生成时间: 2026年07月02日 10:22:39

重点关注论文数量: 0

了解领域论文数量: 5

不相关论文数量: 23

了解领域论文（摘要翻译）

1. 多胞体上多调和方程的可杂交交错间断有限元方法

Hybridizable Staggered Discontinuous Galerkin Methods for Polyharmonic Equations on Polytopes

作者: Long Chen, Xuehai Huang, Yule Sun, Shudan Tian

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00831v1

链接: http://arxiv.org/abs/2607.00831v1

摘要翻译

中文标题: 多胞体上多调和方程的可杂交交错间断有限元方法

摘要翻译: 针对 $\mathbb R^d$ 中形状正则的多胞体网格上的任意阶多调和方程 $(-\Delta)^m u=f$，对任意 $m\ge1$, $d\ge2$ 和多项式次数 $k\ge0$，发展了可杂交交错间断有限元方法。该方法利用混合变量 $\sigma=\nabla^m u$ 以及交错的主-对偶网格，对标量和张量未知量施加互补的连续性，且无需 $d\ge m$ 等限制。局部迹和 bubble 函数加富在无需增加全局未知量的情况下稳定低阶张量空间。杂交化将张量变量局部化，并产生一个等价的无稳定化弱伽辽金形式。证明了适定性和最优能量误差估计，在四边形和四面体网格上的数值实验验证了预测的收敛率。

2. 三角形网格上对流方程的埃尔米特半拉格朗日格式

Hermite Semi-Lagrangian schemes on triangular meshes for advection equations

作者: Ali Elarif, Michel Mehrenberger, Laurent Navoret

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00619v1

链接: http://arxiv.org/abs/2607.00619v1

摘要翻译

中文标题: 三角形网格上对流方程的埃尔米特半拉格朗日格式

摘要翻译: 基于Bell和Argyris有限元，针对对流方程在非结构三角形网格上提出了高阶埃尔米特半拉格朗日格式。节点型半拉格朗日格式同时输送点值、梯度和海森矩阵，而投影型半拉格朗日格式本质上是具有直的剩余后向三角形和共享自由度的半拉格朗日间断伽辽金格式（SLDG）。通过数值实验对不同格式进行了稳定性和收敛性研究。

3. 边界积分方程的多层随机梯度神经求解器

A multilevel stochastic-gradient neural solver for boundary integral equations

作者: Bing-Ze Lu, Richard Tsai

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00560v1

链接: http://arxiv.org/abs/2607.00560v1

摘要翻译

中文标题: 边界积分方程的多层随机梯度神经求解器

摘要翻译: 我们针对第二类边界积分方程开发了一种多层随机梯度神经求解器。未知密度由一个多层感知器表示，该感知器通过在逐渐细化的求积网格阶梯上最小化 Nyström 离散的残量来训练，每一层均从前一层的参数进行热启动。每一步仅需在配置点的 mini-batches 上进行稠密矩阵-向量乘积以及网络的前向传递，这些操作可直接映射到 GPU 硬件上。残量收缩由经验神经正切核 (NTK) 控制，它是单个连续核的离散采样。在固定网格上，一旦残量集中于网络收缩缓慢的模式，训练就会停滞，这一平台现象由频率原理描述；谱分析解释并实验证实了细化求积如何解析更多连续核的频谱并将这些模式带回优化器的可及范围。因此，谱偏置——在其他场合是神经求解器的障碍——在此充当了多重网格型迭代的光滑器，其中求积细化代替了粗网格校正。在网络具有一致正则性有界假设下，总计算量是最细网格上工作量的常数倍，而离散第二类算子的一致良态性使得 NTK 成为唯一决定速率的谱，同时将训练残量转化为后验误差界。针对内部 Dirichlet Laplace/Poisson 问题和外部 Neumann Helmholtz 问题的实验，使用参数化表面和带符号距离函数表面表示，在可比容差下与 GMRES 的比较展示了所提出方法的有效性和效率。

4. 对于具有p-Laplacian型扩散的非线性抛物型方程的非标准有限差分格式

A Nonstandard Finite Difference Scheme for a Nonlinear Parabolic Equation with p-Laplacian-Type Diffusion

