生成时间: 2026年06月19日 08:20:12

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: 18

了解领域论文（摘要翻译）

1. 一种用于Wigner-Poisson系统的守恒自适应秩方法

A conservative adaptive rank method for the Wigner-Poisson system

作者: Andrew Christlieb, Sining Gong, F. Alejandro Padilla-Gomez, Jing-Mei Qiu

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20234v1

链接: http://arxiv.org/abs/2606.20234v1

摘要翻译

中文标题: 一种用于Wigner-Poisson系统的守恒自适应秩方法

摘要翻译: 我们针对1D1V Wigner-Poisson系统提出一种守恒自适应秩方法。该方法针对确定性量子动理学模拟中的一个核心挑战：在降低相空间演化代价的同时保持物理保真度所需的宏观不变量。该格式结合了基于采样的自适应秩Wigner-Poisson更新[7]与一种守恒宏观校正。守恒密度-动量求解提供局部宏观更新，Fermi-Dirac型重构将其转移到动理学解，而全局二次矩校正在动理学层面施加离散总能量约束。与经典动理学中常用的Maxwell-Boltzmann型校正不同，该重构使用受模型量子统计结构启发的Fermi-Dirac型形式。校正状态被纳入ACA SVD表示，允许数值秩适应由非局部Wigner算子和自洽Poisson场所产生的相空间复杂性。针对双流不稳定性、强Landau阻尼和隆起尾不稳定性进行的数值实验表明，该方法在几个量子参数H值下捕捉到了基准Wigner-Poisson动力学，保持了有界自适应秩，并且以接近机器精度的守恒误差保持了指定的全局离散不变量。我们还将此构造（使用局部密度-动量校正加全局总能量校正）与一种相关的质量、动量和能量全局守恒构造[8]进行了比较。对于此处所考虑的周期基准测试，两种方法产生了几乎相同的相空间和诊断结果，表明在校正的1D1V周期设定下，两种校正策略都与Wigner-Poisson动力学的自适应秩压缩相容。

2. 面向 Poisson-Nernst-Planck 和 Poisson-Nernst-Planck-Navier-Stokes 系统的结构导向随机神经网络

Structure-Oriented Randomized Neural Networks for Poisson-Nernst-Planck and Poisson-Nernst-Planck-Navier-Stokes Systems

作者: Yunlong Li, Fei Wang

类别: math.NA, cs.LG, physics.comp-ph

发布日期: 2026-06-18

arXiv ID: 2606.19912v1

链接: http://arxiv.org/abs/2606.19912v1

摘要翻译

中文标题: 面向 Poisson-Nernst-Planck 和 Poisson-Nernst-Planck-Navier-Stokes 系统的结构导向随机神经网络

摘要翻译: 我们针对 Poisson-Nernst-Planck (PNP) 系统与 Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) 系统，开发了一个称为 SO-RaNN 的结构导向随机神经网络框架。解耦的线性化子问题由随机神经网络在时空框架下迭代求解。对于浓度变量，使用逐点截断以在数值层面强制正性，并在选定的校正时刻计算离散质量缩放因子并在时间上插值，从而确保这些时刻的精确质量匹配，并促进其间质量的近似守恒。为引入辅助的离散耗散机制，我们进一步采用 SAV 型后处理校正，该校正使得在理想 SAV 更新下，SAV 辅助变量具有单调性。对于 PNP-NS 系统，对速度场使用保结构随机神经网络 (SP-RaNN)，使得速度逼近在构造上逐点满足不可压缩性约束。在理论方面，我们针对线性化子问题的原始、未校正 RaNN 求解器推导了基于残差的估计，构建了 PNP 系统原始外 Picard 迭代的条件局部时间收敛结果，并分析了数值层面的正性校正以及质量校正和 SAV 后处理步骤。对于 PNP-NS 系统，我们建立了 SP-RaNN 空间的逼近结果，并给出了对应线性化 Oseen 型问题的条件误差陈述。数值实验在源项驱动的人工测试中展示了逼近精度，并说明了预期的数值层面正性校正、选定时刻质量匹配、基于最终规范固定势计算的自由能曲线，以及在基准测试中的无散度逼近。

