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1. 二维Kaup-Broer-Kuperschmidt Boussinesq系统的数值研究
Numerical study of the 2D Kaup-Broer-Kuperschmidt Boussinesq system
作者: Théo Gaudry, Christian Klein, Jean-Claude Saut, Nikola Stoilov
类别: math.AP, math.NA
发布日期: 2026-05-06
arXiv ID: 2605.05183v1
链接: http://arxiv.org/abs/2605.05183v1
摘要: In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.
2. 有限元方法下具奇异势的薛定谔算子的显式双侧特征值界
Explicit Two-Sided Eigenvalue Bounds for Schrödinger Operators with Singular Potentials via Finite Element Method
作者: Xuefeng Liu
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.05177v1
链接: http://arxiv.org/abs/2605.05177v1
摘要: We present, to the best of our knowledge, the first numerical algorithm for explicit, computable two-sided eigenvalue bounds for Schrödinger operators H = -Delta + V on R^N, N = 2,3, in the presence of both an unbounded potential and an unbounded domain. “Explicit” here means that all constants and ingredients are derived in closed form from the mesh, the potential, and a small set of explicit inequalities (Payne-Weinberger, Hardy, and explicit bounded-domain Sobolev embeddings); the conversion to fully verified(IEEE-754-safe, interval-arithmetic) enclosures is a separate verification step and is left for future work. In particular, singular attractive potentials of Coulomb type, V(x) = -Z/|x|, which model the hydrogen atom and the H_2^+ molecular ion, are covered by the theory. The method combines domain truncation to a bounded domain D(R) containing {|x| <= R} with an extension of Liu’s Composite Enriched Crouzeix-Raviart (CECR) finite element method to sign-indefinite potentials. Upper bounds come from the standard conforming Galerkin method; lower bounds come from the CECR construction, whose gap to the exact eigenvalue closes as the mesh is refined. Numerical experiments on the 2D single- and two-centred Coulomb potentials and on the 3D hydrogen atom and H_2^+ molecular ion illustrate the algorithm and confirm the predicted convergence.
3. 理解基于 Transformers 的非线性回归中的上下文学习: 注意力作为特征化器
Understanding In-Context Learning for Nonlinear Regression with Transformers: Attention as Featurizer
作者: Alexander Hsu, Zhaiming Shen, Wenjing Liao, Rongjie Lai
类别: cs.LG, math.NA
发布日期: 2026-05-06
arXiv ID: 2605.05176v1
链接: http://arxiv.org/abs/2605.05176v1
摘要: Pre-trained transformers are able to learn from examples provided as part of the prompt without any weight updates, a remarkable ability known as in-context learning (ICL). Despite its demonstrated efficacy across various domains, the theoretical understanding of ICL is still developing. Whereas most existing theory has focused on linear models, we study ICL in the nonlinear regression setting. Through the interaction mechanism in attention, we explicitly construct transformer networks to realize nonlinear features, such as polynomial or spline bases, which span a wide class of functions. Based on this construction, we establish a framework to analyze end-to-end in-context nonlinear regression with the constructed features. Our theory provides finite-sample generalization error bounds in terms of context length and training set size. We numerically validate the theory on synthetic regression tasks.
4. 微磁学中Landau-Lifshitz-Gilbert方程的BDF2型积分器：先验误差估计
BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: a-priori error estimates
作者: Michele Aldé, Dirk Praetorius, Michael Feischl
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.05129v1
链接: http://arxiv.org/abs/2605.05129v1
摘要: We consider the Landau-Lifshitz-Gilbert equation (LLG), which models time-dependent micromagnetic phenomena. We analyze a fully discrete scheme that combines first-order finite elements in space with a BDF2 method in time. The method requires the solution of only one linear system of equations per time step and does not enforce the pointwise unit-length constraint of the magnetization. While unconditional weak convergence has been analyzed in an earlier work, we now prove optimal-order convergence rates under sufficient regularity assumptions on the exact solution and the external field. In combination with our previous work, this establishes the first higher-order-in-time and linear integrator that converges both to weak and strong solutions of LLG. Numerical experiments confirm first-order convergence in space and second-order convergence in time.