作者: Achraf Zinihi, Matthias Ehrhardt, Moulay Rchid Sidi Ammi

类别: math.NA, physics.comp-ph

发布日期: 2026-07-01

arXiv ID: 2607.00489v1

链接: http://arxiv.org/abs/2607.00489v1

摘要翻译

中文标题: 对于具有p-Laplacian型扩散的非线性抛物型方程的非标准有限差分格式

摘要翻译: 我们针对一维和二维空间域中包含p-Laplacian型扩散算子的非线性抛物型方程，提出并分析了一种非标准有限差分（NSFD）格式。遵循Mickens的设计原则，该离散化采用非线性分母函数phi(.)以及对非线性扩散项Delta_p的非局部逼近，从而产生一个保结构的离散模型。该格式旨在保留连续问题关键的定性性质，包括正性、有界性和稳定性，这些性质可能会被标准有限差分法（FDM）丢失。我们建立了连续模型的适定性，推导了NSFD格式，并研究了其一致性、收敛性和局部截断误差。数值实验证实了理论结果，并表明与标准显式FDM不同，即使对于相对较大的时间步长，所提出的NSFD格式也能避免伪振荡和非物理的负解。

5. 保持松弛原始能量耗散律的相场模型松弛拉格朗日乘子（RLM）格式

Relaxed Lagrange Multiplier (RLM) Schemes for Phase Field Models Preserving the Relaxed Original Energy Dissipation Law

作者: Xiaobo Jing, Jia Zhao

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00355v1

链接: http://arxiv.org/abs/2607.00355v1

摘要翻译

中文标题: 保持松弛原始能量耗散律的相场模型松弛拉格朗日乘子（RLM）格式

摘要翻译: 相场模型通常由自由能泛函的变分原理导出，并广泛用于模拟科学和工程中的复杂多相现象。设计这些模型数值格式的一个核心目标是保持其内在的能量耗散律。本文提出一类用于相场模型的松弛拉格朗日乘子（RLM）格式。与常用的标量辅助变量（SAV）和不变能量二次化（IEQ）方法不同，这些方法耗散一个涉及辅助变量的修正能量，而RLM格式耗散的是原始能量的一个松弛版本，并紧密追踪原始能量耗散率。与经典拉格朗日乘子（LM）方法相比，RLM格式保证了所得离散系统在较宽的时间步长范围内具有唯一可解性。关键思想是在LM公式中增加一个松弛项，从而得到关于乘子的一个标量二次方程，具有显式闭式解。所得格式是线性的且高效，因为每个时间步仅需求解两个常系数线性系统，其计算代价与SAV格式相当。我们构造了一阶和二阶变体，并证明了其能量稳定性。数值实验验证了预期的收敛阶，并表明RLM格式能精确捕捉界面动力学。

不相关论文（基本信息）

1. GAIA: 面向正反问题的几何自适应算子学习

GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems

作者: Meenakshi Krishnan, Pranav Pulijala, Ke Chen, Haizhao Yang, Ramani Duraiswami

类别: cs.LG, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01128v1

链接: http://arxiv.org/abs/2607.01128v1

摘要: Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

2. 原子核中 QRPA 强度函数的 Lanczos 方法

Lanczos Method for QRPA Strength Functions in Atomic Nuclei

作者: Dong Min Roh, Chao Yang, Jonathan Engel, Matthew L. Dai

类别: physics.comp-ph, math-ph, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01114v1

链接: http://arxiv.org/abs/2607.01114v1

摘要: We present a symmetric Lanczos method for computing charge-changing QRPA strength functions in atomic nuclei. Starting from the finite-amplitude-method formulation of the QRPA linear-response problem, we derive equivalent spectral representations and, in the real case, a reduced eigenvalue problem involving the matrix products $MK$ and $KM$, where $M\equiv A+B$ and $K\equiv A-B$ are formed from the usual QRPA matrices $A$ and $B$. The resulting formulation enables a matrix-free Lanczos approximation of the Lorentzian-smeared strength function over a broad energy interval from a single Krylov run, in contrast to conventional frequency-by-frequency response calculations. Numerical tests for $^{112}$Sn and $^{150}$Nd first show that GMRES reproduces the converged iterative FAM strength profiles while requiring fewer iterations. Using GMRES as the frequency-by-frequency reference, we then show that the Lanczos approximation reproduces the same strength profiles with reduced overall cost. These results indicate that symmetric Lanczos projection provides an efficient and accurate approach for QRPA strength-function calculations when spectral information is required over an extended frequency range.