3. 具有分数阶耗散的电子磁流体力学方程的梯度恢复方法

A Gradient Recovery Method for Electron Magnetohydrodynamics with Fractional Dissipation

作者: Hailong Guo, Ruimeng Hu, Qirui Peng, Xu Yang

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.19716v1

链接: http://arxiv.org/abs/2606.19716v1

摘要翻译

中文标题: 具有分数阶耗散的电子磁流体力学方程的梯度恢复方法

摘要翻译: 我们针对周期环面上的$2\tfrac{1}{2}$维（2.5D）具有分数阶耗散的电子磁流体力学系统，提出并分析一种保结构数值方法。该方法直接处理磁场分量，并将这种分量形式与[T. Chu, H. Guo, and Z. Zhang, SIAM J. Numer. Anal., 63 (2025), pp. 23–53]中的梯度恢复算子相结合。我们对半隐式保结构格式建立了离散能量稳定性，并利用显式霍尔积分因子实现，以便在周期网格上进行高效计算。分数阶耗散在Fourier空间中精确处理，面内散度约束通过谱Hodge投影施加。数值实验在多个基准测试中展示了二阶空间收敛性和稳定的霍尔驱动动力学。

4. 基于规范变换的周期dNLS显式傅里叶积分器：离散Bourgain空间中的低正则性估计

Explicit Fourier Integrator for the Periodic dNLS via Gauge Transformation: Low-Regularity Estimates in Discrete Bourgain Spaces

作者: Lun Ji, Hang Li, Alexander Ostermann, Gangfan Zhong

类别: math.NA

发布日期: 2026-06-17

arXiv ID: 2606.19648v1

链接: http://arxiv.org/abs/2606.19648v1

摘要翻译

中文标题: 基于规范变换的周期dNLS显式傅里叶积分器：离散Bourgain空间中的低正则性估计

摘要翻译: 导数非线性薛定谔方程是描述等离子体物理和非线性光学等领域中非线性色散波传播的基本模型。本文在环面上考虑该模型，并研究相应周期问题的滤波显式傅里叶积分器。在应用周期规范变换后，我们考虑一个频率截断模型及其滤波指数-欧拉离散。主要困难在于周期条件下的导数立方非线性，因为局部光滑化不再可用，且共振相互作用比非周期情形更强。为解决这一问题，我们建立了一个适用于规范变换后方程的离散Bourgain空间框架。对于初值 $u_0 \in H^s(\mathbb{T})$，其中 $1/2 < s \le 5/2$，我们证明了在 $H^{1/2}(\mathbb{T})$ 中数值误差的阶为 $\mathcal{O}(τ^{s/2-1/4})$，其中 $τ$ 表示时间步长。数值实验验证了预测的收敛性，并展示了滤波格式对粗糙解的有效性。

不相关论文（基本信息）

1. 用于泊松问题亿亿次级求解的粗网格求解器

CoarseSolvers for Exascale Solution of Poisson Problems

作者: Thilina Ratnayaka, Paul Fischer, Luke Olson

类别: math.NA, cs.DC, cs.MS

发布日期: 2026-06-18

arXiv ID: 2606.20496v1

链接: http://arxiv.org/abs/2606.20496v1

摘要: WepresentatwolevelSchwarzmethodasanalternativetoAlgebraicMultigridmethod(AMG) used as the last level (coarse) solver of the p-multigrid pMG preconditioner for pressure Poission equation resulting from Spectral/Finite element descretization of incompressible Navier-Stokes eqaution. Proposed Schwarz method consits of a local problem in the original pMG coarse space and a global coarse problem. Main contribution of the paper is a novel, structured and a non-nested coarse space for the global coarse problem. Structured nature of the proposed global coarse space enable communication-free interpolation between the original p-multgrid coarse space and the global coarse problem. We demonstrate the effectiveness of the proposed method compared to the state of the art AMG solver BoomerAMG by a series of experiments performed using Nek5000/RS, a suite of highly scalable incompressible Navier-Stokes solvers, on Summit/Frontier supercomputers at Oak Ridge Leadership Computing Facility.