5. 在给定子空间的情况下求半正定矩阵的精确特征值与特征向量
Finding accurate eigenvalues and eigenvectors of positive semi-definite matrices given a subspace
作者: Yuji Nakatsukasa, Zheng Tang
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.05043v1
链接: http://arxiv.org/abs/2605.05043v1
摘要: We revisit a classical problem in numerical linear algebra: given an $k$-dimensional subspace $\mathcal{Q}$ that approximates the leading eigenspace of an $n\times n$ positive semi-definite matrix $A$, the goal is to extract high-accuracy eigenvalues. The Rayleigh-Ritz (RR) method is the standard algorithm for the task, which has been shown to be optimal in several ways (when $A$ is symmetric, not necessarily positive semi-definite $A\succeq 0$). In this paper, we show that when $A \succeq 0$, alternative methods can outperform RR, while having the same computational complexity, that is, the main cost is in computing $AQ$, plus an $O(nk^2)$ term. In particular, we advocate the use of Nystr{ö}m’s method, showing that the approximate eigenvalues always have higher accuracy than RR, and the improvement can be arbitrarily large. The difference is significant, especially when $A$ has a fast-decaying spectrum. A similar improvement is numerically observed for the purpose of approximating the leading eigenvectors. In contrast, when the target eigenvalues are the trailing ones, the situation is reversed, and the Nystr{ö}m method performs poorly; we suggest a remedy for this situation.
6. Strichartz 极值函数的神经网络发现
Neural Discovery of Strichartz Extremizers
作者: Nicolás Valenzuela, Ricardo Freire, Claudio Muñoz
类别: math.AP, cs.LG, math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04918v1
链接: http://arxiv.org/abs/2605.04918v1
摘要: Strichartz inequalities are a cornerstone of the modern theory of dispersive PDEs, but their extremizers are known explicitly only in a handful of sharp cases. The non-convexity of the underlying functional makes the problem hard, and to our knowledge no systematic numerical attack has been attempted. We propose a simple neural-network-based pipeline that searches for extremizers as critical points of the Strichartz ratio, and apply it in three settings. First, on the Schrödinger group we recover the Gaussian extremizers of Foschi and Hundertmark–Zharnitsky in dimensions $d=1,2$ to within $10^{-3}$ relative error, with no analytical prior. Second, on $59$ further admissible pairs in $d=1$ where the answer is conjectural, the method consistently finds Gaussians, supporting the conjecture that Gaussians are the universal extremizers in the admissible range. Third, on the critical Airy–Strichartz inequality at $γ=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;α,1,0,0)$ with growing internal frequency $α$, and the discovered ratio approaches the Frank–Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\simα^{-0.9}$. We confirm the same picture with an independent Hermite-basis ansatz. We propose a precise conjecture: the supremum equals $\widetilde A_{q,r}$ and is approached, but not attained, along the breather family. The pipeline thus serves both as a validator on known cases and as a discovery tool when no extremizer exists.
7. 更新幅度状态重分配（UM-SRD）：加权SRD在切割单元法中的关断扩展
Update-Magnitude State Redistribution (UM-SRD): A Shut-off Extension of Weighted SRD for Cut-Cell Methods
作者: Justo E. Karell
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04863v1
链接: http://arxiv.org/abs/2605.04863v1
摘要: Berger & Giuliani (2024) developed a provably stable weighted state redistribution (SRD) algorithm for cut-cell meshes. A key limitation of their method is that, although flux redistribu- tion naturally vanishes when updates are small, SRD continuously applies redistribution even when the flux balance is zero, preventing exact steady-state preservation and potentially in- troducing unnecessary dissipation in smooth regions. This work introduces Update-Magnitude State Redistribution (UM-SRD), which blends the SRD operator with the identity operator via a smooth, locally-defined scalar indicator of the finite-volume update magnitude. UM-SRD preserves conservation and reduces exactly to the base scheme when the finite-volume update is exactly zero in a small-cell neighborhood. For a one-dimensional model problem with a single small cut cell, we prove UM-SRD is total variation diminishing under the same CFL condition as the base upwind scheme, show the local truncation error modification is higher-order in smooth regions with the unnormalized indicator, and show that the normalized implementation pre- serves first-order accuracy. Numerical experiments demonstrate convergence toward first order on smooth 1D and 2D advection tests, confirm shut-off behaviour, verify non-oscillatory proper- ties, provide numerical evidence that UM-SRD stabilizes the base scheme near a small cut cell where the base scheme diverges, and confirm exact steady-state preservation. The algorithm reuses existing weighted SRD infrastructure, adding only a local blending mechanism, making it practical for cut-cell finite-volume codes.