3. 通过单调输入凸神经网络压缩各向同性函数的多凸包络

Compression of Polyconvex Envelopes of Isotropic Functions via Monotonic Input Convex Neural Networks

作者: Timo Neumeier, Julian Salmon

类别: math.NA, cs.CE

发布日期: 2026-07-01

arXiv ID: 2607.01055v1

链接: http://arxiv.org/abs/2607.01055v1

摘要: This work presents a novel neural-network compression approach for polyconvex envelopes of isotropic functions. The approach relies on a classical sufficient criterion for polyconvexity and is particularly suited for the representation of determinant-constrained energy densities arising in non-linear elasticity. Compared with existing compression methods based on the necessary and sufficient characterisation of polyconvex isotropic functions, the proposed framework reduces computational costs, due to the domain reduction through the restriction to the positive octant in the singed singular value space. The underlying neural-network architecture employs input-convex neural networks (ICNNs) with non-negative weight constraints to enforce the required convexity and monotonicity properties. The additional symmetry and inequality conditions characterising the polyconvex envelope are incorporated weakly through the loss function during training. Although the employed criterion is only sufficient and thus generally yields only a lower bound on the polyconvex envelope, numerical experiments based on the classical Saint Venant–Kirchhoff energy demonstrate that the proposed approach produces accurate approximations in practice while offering a computationally more efficient alternative to existing methods.

4. PICNN辅助的保物理格式用于多孔介质热力学一致两相流

The PICNN-Assisted Physics-Preserving Scheme for Thermodynamically Consistent Two-Phase Flow in Porous Media

作者: Yuanshuo Kong, Xue Wang, Yujing Yan

类别: physics.flu-dyn, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.01020v1

链接: http://arxiv.org/abs/2607.01020v1

摘要: In this paper, we develop a physics-informed convolutional neural network (PICNN) assisted physics-preserving method for a thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media. Following the physics-preserving prediction-correction scheme of Li et al. \cite{li2025class}, the prediction step is performed by a PICNN trained with finite-volume residuals, where the interfacial fluxes are evaluated by the two-point flux approximation (TPFA) using two-point difference quotients of neighboring cell-centered unknowns to approximate interfacial normal gradients. The PICNN output is further corrected by a post-processing procedure to obtain energy-stable, mass-conservative, and bounds-preserving solutions. Numerical results show that the finite-volume residuals trained PICNN can replace the traditional prediction solver within the physics-preserving framework. Compared with conventional physics-informed neural networks (PINNs), the PICNN better captures local spatial interactions between each control volume and its neighboring cells, while the finite-volume residuals accommodate discontinuous permeability fields and interfacial flux continuity.

5. 基于分层半可分矩阵的二维第二类非均匀离散傅里叶逆变换的直接超快速求解器

A Superfast Direct Solver for 2D Type-II Inverse Nonuniform Discrete Fourier Transform Based on Hierarchically Semiseparable Matrix

作者: Yingzhou Li, Jingyu Liu

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00928v1

链接: http://arxiv.org/abs/2607.00928v1

摘要: This paper proposes a direct inversion method for the 2D type-II nonuniform discrete Fourier transform~(NUDFT). The NUDFT matrix $A$ is factored as $A = G F$, where $G$ can be expressed as a kernel matrix and $F$ is the 2D DFT matrix. We show that $G$ can be approximated by a hierarchically semiseparable~(HSS) matrix and give an estimate of the HSS rank. Then, using the least-squares solver for HSS matrix and the two-dimensional inverse fast Fourier transform, the inverse NUDFT problem can be solved efficiently. Our algorithm has an offline complexity of $O\bigl(M+ N^{3 / 2} \log^{3} N\bigr)$ where $M$ and $N$ are the size of rows and columns of the NUDFT matrix, respectively. Once the direct solver is built, it can be applied to a vector with an online complexity of $O\bigl(M+ N \log^{3} N\bigr)$. The proposed method can be used as a preconditioner for iterative methods, especially when the sample points are distributed on a grid such that $A$ is ill-conditioned. Numerical results are provided to show the scaling performance of the inversion method and demonstrate the efficiency and robustness of it as a preconditioner.