2. 代理式符号搜索：表征超越手工表达式、网格和神经网络的偏微分方程

Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

作者: Zongmin Yu, Liu Yang

类别: cs.LG, math.NA, physics.comp-ph

发布日期: 2026-06-18

arXiv ID: 2606.20467v1

链接: http://arxiv.org/abs/2606.20467v1

摘要: Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

3. 面向物理信息神经网络的进化两阶段超参数优化策略

Evolutionary Two-Stage Hyperparameter Optimization Strategies for Physics-Informed Neural Networks

作者: Fedor Buzaev, Dmitry Efremenko, Egor Bugaev, Andrei Ermakov, Denis Derkach, Daria Pugacheva, Fedor Ratnikov

类别: cs.LG, cs.NE, math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20442v1

链接: http://arxiv.org/abs/2606.20442v1

摘要: Physics-Informed Neural Networks (PINNs) solve Partial Differential Equations (PDEs) by embedding physical laws into neural network training. However, their performance suffers from unstable convergence, training plateaus, and strong sensitivity to architectural and optimization hyperparameters due to the highly non-convex and multi-term structure of the physics-informed loss. In this setting, the outer-loop hyperparameter search is a noisy and black-box optimization problem over heterogeneous parameters, where classical local or gradient-based strategies are easily trapped in suboptimal regions. Evolutionary algorithms, with their population-based exploration and ability to handle mixed, non-differentiable search spaces, provide a more robust mechanism for discovering promising configurations. We propose and investigate a two-stage approach based on evolutionary algorithms that combines exploration and exploitation parts of PINNs training to improve solution accuracy and robustness under fixed computational budgets. In the first stage, we perform low-fidelity training runs with truncated epochs to rapidly screen candidate configurations, treating hyperparameter selection as a black-box outer-loop problem. In the second stage, only the most promising candidates are fully trained with standard gradient-based optimizers to refine the solution. Evaluated on three popular problems, namely Advection, Klein-Gordon and Helmholtz equations, our method consistently outperforms standard training and achieves significantly lower mean error within constrained computational resources.

4. 电子薛定谔方程的稀疏组态相互作用方法再探：完备基组极限复杂度与量子编码影响

Sparse Configuration Interaction for the Electronic Schrödinger Equation Revisited: Complete Basis Set Limit Complexity and Quantum-Encoding Impact

作者: Michael Griebel, Jan Hamaekers

类别: quant-ph, math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20385v1

链接: http://arxiv.org/abs/2606.20385v1

摘要: In this article we revisit regularity results for eigenfunctions in the discrete spectrum of the electronic Schrödinger equation and study their consequences for approximation complexity. In particular, for the convergence to the complete basis set limit, it can be shown that the curse of dimensionality in the leading algebraic exponent can be mitigated. That is, for general sparse grid constructions, the main term of the convergence rate with respect to the number of degrees of freedom is independent of the number of electrons. These insights indicate potential benefits for classical numerical solvers of the electronic Schrödinger equation and also for quantum-computing approaches through new qubit-efficient wavefunction encodings.

5. 基于多边形单元光滑有限元框架的非线性岩土分析

Nonlinear Geotechnical Analysis Using a Polygonal Cell-Based Smoothed Finite Element Framework

作者: Mingjiao Yan, Yang Yang, Zongliang Zhang, Yinpeng Yin, Miao Zhang, Yijia Dong, Dong Pan, Xiaozi Lin, Tiankai Yang

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20384v1

链接: http://arxiv.org/abs/2606.20384v1

摘要: Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain–displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr–Coulomb model and the Duncan–Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

6. 通过形状参数的自适应修改的数据依赖谢泼德逼近

Data dependent Shepard approximation through and adaptive modification of the shape parameter