8. 退化椭圆-抛物方程的Schwarz型方法收敛性分析
Convergence analysis of Schwarz-like methods for degenerate elliptic-parabolic equations
作者: Monika Eisenmann, Eskil Hansen
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04843v1
链接: http://arxiv.org/abs/2605.04843v1
摘要: Convergence is proven for Schwarz-like methods applied to degenerate elliptic-parabolic equations with a $p$-structure. This family of PDEs, e.g., arises when modelling nonlinear diffusion processes. The Schwarz-like approximation methods are based on decomposing the space-time domain into overlapping subdomains, which enables parallel implementations. The methods are derived by introducing a pseudo-time component and applying time integrators of splitting type, which are time stepped towards infinity. This approach of decomposing the space-time domain is related to Schwarz waveform relaxation methods, but the methods considered here have the advantage that they can be proven to converge when applied to nonlinear parabolic, or even degenerate elliptic-parabolic, PDEs. We prove convergence by deriving a nonlinear framework based on the abstract theory for monotone operators and the existence theory for degenerate elliptic-parabolic equations.
9. 基于广义重心坐标的自适应有限元方法
An Adaptive Finite Element Method Based on Generalized Barycentric Coordinates
作者: Yihui Zhou, Yuwen Li
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04805v1
链接: http://arxiv.org/abs/2605.04805v1
摘要: This work derives a posteriori error estimate of polygonal finite element methods based on Wachspress barycentric coordinates. In particular, we prove that the classical residual-based a posteriori error estimator is both an upper and lower bounds for the discretization error. The analysis relies a Scott-Zhang type interpolation and homogeneity arguments for rational functions on polygonal elements. Numerical experiments on square and L-shaped domains demonstrate the effectiveness of the adaptive algorithm.
10. 有限元方法中基于光滑化的超收敛
Superconvergence in finite element method by smoothing
作者: Yuwen Li, Han Shui, Ludmil Zikatanov
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04786v1
链接: http://arxiv.org/abs/2605.04786v1
摘要: This paper develops a smoothing-based postprocessing method for superconvergence in finite element methods. The method applies a few smoothing iterations, such as damped Jacobi, Gauss-Seidel, or conjugate gradient, with initial guess being the current finite element solution embedded in an enriched finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we prove superconvergence of the smoothed solutions under additive and multiplicative smoothers. Effectiveness of the proposed method is demonstrated by numerical experiments for the Poisson, Maxwell, biharmonic and Helmholtz equations.
11. 用于求解定常 Navier-Stokes 方程的稳态增量黏性分裂方法
Steady Incremental Viscosity Splitting Method for solving the stationary Navier-Stokes equation
作者: Aziz Takhirov, Driss Yakoubi
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04778v1
链接: http://arxiv.org/abs/2605.04778v1
摘要: We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each nonlinear iteration, the scheme requires solving an elliptic PDE for the velocity variable and a system with an SPD matrix for the pressure variable, which remains the same across all nonlinear iterations. The method can also be interpreted as an algebraic splitting approach. We prove boundedness and geometric convergence. Numerical tests illustrate the efficiency of the proposed algorithm.