6. 具有分离空间变量的PGD降阶模型的认证

Certification of PGD reduced-order models with separated spatial variables

作者: Jean Ruel, Ludovic Chamoin, Frédéric Legoll, Arthur Lebée

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00896v1

链接: http://arxiv.org/abs/2607.00896v1

摘要: Model order reduction techniques have become an attractive approach for obtaining fast approximations of multidimensional problems. Besides computational efficiency, ensuring the reliability of the resulting approximations is of primary importance. This work focuses on the certification of PGD-based reduced-order models based on the separation of spatial variables, which are particularly well suited to plate and shell geometries. Considering diffusion problems defined in plate-like domains, we introduce a guaranteed global error estimate associated with the PGD approximation. To this end, the error bounds are derived from the Constitutive Relation Error (CRE) method. The main difficulty of this approach lies in the construction of equilibrated fluxes, for which a dedicated procedure is proposed. Based on the resulting estimator, an adaptive strategy is developed to control both the discretization error and the number of PGD modes. This certification procedure is further extended to the error control in quantities of interest. We provide several numerical examples illustrating the reliability and efficiency of our procedure.

7. 用于无网格有限差分法的重叠区域分解

Overlapping Domain Decomposition for Meshless Finite Difference Methods

作者: Alexander Westermann, Oleg Davydov, Stefan Turek

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00842v1

链接: http://arxiv.org/abs/2607.00842v1

摘要: Schwarz type domain decomposition methods generally require a partition of unity to combine solutions on subdomains. However, in mesh-based methods it is common to organize subdomains with minimal overlap, if any, which is facilitated by the availability of a mesh. This study analyzes how the continuity of the partition of unity affects the algebraic Schwarz method for Poisson and Stokes equations from a meshless point of view, whereby the underlying differential operators are discretized using the radial basis function finite difference (RBF-FD) method. We demonstrate numerically that, in this setting, small overlaps improve the performance of the domain decomposition, leading to smaller iteration counts, and therefore no disjoint partitioning technique is required.

8. 基于代理面的分布式内存系统 TE 模式散射问题快速直接求解器

Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems

作者: Yasuhiro Matsumoto, Rio Yokota

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00790v1

链接: http://arxiv.org/abs/2607.00790v1

摘要: This paper describes an MPI/OpenMP hybrid parallelized fast direct solver for the scattering problem of transverse electric (TE)-mode electromagnetic waves. Because TE-mode scattering can be reduced to the two-dimensional Helmholtz equation, solvers based on the hierarchically semiseparable (HSS) representation are highly attractive due to their high parallel efficiency. However, as the HSS representation applies low-rank approximations to all off-diagonal blocks, it exhibits poor compatibility with high-order discretization methods. We developed a fast direct solver with $O(h^3)$ convergence for Helmholtz transmission problems, whereas conventional HSS solvers typically yield only $O(h)$ convergence (where $h$ represents intervals between the quadrature nodes). It is based on the weakly singular Burton-Miller boundary integral equation and the Nyström method with a one-point correction. Furthermore, recognizing that matrix component calculation, rather than matrix factorization, dominates the total computational time of HSS-type boundary integral solvers, we introduced a load-balancing method to maximize parallel efficiency. Numerical results demonstrate that the direct solver achieves high-accuracy convergence and nearly ideal strong and weak scalabilities.

9. 使用时间并行采样加速离散扩散模型

Accelerating Discrete Diffusion Models with Parallel-In-Time Sampling

作者: Yu Yao, Huanjian Zhou, Andi Han, Wei Huang, Masashi Sugiyama

类别: cs.LG, cs.DC, cs.DS, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00773v1