作者: José Kuruc, Juan Ruiz-Álvarez, Bo Wang, Dionisio-Félix Yáñez

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20332v1

链接: http://arxiv.org/abs/2606.20332v1

摘要: In this article, we introduce a novel data-dependent Shepard interpolation method inspired by the adaptive strategies proposed in [2]. In this case, as Shepard interpolation does not produce oscillations, our approach has the core objective of reducing the smearing near jump discontinuities in the data in one and two dimensions. While the original work in [2] focuses in on Radial Basis Function (RBF) interpolation, we extend these ideas to the Shepard framework by incorporating a data-dependent adaptation mechanism. Specifically, we modify the classical Shepard interpolation by adaptively adjusting the influence weights based on local smoothness indicators that modify the shape parameter. These indicators, similar to those used in [2], are designed to detect discontinuities: for grid-based data, we use squared undivided second-order differences, and for scattered data, we employ squared least-squares approximations of the Laplacian scaled by the square of the mean local separation of stencil points. The resulting data-dependent weighting scheme forces the kernels close to a discontinuity to behave like a local delta function, effectively reducing the smearing of the discontinuities introduced by the classical Shepard approach. We establish the theoretical foundation of the method, including the properties of the new interpolation and we theoretically prove that the reduction of the smearing of discontinuities is possible. Numerical experiments in one and two dimensions confirm that the proposed data-dependent Shepard interpolation significantly reduces the smearing of jump discontinuities while maintaining high accuracy in smooth regions.

7. 随机化草图对GPU低精度舍入具有鲁棒性

Randomized Sketching is Robust to Low-Precision Rounding on GPUs

作者: Aryaman Jeendgar, Clément Flint, Hartwig Anzt

类别: cs.PF, math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20195v1

链接: http://arxiv.org/abs/2606.20195v1

摘要: Randomized sketching is a core primitive in randomized numerical linear algebra. On modern hardware architectures, in particular on GPUs, the performance of sparse sketches is limited by memory traffic and atomic accumulation rather than floating-point throughput. This makes sketching a natural target for mixed precision, provided that low-precision accumulation does not degrade the embedding quality. We study mixed-precision GPU implementations of sparse oblivious subspace embeddings, focusing on a SparseStack generalization of the GPU CountSketch kernel of Higgins et al. SparseStack improves embedding quality relative to CountSketch on coherent inputs, but its additional nonzeros per column increase atomic-update contention and reduce throughput. We therefore implement FP16 SparseStack variants using deterministic round-to-nearest, exact stochastic rounding, and dithered rounding, and compare them with FP32 SparseStack, CountSketch, mixed-precision CountSketch, and FlashSketch. Our main empirical finding is that, for the tested regimes, SparseStack embedding quality is insensitive to the FP16 rounding rule. Deterministic, stochastic, and dithered rounding FP16 SparseStack produce nearly identical subspace distortion and sketch-and-solve least-squares accuracy across incoherent, coherent, and adversarial test problems. The dominant accuracy factor is the sketch distribution rather than the quantization rule: SparseStack variants substantially improve distortion on coherent inputs, while all methods behave similarly on incoherent inputs. Since deterministic rounding has the lowest overhead, it provides the best performance–accuracy tradeoff among the FP16 SparseStack variants.

8. 基于$C_0$半群的拟抛物方程的后验误差界

A posteriori error bounds for pseudo-parabolic equations using $C_0$ semigroups

作者: Martin Ossadnik, Torsten Linß

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20073v1

链接: http://arxiv.org/abs/2606.20073v1

摘要: A class of pseudo-parabolic partial differential equations is considered. We derive a posteriori error bounds for approximations obtained by FEMs in space and a BDF formula in time. The analysis is based on the $C_0$ semigroup theory and an adaptation of the concept of elliptic reconstruction to pseudo-parabolic problems. The analysis is complemented with numerical experiments.