12. FC-Gram近似算法的变体与推广的计算与分析研究
Computational and Analytical Study of Variations and Generalizations of the FC-Gram Approximation Algorithm
作者: Prakash Nainwal, Akash Anand
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04765v1
链接: http://arxiv.org/abs/2605.04765v1
摘要: The FC-Gram algorithm approximates non-periodic functions to high order by constructing a periodic extension with controlled boundary behavior and applying trigonometric interpolation. In this paper we introduce a generalized FC-Gram framework (GenFC), which provides greater flexibility in the construction of the blending continuation of Gram polynomials. This flexibility gives better control over the shape of the periodic extension and leads to improved approximation accuracy. We establish a convergence theorem showing that the trigonometric interpolant converges at the rate $\mathcal{O}(n^{-\min(r+β,\,d)})$ in the supremum norm on the original interval, where $r$ is the smoothness of the target function, $d$ the number of Gram polynomials, and $β\in [0,1]$ a Fourier-decay parameter. The framework and its analysis are developed so that the modified FC-Gram method of [J. Sci. Comput., 105(1):8, 2025] is recovered as a particular case. Numerical experiments confirm the predicted convergence rates and show that the added flexibility of the GenFC framework leads to improved approximation accuracy, with the gains carrying over to a Fourier continuation solver for two-point boundary value problems.
13. 顺序拓扑优化: 用于水平集边界细化的 SIMP 初始化
Sequential topology optimization: SIMP initialization for level-set boundary refinement
作者: Ondřej Ježek, Ján Kopačka, Martin Isoz, Dušan Gabriel
类别: cs.CE, math.NA, math.OC
发布日期: 2026-05-06
arXiv ID: 2605.04735v1
链接: http://arxiv.org/abs/2605.04735v1
摘要: Density-based topology optimization methods such as SIMP enable efficient topological exploration but produce diffuse material boundaries that require interpretation before manufacturing. Level-set methods maintain sharp interfaces but are sensitive to the initial design. This paper presents a sequential framework that addresses these complementary limitations through a signed distance function (SDF)-based geometry transfer, formulated for three-dimensional meshes. The SIMP density distribution is converted into an SDF that initializes subsequent level-set boundary refinement. From the level-set perspective, the SIMP-derived initialization mitigates sensitivity to the initial design. From the SIMP perspective, the level-set stage acts as optimization-driven post-processing that produces manufacturing-ready boundaries. Validation on three-dimensional cantilever and MBB benchmarks demonstrates compliance comparable to standalone level-set optimization, with up to 4.6x wall-clock speedup on the cantilever case. The full implementation is released under an open-source license to support reproducibility.
14. 用于不可压缩混合降阶优化的哈密顿界面动力学
Hamiltonian Interface Dynamics for Reduced-Order Optimization of Incompressible Mixing
作者: Ziqian Li, Enrique Zuazua
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04688v1
链接: http://arxiv.org/abs/2605.04688v1
摘要: We develop a reduced-order framework for optimizing mixing in two-dimensional incompressible flows. Instead of optimizing the full transport PDE, the method maximizes the length of advected material interfaces, leading to a finite-dimensional Hamiltonian control problem based on parametrized stream functions. We derive the continuous adjoint equations and reduced gradients, and discretize the forward and adjoint dynamics with the implicit midpoint rule. The resulting discrete adjoint is algebraically consistent with the derivative of the fully discrete objective, up to the tolerance of the nonlinear midpoint solves. The approach applies to bounded two-dimensional domains with smooth finite-dimensional stream-function parametrizations. Numerical experiments on cellular-flow and Doswell frontogenesis benchmarks show that the optimized time-dependent Hamiltonians generate near-exponential interface stretching and substantially faster decay of the $\dot{H}^{-1}$ mix-norm, in contrast with the polynomial behavior observed for stationary flows. When evaluated on a common reference transport solver, the interface-based controls produce faster $\dot{H}^{-1}$ decay than a Eulerian Sobolev-norm optimizer under a matched setup, while substantially reducing computational cost. We also identify a limitation of the reduced model: increasing the control basis may further improve the interface-length objective without yielding proportional gains in $\dot{H}^{-1}$ mixing, confirming that interface length is an effective but not fully faithful proxy for mixing in geometrically complex regimes.