链接: http://arxiv.org/abs/2607.00773v1

摘要: Discrete diffusion models are widely used for learning and generating discrete distributions. As the generation process is inherently sequential, the acceleration of sampling is of significant importance. In this work, we parallelize the mainstream $τ$-leaping algorithm for absorbing discrete diffusion in a Continuous-Time Markov Chain (CTMC) framework. By leveraging the continuous-time stochastic integral form of the $τ$-leaping algorithm and the Picard iteration method, we achieve parallel-in-time sampling acceleration and provide a proof of exponential-factorial convergence for our algorithm. We improve the overall time complexity of $τ$-leaping under absorbing settings from ${\mathcal{O}}(d \log S)$ to ${\mathcal{O}}(\log (d\log S)\cdot \log d)$ with respect to NFE. Empirically, our method shows consistent acceleration across synthetic and real-data settings. The new sampler achieves at most $7$–$9\times$ runtime speedup for synthetic distribution, and maintains the same quality with $50\%$ fewer NFE and $1.45$–$1.86\times$ runtime speedups in image/text tasks on a single GPU. Our research expands the potential of discrete diffusion models for efficient parallel inference, with broader implications for applications such as molecular structure and language generation.

10. 用于多物种Keller-Segel趋化系统的线性、解耦保正的时间交错块中心有限差分方法

A linear, decoupled and positivity-preserving time-staggered block-centered finite difference method for the multi-species Keller-Segel chemotaxis system

作者: Ao Zhang, Bingyin Zhang, Hongfei Fu

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00713v1

链接: http://arxiv.org/abs/2607.00713v1

摘要: In this paper, we present a linearly implicit, second-order block-centered finite difference (BCFD) prediction-then-projection scheme for the multi-species Keller-Segel chemotaxis system on non-uniform spatio-temporal grids. The proposed scheme integrates a standard Crank-Nicolson time-marching algorithm with an $L^2$ projection step to enforce positivity and mass conservation. The use of variable time stepsize and time-staggered discretization fully decouples the solutions of the multi-species cell density variables and the chemoattractant concentration variable while facilitating linearization, thereby greatly enhancing computational efficiency. Notably, the variable time-stepping algorithm and non-uniform grid BCFD discretization jointly enable adaptive resolution and local refinement near blow-up, thereby improving efficiency and accuracy without compromising the desired physical property-preserving in the simulation. Furthermore, using the mathematical induction method and the energy analysis approach, the unique solvability of the proposed scheme is rigorously proved, and we show that cell densities achieve second-order convergence in both time and space in the discrete $L^2$ norm, while the chemoattractant concentration achieves second-order convergence in the discrete $H^1$ norm. Representative numerical experiments are presented to validate the theoretical findings and demonstrate the reliability of the proposed scheme in simulating the blow-up phenomenon.

11. 基于对偶加权残量法的Navier–Stokes方程目标导向时空自适应

Goal-oriented space-time adaptivity for the Navier–Stokes equations based on the dual weighted residual method

作者: Marius Paul Bruchhäuser, Nils Margenberg, Markus Bause

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00686v1

链接: http://arxiv.org/abs/2607.00686v1

摘要: This work presents a goal-oriented a posteriori error estimator based on the Dual Weighted Residual (DWR) method together with space-time mesh adaptivity for the Navier–Stokes equations. The resulting nonlinear algebraic systems on the space-time slabs are solved by Newton’s method with GMRES, preconditioned by a slab-wise geometric multigrid method. This combination yields reliable control of target quantities on computationally feasible space-time meshes together with a robust and efficient solution of the algebraic systems. The implementation is based on a MPI-parallel programming model in the deal.II library. Further ingredients are a discontinuous Galerkin discretization in time and inf-sup stable finite element pairs with discontinuous pressure on tensor-product meshes. The performance of the approach is investigated in benchmark computations with regard to accuracy, efficiency, and stability.

