http://arxiv.org/abs/2605.23290v1
摘要: We develop and analyze high-order ensemble schemes for the unsteady Navier–Stokes–Darcy system with uncertain initial conditions, forcing terms, hydraulic conductivity tensors, and Lions-Beavers-Joseph-Saffman interface conditions. The proposed schemes which are called GSAV-GBDF$k$-Ensemble schemes integrate a partitioned decoupling strategy, the generalized scalar auxiliary variable (GSAV) approach, and generalized BDF$k$ discretizations. This framework achieves high-order temporal accuracy and long-time stability, permits explicit treatment of the nonlinear term, and facilitates an efficient ensemble implementation for multiple parameter realizations by sharing a single, unified coefficient matrix at each time step. Moreover, the numerical solutions are shown to satisfy uniform-in-time bounds without time-step restrictions. Owing to the ensemble formulation, the resulting linear systems share common coefficient matrices, which significantly improves computational efficiency. We further establish optimal-order error estimates for the proposed high-order schemes. Numerical results are included to confirm the theoretical analysis and to illustrate the accuracy, stability, and efficiency of the proposed methods.
12. 定义在$\mathbb R^n$上的函数的分数阶拉普拉斯算子和分数阶$p$-拉普拉斯算子数值逼近的基于矩阵的谱方法
A matrix-based spectral method for the numerical approximation of the fractional Laplacian and the fractional $p$-Laplacian of functions defined on $\mathbb R^n$
作者: Loïc Constantin, Carlota M. Cuesta, Francisco de la Hoz
类别: math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23252v1
链接: http://arxiv.org/abs/2605.23252v1
摘要: Given a function $u$ defined on $\mathbb R^n$, its fractional $p$-Laplacian is given by
$$(-Δ)_p^su(\vec x)=C_1(n,s,p)\int_{\mathbb R^n}\frac{|u(\vec x)-u(\vec y)|^{p-2}(u(\vec x)-u(\vec y))}{\|\vec x-\vec y\|_2^{n+sp}}d\vec y,\quad\vec x\in\mathbb R^n,$$where the integral is understood in the principal value sense, $p\in(1,\infty)$, $s\in(0,1)$, and $C_1(n,s,p)$ is a normalization constant. A formally equivalent nonlinear Balakrishnan formulation is given by
$$(-Δ)_p^su(\vec x)=C_4(n,s,p)\int_0^\inftyΔ(t-Δ)^{-1}\left[Φ_p(u(\vec x)-u(\cdot))\right](\vec x)\frac{dt}{t^{1-sp/2}},$$where $C_4(n,s,p)$ is another normalization constant, and $Φ_p(t)=|t|^{p-2}t$. In this paper, we present a matrix-based spectral method to approximate numerically the fractional Laplacian (i.e., the linear case, where $p = 2$) and the fractional $p$-Laplacian for functions defined on $\mathbb R^n$. Our approach builds on the Balakrishnan representation, where we discretize the 2nd-order derivatives in $Δ$ using spectrally accurate differentiation matrices. A key advantage is that these matrices can be diagonalized in a well-conditioned manner, enabling a stable and robust numerical scheme that naturally extends to arbitrary spatial dimensions $n$. In particular, this diagonalization allows the fractional operator to act directly on the eigenvalue spectrum, effectively reducing the Balakrishnan integral to an analytical evaluation at the spectral level and thereby avoiding costly multidimensional quadrature. The resulting method also avoids domain truncation and variational formulations, making it both computationally efficient and conceptually straightforward. As a practical application, we simulate the evolution of
$$\frac{\partial u}{\partial t}+(-Δ)^s_pu=0,$$in one and two spatial dimensions, being able to capture the self-similar solutions that arise as $t\to\infty$.
13. 多扩展，少收缩：塑造有效秩动态以实现推荐中的稠密缩放
Expand More, Shrink Less: Shaping Effective-Rank Dynamics for Dense Scaling in Recommendation
作者: Guoming Li, Shangyu Zhang, Junwei Pan, Wentao Ning, Jin Chen, Gengsheng Xue, Chao Zhou, Shudong Huang, Haijie Gu, Menglin Yang
类别: cs.LG, cs.IR, math.NA
发布日期: 2026-05-22
arXiv ID: 2605.23191v1
链接: http://arxiv.org/abs/2605.23191v1
摘要: Scaling recommendation models is a central challenge in recommender systems. Recently, RankMixer has emerged as an effective solution, operating on a unified token representation and alternating between token mixing and per-token feedforward networks (P-FFNs) to achieve scalable performance. However, RankMixer suffers from \textit{embedding collapse}, where learned representations have low effective rank, limiting expressivity and underutilizing the expanded representation space. Through empirical analysis and theoretical insights, we identify rigid token mixing and P-FFN modules as the primary causes of this phenomenon, jointly inducing a \textbf{damped oscillatory trajectory} in effective-rank evolution across layers. To address it, we propose RankElastor, a novel architecture that produces spectrum-robust representations with provable collapse mitigation. RankElastor introduces two components: (i) \textbf{parameterized full mixing}, which enables expressive token mixing with improved spectral robustness; and (ii) \textbf{GLU-improved P-FFNs}, which stabilize representation spectra through GLU-style FFN modules. Extensive experiments on large-scale industrial datasets demonstrate that RankElastor consistently improves recommendation performance, mitigates embedding collapse, and exhibits robust scaling behavior. Code is available at this GitHub repository: https://github.com/vasile-paskardlgm/RankElastor
14. 使用物理约束高斯过程回归的模态振型展开
Mode-Shape Expansion Using Physics-Constrained Gaussian Process Regression
作者: Farid Ghahari
类别: math.NA, stat.ML
发布日期: 2026-05-21
arXiv ID: 2605.23101v1
链接: http://arxiv.org/abs/2605.23101v1
摘要: This paper addresses the challenge of reconstructing full-field structural mode shapes from sparse sensor data. While Gaussian Process Regression (GPR) offers a robust non-parametric framework for spatial interpolation and uncertainty quantification, standard formulations often yield physically inconsistent mode-shape reconstructions under sparse sensing conditions. A Physics-Constrained Single-Output Gaussian Process (CONS-SOGP) framework is derived that utilizes independent modal kernels while coupling the optimization via a mass-orthogonality penalty. The paper presents derivations for the marginal likelihood, hyperparameter gradients, and penalty coupling. Numerical verification on a multi-degree-of-freedom structure demonstrates that the proposed method overcomes existing limitations in GP-based prediction, providing more accurate and reliable expanded mode shapes.