15. 带精确二维节点求解器的三阶多矩格心拉格朗日流体动力学格式
A third-order multi-moment cell-centered Lagrangian scheme for hydrodynamics with an accurate 2D nodal solver
作者: Xiaoteng Zhang, Xun Wang, Zhijun Shen, Chao Yang
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04660v1
链接: http://arxiv.org/abs/2605.04660v1
摘要: This paper presents a novel high-order cell-centered Lagrangian scheme for 2D compressible hydrodynamics by bridging the multi-moment constrained finite volume method (MCV) [16, 51, 52] with a nodal Riemann solver. This scheme (denoted by LMCV) not only maintains high-order accuracy as MCV but also inherits the conservation and robust properties of the nodal Riemann solver. On the one hand, the MCV employs and evolves both the point-values (PV) at cell vertexes and the volume-integrated averages (VIA) on computational mesh, which ensures the rigorous numerical conservation and establishes an adequate foundation for the computation of Lagrangian fluxes with high accuracy. On the other hand, we developed a 2D Riemann solver based on EUCCLHYD [24], it takes fully advantage of numerical formulations from high-order scheme and accomplishes the compatibility between the mesh movement and numerical fluxes. The main new features of the solver are the introduction of a new set of jump and balance conditions. The jump condition provides a high-accurate formulation linking the surface pressure of each cell to its nodal velocity, while the balance condition ensures nodal conservation and stabilizes the velocity field without losing accuracy. More intriguing is that our nodal solver can be regarded as a natural high-order extension of the HLLC and the HLLC-2D [41] solvers. The comparison between these solvers better demonstrates our innovative approach in addressing the difficulties encountered in constructing 2D high-order Lagrangian schemes. A variety of numerical experiments are carried out to illustrate the accuracy and robustness of the algorithm.
16. 织物中的热质传递：一个加热圆柱体织物干燥模型
Heat and mass transfer through fabric: a model for fabric drying with heated cylinders
作者: Stefania Bellavia, Nicolò Fiorini, Adriano Milazzo, Alessandra Papini
类别: math.NA, math.OC
发布日期: 2026-05-06
arXiv ID: 2605.04644v1
链接: http://arxiv.org/abs/2605.04644v1
摘要: Textile drying is a key operation in the textile production cycle as it represents one of the most energy-intensive stages and plays a critical role in determining both product quality and overall process efficiency. In this work we propose a mathematical model for the drying process of a generic textile material using heated cylinders, operating under low-pressure conditions. The model’s parameters are estimated by nonlinear least squares regression. Given a specific fabric, the developed model allows to predict the drying time and the residual moisture content. The model is validated using real world data provided by a major Italian textile company.
17. 采用不连续转动与离散曲率的几何精确梁混合有限元法
Mixed Finite Elements for Geometrically Exact Beams using Discontinuous Rotations and Discrete Curvature
作者: Alexander Humer, Ivo Steinbrecher, Astrid Pechstein
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04573v1
链接: http://arxiv.org/abs/2605.04573v1
摘要: We propose a novel mixed finite-element formulation for geometrically exact (Simo–Reissner) beams that introduces the moment vector as additional independent field. The specific mixed form allows for an element-local, discontinuous approximation of rotations, which is key to a simple and efficient discretization framework. The concept of discrete curvature provides a mathematically consistent treatment of rotation discontinuities. For linear constitutive laws, the mixed form is derived via a Legendre transform of the curvature-related strain energy. Objectivity is retained at the discrete level by interpolating relative rotations through a multiplicative split of the rotation field; path-independence is inherent to the total Lagrangian setting and verified numerically. Several benchmarks demonstrate optimal rates of convergence and accuracy, irrespective of the beam’s slenderness and order of approximation. Notably, the lowest-order element entirely avoids rotation interpolation by employing element-constant rotations only.