12. 基于de Rham复形的无惩罚自然深度里茨方法求解高维狄利克雷边值问题

Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems

作者: Jiarong Chen, Xia Ji, Haijun Yu, Shuo Zhang

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00676v1

链接: http://arxiv.org/abs/2607.00676v1

摘要: Deep neural networks show great promise for high-dimensional PDEs, yet enforcing essential boundary conditions remains challenging, especially as penalty parameters require problem-specific retuning with increasing dimensionality. In this work, we extend the Natural Deep Ritz Method (NatDRM) [H. Yu and S. Zhang, J. Comput. Phys., 537 (2025)] to a unified framework for all dimensions $d \geq 2$ based on the de Rham complex and its penalty-free boundary decomposition: curl-type operators act on scalar potentials in 2D, vector potentials in 3D, and antisymmetric second-order tensor potentials in $d \geq 4$, respectively. This method converts Dirichlet constraints into three coupled natural (Neumann-type) subproblems with corresponding Ritz-type losses, eliminating the need for a boundary penalty parameter $β$. We derive dimension-unified discrete losses, lightweight boundary-based gauge-fixing regularizations to resolve curl-kernel non-uniqueness, and a joint training procedure; extensions to variable-coefficient elliptic and semilinear Poisson problems are formulated at the first subproblem level. Numerical experiments on smooth benchmarks up to 6D show that NatDRM, without any penalty tuning, matches or exceeds the accuracy of optimally tuned DRM and PINN in most cases. It converges stably in 6D where penalized DRM fails for most penalty values, and exhibits synchronous decay of interior and boundary errors, resolving the inherent imbalance of penalty-based methods.

13. 卷积对称自编码器：通过微分几何增强隐空间稳定性

Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry

作者: G. Li Causi, N. Tonicello, L. Magri, G. Rozza

类别: math.NA, stat.ML

发布日期: 2026-07-01

arXiv ID: 2607.00669v1

链接: http://arxiv.org/abs/2607.00669v1

摘要: Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov $n$-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs). However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by arXiv:2307.15288v2 and arXiv:2506.11641v1 have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, the Viscous Burger and the Kuramoto Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.

14. 一种用于随机Lyapunov矩阵方程的带最优参数的内外迭代算法

An Inner-Outer Iteration Algorithm with Optimal Parameters for Stochastic Lyapunov Matrix Equation

作者: Donghuan He, Xiaowen Su, Feng Wang, Xuesong Chen

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00451v1

链接: http://arxiv.org/abs/2607.00451v1

摘要: This paper proposes an inner–outer (IO) iterative algorithm with optimal parameters for solving stochastic Lyapunov matrix equation associated with discrete-time stochastic linear system. First, under the assumption that the underlying stochastic linear system is asymptotically mean-square stable, the monotonicity and boundedness of the iterative sequence generated by the proposed algorithm are analyzed. On this basis, a sufficient convergence result is established for the zero initial condition. Second, by deriving the spectral radius of the corresponding iteration matrix, several necessary and sufficient convergence conditions are obtained for arbitrary initial conditions. In addition, the optimal parameter-selection strategies are developed to improve the convergence performance of the algorithm. Finally, numerical examples are presented to verify the theoretical results and demonstrate the advantages of the proposed algorithm over several existing iterative methods.

15. 通过匹配传输算子与不变测度的数据自适应动力系统学习

Data-Adaptive Learning of Dynamical Systems by Matching Transfer Operators and Invariant Measures

作者: Yinong Huang, Jonah Botvinick-Greenhouse, Yunan Yang

类别: math.NA, math.DS

发布日期: 2026-07-01

arXiv ID: 2607.00391v1

链接: http://arxiv.org/abs/2607.00391v1

摘要: Trajectory-based learning of dynamical systems is often fragile in the presence of noise, chaos, or sparse observations, as small pointwise errors can rapidly amplify. We introduce a transition-statistics approach to system identification that learns dynamics by matching the induced motion of probability mass across a data-adaptive mesh. Given trajectory data, we build an unstructured partition of state space and approximate the Perron–Frobenius operator with a regularized Ulam transition matrix. We replace hard cell indicators with continuous, piecewise-smooth partition-of-unity weights, yielding a Markov matrix supporting gradient-based optimization with respect to the parameters of a learned vector field. This enables two related training objectives: matching invariant measures through the stationary eigenvectors of the transition matrices, and matching the full transition matrices to capture transport between regions of state space. Numerical experiments on Lorenz-63, Lorenz-96, and a reduced-order NOAA sea surface temperature forecasting problem show that transition-statistics matching gives more reliable long-time dynamics than pointwise trajectory matching, particularly under measurement noise and sparse sampling. The approach provides a robust operator-theoretic alternative to trajectory-level losses for learning chaotic and partially observed dynamical systems.