9. 平面三维 ILSA 水力压裂模型的加速方法

Acceleration methods for the planar 3D ILSA hydraulic fracturing model

作者: V. I. Shukalo, A. V. Valov, A. N. Baykin

类别: physics.geo-ph, math.NA

发布日期: 2026-06-18

arXiv ID: 2606.20025v1

链接: http://arxiv.org/abs/2606.20025v1

摘要: Planar 3D models of hydraulic fracturing provide a practical balance between models with restrictive geometric assumptions and fully 3D simulators, capturing fractures with arbitrary planar footprints at moderate computational cost. Nevertheless, applications such as treatment design optimization and mini-frac test interpretation require large ensembles of simulations, for which the cost of planar 3D models remains a significant bottleneck. This work presents acceleration strategies for the planar 3D Implicit Level Set Algorithm (ILSA) to reduce simulation runtime while preserving numerical accuracy. A unified planar 3D ILSA scheme that consolidates the nested loops of the elastohydrodynamic solver and the front tracking algorithm into a single iterative process is introduced. A matrix splitting approach is applied to the linearized elastohydrodynamic system, moving the dense part of the elasticity operator to the right-hand side and yielding a sparse system matrix that can be solved more efficiently. Anderson acceleration is incorporated into the solution of the elastohydrodynamic system to improve convergence under varying fracture geometry. Additionally, a predictor–corrector scheme is examined with the proposed methods to assess their combined effect. Each technique is evaluated individually and in combination on both the reference and unified planar 3D ILSA schemes across five benchmark cases. Numerical experiments demonstrate that the unified scheme alone delivers an average 2.5x speedup, reaching 5.7x for the sandglass geometry. The combined application of all techniques achieves an average 4x speedup and up to 11x for the sandglass case, with the relative discrepancy in fracture aperture below 5% compared with the reference scheme.

10. 基于快速直接求解器的神经网络求解偏微分方程

A fast direct solver based neural network for solving PDEs

作者: Jashwanth Reddy Kadaru, Vaishnavi Gujjula

类别: math.NA, cs.LG

发布日期: 2026-06-18

arXiv ID: 2606.19895v1

链接: http://arxiv.org/abs/2606.19895v1

摘要: The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions. HODLR (Hierarchical Off-Diagonal Low-Rank) matrices are a subclass of hierarchical matrices in which all off-diagonal submatrices at every level of a recursive binary partition are low-rank. In this article, we present a neural network that learns the inverse operation of HODLR matrices based on the fast direct solver for HODLR matrices developed by Ambikasaran and Darve (2013). We further extend the architecture to learn nonlinear solution operators associated with PDEs by replacing some of the linear layers with deep sub-networks. We demonstrate the performance of the proposed architecture by performing a comprehensive set of experiments that include (i) solving a linear problem such as the Fredholm integral equation of the second kind, (ii) solving PDEs such as the nonlinear Schrödinger equation, Burgers’ equation, and the steady-state Darcy’s flow equation, (iii) generalization study across varying parameter values, (iv) comparing the inference time of the proposed network with the run time of a classical numerical solver, and (v) comparing the proposed network with some of the existing neural operator learning networks.

11. 具有超线性增长系数的随机微分方程的随机θ方法的不变测度

Invariant measures of the stochastic theta method for stochastic differential equations with super-linearly growing coefficients

作者: Xiaotong Li, Wei Liu, Wenjie Xiao

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.19886v1

链接: http://arxiv.org/abs/2606.19886v1

摘要: The stochastic theta method is proposed to approximate invariant measures of stochastic differential equations (SDEs), both of whose drift and diffusion coefficients may grow super-linearly. For the numerical solution generated by the stochastic theta method, we show the existence and uniqueness of the numerical invariant measure first. Then, we prove that the numerical invariant measure is convergent to the exact invariant measure of the underlying SDE. We also provide some numerical simulations to illustrate our theoretical results. This work could be regarded as an extension of the results in [Y. Jiang et al, Numer. Algorithms 83(4)(2020), pp. 1531-1553] to the case of super-linearly growing diffusion coefficient. As the backward Euler-Maruyama (EM) method is a special case of the stochastic theta method, the results derived in this work could also be regarded as a generalization of the results for the backward EM method in [W. Liu et al. Appl. Numer. Math. 184(2023), pp. 137-150] to the stochastic theta method.