18. 神经网络引导的区域限制加速结构化非正规带状矩阵的伪谱计算
Neural-Guided Domain Restriction to Accelerate Pseudospectra Computation for Structured Non-normal Banded Matrices
作者: Amit Punia, Rakesh Kumar, Madan Lal
类别: math.NA, cs.LG
发布日期: 2026-05-06
arXiv ID: 2605.04550v1
链接: http://arxiv.org/abs/2605.04550v1
摘要: Computing pseudospectra of non-normal matrices is essential for understanding the stability and transient behavior of dynamical systems. Such analysis is critical in applications including fluid dynamics, control systems, and differential operators, where non-normality can lead to significant transient amplification and sensitivity to perturbations that are not captured by eigenvalue analysis alone. At large scales, commonly used numerical approaches for pseudospectra computation can become computationally demanding, as they require repeated auxiliary computations to identify spectrally sensitive regions in the complex plane. We present a neural network-based approach that predicts sensitive regions directly from matrix features, thereby avoiding exhaustive pseudospectra evaluation across the entire complex plane. We calibrate the prediction threshold on validation data to ensure reliable coverage of sensitive regions. The trained neural network guides the selection of grid points requiring full computation, enabling focused computation only where necessary. The approach provides a practical preprocessing strategy for efficient pseudospectra computation. Numerical experiments on non-normal banded matrices demonstrate substantial speedup compared to full grid-based numerical evaluation while maintaining high accuracy in identifying sensitive regions.
19. 从视频到PDE：非线性染料羽流动力学的数据驱动发现
From Video-to-PDE: Data-Driven Discovery of Nonlinear Dye Plume Dynamics
作者: Cesar Acosta-Minoli, Sayantan Sarkar
类别: cs.LG, math.NA, physics.comp-ph, stat.AP, stat.ML
发布日期: 2026-05-06
arXiv ID: 2605.04535v1
链接: http://arxiv.org/abs/2605.04535v1
摘要: Inferring continuum models directly from video is hampered by two facts: the recorded field is uncalibrated image intensity rather than a physical state, and direct numerical differentiation of noisy frames is unstable. We develop a video-to-PDE pipeline that converts grayscale recordings of an ink plume into a normalised scalar field $u(x,y,t)$, isolates a bulk drift $\mathbf{v}(t)$ from intrinsic spreading via the intensity-weighted centroid, and identifies an effective transport law by weak-form sparse regression. Conditioning, threshold-sweep and random-centre diagnostics show that overcomplete libraries are strongly collinear; the search is therefore restricted to compact gradient-based libraries. Coefficients are refined by an inverse physics-informed network and recalibrated against forward rollouts, with a chronological block bootstrap quantifying uncertainty. The selected reduced model $u_t+\mathbf v(t)\!\cdot\!\nabla u = 9.005\,|\nabla u|^{2}+0.666\,Δu$ outperforms advection–diffusion baselines on held-out frames, retains a positive Laplacian coefficient, and admits a Cole–Hopf reduction to a linear advection–diffusion equation. The framework demonstrates that uncalibrated visual data can yield compact, predictive and structurally interpretable continuum models when discovery, calibration and uncertainty are treated as distinct stages.
20. 计算旋转非线性薛定谔方程散焦作用基态的梯度流分析
Analysis of gradient flow for computing defocusing action ground states of rotating nonlinear Schrödinger equations
作者: Wei Liu, Tingfeng Wang, Yongjun Yuan, Xiaofei Zhao
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04485v1
链接: http://arxiv.org/abs/2605.04485v1
摘要: This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schrödinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.