16. 非线性薛定谔方程的指数低正则性Parareal算法

Exponential Low-Regularity Parareal Algorithms for Nonlinear Schrödinger Equations

作者: Qingle Lin, Zhi Zhou

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00384v1

链接: http://arxiv.org/abs/2607.00384v1

摘要: The parareal algorithm is one of the most widely studied parallel-in-time methods for the numerical approximation of time-dependent problems. For non-diffusive equations, however, standard parareal methods may converge slowly or even become unstable due to the absence of damping, while nonlinear interactions can transfer and amplify phase errors across Fourier modes. In this work, we consider the nonlinear Schrödinger equation (NLS) as a representative non-diffusive model and analyze parareal algorithms with an exact fine propagator, with particular emphasis on the design of suitable coarse propagators. We establish a general convergence framework, valid for solutions with limited regularity, under stability and local truncation error assumptions on the coarse propagator. These assumptions are verified for selected exponential low-regularity integrators designed for one-dimensional quadratic and cubic NLS equations, which achieve optimal approximation orders without derivative loss. To the best of our knowledge, this is the first construction of parareal algorithms for NLS equations that are provably linearly convergent, with a contraction factor proportional to the coarse time-step size even for solutions of limited regularity. Numerical experiments on quadratic, cubic, and quintic NLS equations demonstrate rapid convergence and improved performance over parareal variants using classical coarse propagators, including Lie and Strang splitting methods and first- and third-order exponential Runge–Kutta integrators.

17. 物理信息神经网络用于非线性延迟微分方程

Physics Informed Neural Networks for Nonlinear Delay Differential Equations

作者: Stone Yao, Vipin Kumar, Roberto Guglielmi

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00380v1

链接: http://arxiv.org/abs/2607.00380v1

摘要: In this paper we propose a novel physics-informed neural network framework for solving general first-order delay differential equations. Our approach combines a differentiable history switch, a trial-solution formulation that explicitly enforces history constraints, and a segmented collocation strategy to stabilize gradient propagation across large temporal domains. The method enables a scalable and physics-consistent approximation of delay differential equation solutions while maintaining continuity across subintervals. Numerical experiments demonstrate the effectiveness of the proposed method.

18. WarpagePINN: 基于两阶段物理信息神经网络的先进封装热翘曲预测

WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks

作者: Xinyu Li, Min Tang, Zeyu Sun, Wenxing Zhu, Jianhua Zhang, Liang Chen

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00364v1

链接: http://arxiv.org/abs/2607.00364v1

摘要: Thermal warpage has become a critical issue in advanced packaging, primarily caused by the mismatch in coefficients of thermal expansion (CTE) among heterogeneously integrated materials. However, only a limited number of studies have focused on developing computational methods for coupled thermal-warpage prediction in the chiplet. This paper proposes a two-stage physics-informed neural network (WarpagePINN) framework to compute both temperature profile and warpage deformation of chiplets. The neural networks are trained without relying on labeled datasets generated by conventional simulators. In the first stage, the temperature field is modeled using a Fourier series representation that inherently satisfies boundary conditions, and the network is trained solely through a loss function derived from the governing equation. In the second stage, a multilayer perceptron (MLP) is employed for warpage prediction, utilizing a novel hybrid supervisory strategy to optimize the energy-based loss function instead of residual loss. A parametric WarpagePINN is also developed to quantify uncertainties associated with the CTE. Numerical results show that the proposed WarpagePINN framework achieves excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 μm, while achieving a speedup of approximately 1000 {\times} in CTE parameterization studies.

19. 半经典Dirac方程的时间分裂方法的严格分析

Rigorous analysis of the time-splitting methods for the semiclassical Dirac equation

作者: He Wang, Jia Yin

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00335v1

链接: http://arxiv.org/abs/2607.00335v1

摘要: We provide rigorous error analysis of the mass-preserving time-splitting methods for solving the semiclassical Dirac equation. The scaled Planck constant $ε$ in the equation gives rise to rapid oscillations in both space and time when $0<ε\ll 1$ with wavelengths of order $O(ε)$. %We prove that the first-order splitting $S_1$ and the second-order splitting $S_2$ schemes preserve the total discretized mass. Rigorous error estimates reveal the precise dependence of the approximation errors on the time step $τ$, the spatial mesh size $h$, and the parameter $ε$. Specifically, the temporal error scales as $O\left(τ/ε^2\right)$ for the first-order splitting $S_1$ and as $O\left(τ^2/ε^3\right)$ for the second-order splitting $S_2$, while the spatial error scales as $O(h^m/ε^m)$ for both methods, where $m$ is related to the regularity of the solution. In addition, we obtain error bounds for key physical observables, including the total probability density $ρ$ and the current density $\mathbf{J}$. Compared with finite difference time domain (FDTD) methods, time-splitting approaches exhibit spectral accuracy in space and retain a relatively low computational cost. Furthermore, we demonstrate that higher accuracy can be achieved by employing the fourth-order compact time-splitting ($S_\text{4c}$) method. Numerical experiments are conducted to verify the reliability of the error estimates.