12. 流体-多孔弹性结构相互作用问题的二阶显式分裂格式

Second order explicit splitting scheme for fluid-poroelastic structure interaction problems

作者: Yifan Wang, Jeonghun Lee, Suncica Canic

类别: math.NA, math-ph

发布日期: 2026-06-18

arXiv ID: 2606.19811v1

链接: http://arxiv.org/abs/2606.19811v1

摘要: Efficient and provably accurate partitioned methods for fluid-poroelastic structure interaction remain challenging because explicit treatment of the Stokes-Biot interface coupling condition can compromise stability. In this work, we develop and analyze a fully discrete, second-order, explicit splitting scheme for the time-dependent Stokes-Biot problem on fixed domains. The method combines BDF2 time stepping with second-order Adams-Bashforth extrapolation of interface data through a Robin reformulation, yielding a partitioned algorithm in which the Stokes and Biot subproblems are solved independently and in parallel at each time step. The main analytical contribution is a rigorous stability and error analysis for this second-order explicit coupling strategy. Using BDF2 energy identities, a sharp decomposition of the extrapolated interface terms, and discrete trace estimates, we prove a closed stability bound under a parabolic CFL condition. We then derive an a priori error estimate through a projection-based framework using a Fortin projection for the fluid variables and Ritz-type projections for the poroelastic variables. The analysis identifies consistency defects from BDF2 time discretization, Adams-Bashforth interface extrapolation, and the projected kinematic relation. It shows that the total errors in fluid velocity, structure velocity, pore pressure, and elastic displacement are bounded by C times the sum of the kth power of the mesh size and the square of the time step, for k from 1 to 3, in bulk energy norms. Numerical experiments with manufactured solutions confirm second-order temporal convergence and optimal-order spatial convergence. We also include a moving-domain example with Navier-Stokes fluid flow, demonstrating applicability beyond the fixed-domain Stokes-Biot setting analyzed.

13. 可压缩大气欧拉方程的良好平衡二阶近似

Well-balanced second-order approximation of the compressible atmospheric Euler equations

作者: Crystal Farris, Matthias Maier, Eric J. Tovar

类别: math.NA

发布日期: 2026-06-18

arXiv ID: 2606.19764v1

链接: http://arxiv.org/abs/2606.19764v1

摘要: We introduce a second-order approximation to the compressible atmospheric Euler equations with gravity that is invariant domain preserving and well-balanced with respect to rest states. The approximation is built upon discrete auxiliary states derived from a hydrostatic reconstruction of the density. These auxiliary states, together with an affine shift of the numerical state, provide local bounds needed for maintaining well-balancing and invariant domain preserving properties of the method. The numerical method is then verified and validated with analytic solutions, well-balancing tests, and typical benchmark problems for atmospheric flows.

14. 基于物理信息宽度学习系统学习偏微分方程的通用逼近

Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

作者: Zhiwen Yu, Derong Yang, Liujian Zhang, Kaixiang Yang, Peilin Zhan, Jianmin Lv, Jane You, C. L. Philip Chen

类别: cs.LG, math.NA

发布日期: 2026-06-18

arXiv ID: 2606.19754v1

链接: http://arxiv.org/abs/2606.19754v1

摘要: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems. While traditional numerical solvers are robust, they often incur prohibitive computational costs due to mesh dependencies, whereas recent Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative but frequently suffer from slow convergence and optimization instability. To bridge this gap, this article proposes the Physics-Informed Broad Learning System (PIBLS), a novel backpropagation-free framework that reformulates PDE solving as a direct least-squares optimization. We improved an algorithm within this framework to handle nonlinear PDEs efficiently and provide a rigorous mathematical proof establishing the universal approximation property of PIBLS for these equations. Experiments on linear and nonlinear PDEs demonstrate that PIBLS is one to three orders of magnitude faster than conventional PINNs while achieving significantly higher solution accuracy. This framework provides a computationally efficient paradigm for scientific machine learning, offering a practical, high-speed alternative for real-time simulation and design optimization tasks.

15. 关于一类修正的 Cayley–Magnus 方法

On a class of modified Cayley–Magnus methods

作者: Sergio Blanes, Fernando Casas, Arieh Iserles

类别: math.NA

发布日期: 2026-06-17

arXiv ID: 2606.19614v1

链接: http://arxiv.org/abs/2606.19614v1

摘要: We introduce a new class of numerical integrators for the time integration of non-autonomous linear ordinary differential equations whose coefficient matrix is sparse and evolves within a quadratic matrix Lie group. In contrast to standard Lie group integrators, the proposed methods avoid the evaluation of matrix exponentials acting on vectors and instead rely on solving a sequence of linear systems with sparse coefficient matrices. Moreover, they are well suited for problems arising from unbounded operators, as they inherently produce bounded solutions. We construct optimised schemes of orders four and six and assess their performance on a representative numerical example, demonstrating clear advantages over existing Lie-group integrators.