21. SO(n) 与 SE(n) 上随机微分方程的几何 Milstein 格式
Geometric Milstein Scheme for Stochastic Differential Equations on SO(n) and SE(n)
作者: Xi Wang, Victor Solo
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04480v1
链接: http://arxiv.org/abs/2605.04480v1
摘要: In the paper, we propose a higher-order geometry-preserving numerical method for stochastic differential equations (SDEs) evolving on the Lie groups SO(n) and SE(n). Most existing Lie group integrators rely on Magnus expansion of the exponential map, which makes the construction of higher-order stochastic schemes difficult. To overcome this limitation, we develop a tangent-space parameterization corrected Milstein method (TaSP-CM), extending the tangent space parameterization (TaSP) framework from Lie-group ODEs to the stochastic setting. Although TaSP is a well-established method for Lie ODEs, the extension to SDEs is non-trivial and requires new stochastic corrections that ensure both geometric consistency and higher-order accuracy. We prove that the proposed scheme achieves strong convergence of order 1 under both commutative and non-commutative noise. Numerical experiments illustrate the theoretical results and demonstrate the efficiency and robustness of the proposed method.
22. 二维不可压缩Navier-Stokes方程族的DLN方法的长时间$L^2$&$H^1$稳定性
Long-time $L^2$&$H^1$-stability of the Family of DLN Methods for the Two-dimensional Incompressible Navier-Stokes Equations
作者: Isabel Barrio Sanchez, Wenlong Pei, Catalin Trenchea
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04455v1
链接: http://arxiv.org/abs/2605.04455v1
摘要: In this report, we study the long-time stability of the family of one-leg DLN methods for the two-dimensional incompressible Navier-Stokes equations. The family of DLN methods (with one parameter $θ$), non-linear energy stable ($G$-stable) and second-order accurate under arbitrary time grids, has been widely applied to the simulations of various fluid models with success. We derive a new version of the $G$-stability identity for the family of DLN methods under uniform time grids and mild time constraints. Then we utilize this crucial auxiliary tool and the discrete uniform Grönwall inequality lemma to prove the uniform-in-time stability of the numerical solutions. Essentially, the bounds are independent of the time interval and the initial conditions, consistent with the theories of the continuous case.
23. 不可压缩Oseen问题的结构保持与压力鲁棒PINNs
Structure-Preserving and Pressure-Robust PINNs for Incompressible Oseen Problems
作者: Shiv Mishra, Arbaz Khan
类别: math.NA
发布日期: 2026-05-06
arXiv ID: 2605.04427v1
链接: http://arxiv.org/abs/2605.04427v1
摘要: We develop a new class of physics-informed neural network approximations for the stationary Oseen equations based on stability-consistent loss constructions. In contrast to standard PINN formulations, which are typically heuristic, the proposed consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. Within this setting, we introduce two fundamentally new approaches. First, we design standard CPINN formulations that exhibit clear improvements over conventional PINNs. Second, we propose pressure-robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors that depend solely on the divergence-free component of the forcing and are independent of the pressure. The framework accommodates both exactly divergence-free architectures and unconstrained velocity approximations, providing a unified treatment of these two paradigms. Using techniques from optimal recovery theory, we establish, for the first time in the PINN setting for Oseen-type problems, quantitative recovery estimates and optimal error bounds for both velocity and pressure under suitable Besov regularity assumptions. In particular, we obtain optimal rates for the velocity in $\boldsymbol{H}^1(Ω)$ and for the pressure in $L^2(Ω)$. The proposed methodology introduces a pressure-robust CPINN paradigm for incompressible flows, combining structural consistency, robustness with respect to irrotational forces, and rigorous accuracy guarantees. Numerical experiments corroborate the theoretical findings and demonstrate the effectiveness of the approach.
24. 用于从代数簇中采样的刚性同伦：一个Waring结构复杂度模型
Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model
作者: Abigail R. Jones, Kisun Lee, Jose Israel Rodriguez
类别: math.NA, cs.CC, math.AG
发布日期: 2026-05-05
arXiv ID: 2605.04302v1
链接: http://arxiv.org/abs/2605.04302v1
摘要: Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale’s 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation.