20. 从谱方法到傅里叶神经算子的样本复杂度界

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

作者: Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

类别: stat.ML, cs.LG, math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00320v1

链接: http://arxiv.org/abs/2607.00320v1

摘要: We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-$T$ solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier–Stokes, Allen–Cahn, and Cahn–Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

21. 通过薛定谔化与人工压缩性对斯托克斯流的量子模拟

Quantum Simulation of Stokes Flow via Schrödingerisation and Artificial Compressibility

作者: Shi Jin, Jiaqi Tang, Qilong Zhai, Lei Zhang

类别: math.NA

发布日期: 2026-07-01

arXiv ID: 2607.00281v1

链接: http://arxiv.org/abs/2607.00281v1

摘要: Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem. In this work, we present a quantum algorithm based on the Schrödingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization. The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system. The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schrödingerisation procedure. A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality. The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.

22. 识别无限周期点源阵列中的缺陷单元

Identifying defective units in infinite periodic arrays of point sources

作者: Dinh-Liem Nguyen, Nhung H. Nguyen, Thi-Phong Nguyen

类别: math.NA

发布日期: 2026-06-30

arXiv ID: 2607.00241v1

链接: http://arxiv.org/abs/2607.00241v1

摘要: This paper focuses on identifying defective units in unbounded periodic arrays of point sources using boundary data. The study is motivated by the noninvasive evaluation of large-scale periodic source systems. Unlike classical inverse source problems in free space, the key challenge here lies in the disruption of periodicity caused by defective sources in the infinite array. To address this, we employ the Floquet - Bloch transform to reformulate the original inverse source problem as a quasi-periodic inverse source problem. We first establish uniqueness theorems for both the original and the quasi-periodic formulations. Then, we develop a new numerical method for identifying defective sources. This method combines a sampling indicator function with an algebraic technique to determine not only the number of defective sources, but also their locations and intensities. Numerical experiments are presented to validate the effectiveness of the proposed method.

23. 耦合复制-突变动力学中的执行滤波与延迟预算不稳定性

Implementation Filters and Delay-Budget Instability in Coupled Replicator–Mutator Dynamics

作者: Alexander Omelchenko

类别: math.DS, math.NA

发布日期: 2026-06-30

arXiv ID: 2607.00227v1

链接: http://arxiv.org/abs/2607.00227v1

摘要: We model an adaptive contest in which two antagonistically coupled populations continually reallocate effort among competing methods, but decisions are not fielded instantly. Each side has an intended portfolio and a deployed portfolio: intended reallocations follow delayed observations of the opponent, while deployment follows intent through a first-order implementation filter. Under barycentric balance and uniform exploration, the linearized scalar branches have a characteristic factor in which hard observation and deployment lags enter only through their total sum, whereas implementation rates enter through real filter factors that cannot be absorbed into selection or exploration. In the strictly antagonistic class, negative spectral branches split into three regimes: weak branches have no positive-frequency crossing, intermediate branches lose stability through a delay-induced Hopf bifurcation, and strong branches are at or beyond the implementation-filter instability margin already at zero hard delay. This gives an operational delay-budget rule: in the delay-induced window, reducing any hard lag has the same first-order stabilizing leverage at onset; in the filter-induced regime, hard-lag reduction alone cannot restore stability. Balanced scalar performance observables generically show a mean shift and a second harmonic at twice the compositional frequency, and under strict antagonism the two performance signals are locked in antiphase with fixed amplitude ratio. For a baseline branch, a finite-dimensional Hopf normal-form calculation gives a negative cubic coefficient, and direct simulations reproduce the predicted threshold, amplitude scaling, and observable signatures. Motivating applications include cybersecurity and rapid technological countermeasure adaptation.