16. 正则化平稳平均场博弈的Bregman投影镜像方法

Bregman-projected mirror methods for regularized stationary mean-field games

作者: Hussain Al Abdulaziz, Yuri Ashrafyan, Yeva Gevorgyan, Diogo Gomes

类别: math.NA, math.AP

发布日期: 2026-06-17

arXiv ID: 2606.19611v1

链接: http://arxiv.org/abs/2606.19611v1

摘要: We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form (H(x,p,m)=H_0(x,p)-g(m)), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order (\barγ)-Laplacian regularization of the stationary MFG system as a variational inequality on (L^{\barβ}(\mathbb T^d)\times W^{1,\barγ}(\mathbb T^d)). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue–Sobolev exponents of the problem and analyze a constrained two-step mirror method with frozen operator evaluation. For the exact constrained iteration and each fixed regularization parameter (\epsi>0), we derive a one-step Bregman inequality and use it to prove that the constrained iteration converges strongly to the unique solution of the regularized variational inequality under natural summability conditions on the step sizes. Numerical experiments on one- and two-dimensional models, validated against exact test solutions, illustrate residual decay under mesh refinement and suggest improved practical performance of the two-step implementation in the tested discretizations.

17. 纳维-斯托克斯方程的高精度模块化数据同化

Higher Accuracy Modular Data Assimilation for the Navier-Stokes Equations

作者: Troy Yang

类别: math.NA

发布日期: 2026-06-17

arXiv ID: 2606.19508v1

链接: http://arxiv.org/abs/2606.19508v1

摘要: This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation \begin{align} \text{Forecast step: } \quad &\frac{3\widetilde{v}^{n+2}-4v^{n+1}+v^n}{2Δt}+\widetilde{v}^{n+2} \cdot \nabla \widetilde{v}^{n+2} - νΔ\widetilde{v}^{n+2} + \nabla q^{n+2}=f(x) \notag \ &\nabla \cdot \widetilde{v}^{n+2} = 0 \notag \ \text{Analysis step: } \quad &\frac{3v^{n+2}-3\widetilde{v}^{n+2}}{2Δt}-χI_H(u(t^{n+2})-v^{n+2})=0. \notag \end{align} If $I_H=I_H^2$, the analysis step can be made explicit, taking the form \begin{align} v^{n+2}=\widetilde{v}^{n+2}+\frac{2Δtχ}{3+2Δtχ}I_H(u^{n+2}-\widetilde{v}^{n+2}). \notag \end{align} This implies the analysis step has the stability property of an implicit step and lower complexity than an explicit analysis step. Stability and error estimates for the BDF2 scheme are presented along with their proofs. Numerical experiments are conducted to assess the performance of BDF2 modular assimilation algorithm. The results of the experiments support the conclusion that modular data assimilation has comparable accuracy to standard, fully coupled data assimilation while greatly reducing computational complexity and cost.

18. 基于Moreau-Yosida的周期系统Kohn-Sham反演

Moreau-Yosida-based Kohn-Sham Inversion for Periodic Systems

作者: Vebjørn H. Bakkestuen, Michael F. Herbst, Vegard Falmår, Markus Penz, Andre Laestadius

类别: math.NA, cond-mat.mtrl-sci, math.FA, physics.chem-ph

发布日期: 2026-06-17

arXiv ID: 2606.19471v1

链接: http://arxiv.org/abs/2606.19471v1

摘要: Density-potential inversion for periodic systems within Moreau-Yosida-regularised density-functional theory is investigated, both theoretically and numerically. We develop the framework in a periodic homogeneous Sobolev space and use it to recover the exchange-correlation potential of Kohn-Sham theory through a limiting procedure. A key analytical ingredient is the proof of lower semicontinuity of the non-interacting kinetic-energy functional in the chosen topology. The proximal mapping, together with its algorithmic evaluation, plays a central role in the resulting inversion scheme. Numerical experiments illustrate the performance and properties of the method for both the Kohn-Sham and Gross-Pitaevskii equations.