25. 基于集中不等式的概率浮点舍入误差分析
Probabilistic Floating-Point Round-Off Analysis via Concentration Inequalities
作者: Yichen Tao, Hongfei Fu, Jiawei Chen, Jean-Baptiste Jeannin
类别: cs.LO, cs.PL, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.04232v1
链接: http://arxiv.org/abs/2605.04232v1
摘要: Floating-point round-off errors are ubiquitous in numerically intensive programs arising in fields such as scientific computing and optimization. As floating-point errors potentially lead to unexpected and catastrophic program failures, one must derive guaranteed round-off thresholds to ensure the correctness of these programs. However, deterministic round-off thresholds tend to be too conservative to be usable in practice, since they often involve large round-off errors that occur with small probability. Probabilistic thresholds relax deterministic ones by specifying that the probability of the round-off error exceeding a threshold is below a given confidence. In this work, we propose a novel approach to probabilistic round-off analysis, by applying concentration inequalities over the Taylor expansion from FPTaylor (TOPLAS 2018). A major obstacle in applying concentration inequalities is that the Taylor expansion involves absolute value operators that make the calculation of the expected values of the first order partial differential terms difficult. Our first step to overcome this obstacle is a sound over-approximation that removes the absolute value operators in polynomial expressions. Then, we show how to handle fractional expressions by a transformation into polynomial case. Finally, we show how to improve our approach with range partitioning. Our approach is scalable since the key computational part is the calculation of expected values of polynomial expressions with independent variables, for which the linear and independence properties of expectation boost the computation. Experimental results show that our approach is orders of magnitude more time efficient, while producing thresholds with comparable precision against the state of the art.
26. 基于最缓上升动力学的电力系统吸引域边界计算
Calculating Domain of Attraction Boundary of Power Systems Based on the Gentlest Ascent Dynamics
作者: Sixu Wu, Chenmin Zhang, Aiqing Zhu, Yang Liu, Jianxi Lin, Yifa Tang
类别: math.DS, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.04197v1
链接: http://arxiv.org/abs/2605.04197v1
摘要: The power system, a fundamental public utility, is increasingly important due to growing global electricity demand. Recent large-scale blackouts (e.g., Iberian Peninsula, UK) have raised concerns about transient stability under impact faults. Transient stability is determined by post-disturbance synchronizing capability of synchronous generators, formulated as identifying the domain of attraction (DOA) boundary of the asymptotically stable equilibrium. Using a benchmark model of synchronous-generator-dominated power systems, this report employs a gentlest ascent dynamics (GAD) method for 1-saddle points, an adjoint operator method for periodic orbits, and stable manifold algorithms to compute the DOA boundary. These algorithms transform DOA boundary determination into constructing unstable critical elements (saddle points and periodic orbits) and their stable manifolds. Theoretically, under certain assumptions we prove that the DOA boundary is the closure of the union of stable manifolds of index-1 critical elements, and establish a stability theory for a perturbed GAD system. Numerical experiments on two-machine and three-machine systems (with only saddle points or with periodic orbits) validate the effectiveness and accuracy. Results show the algorithms accurately capture the geometric structure of the DOA boundary, providing a new numerical tool for transient stability analysis.
27. 学习分数阶随机微分方程的误差分析及其在神经逼近中的应用
Error analysis for learning fractional stochastic differential equations with applications in neural approximations
作者: Mahdi Dehshiri, Kerlyns Martinez, Lauri Viitasaari
类别: math.PR, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.04168v1
链接: http://arxiv.org/abs/2605.04168v1
摘要: This paper develops a framework for the error analysis in nonparametric model fitting of fractional stochastic differential equations based on discrete observations. We identify and quantify the main error sources – time discretization, coefficient approximation, and model fitting error – within a unified framework. Through Sobolev-type norms, we derive convergence rates that incorporate the regularity of trajectories, thereby capturing the interaction of these error components. To demonstrate the applicability of the theory, we introduce a training scheme for coefficient function estimation based on shallow neural networks and a recurrent architecture. Numerical experiments validate the theoretical findings and illustrate the effectiveness of the approach.