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1. 参数化变分不等式的模型降阶：在人群运动中的应用
Model order reduction for parametrized variational inequalities: application to crowd motion
作者: Giulia Sambataro, Virginie Ehrlacher
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.04037v1
链接: http://arxiv.org/abs/2605.04037v1
摘要: This work investigates model order reduction for time-dependent parametrized variational inequalities, with a focus on discrete contact problems. As a prototypical example, we consider an agent-based crowd model [Maury et al., 2011] in which agent velocities are obtained at each time step from a constrained least-squares problem. Geometric parameter variations induce significant variability in both agent positions and contact forces, leading to a slowly decaying Kolmogorov $n$-width of the solution manifold. We propose a nonlinear approach that combines a linear reduced-order model with a deep-learning-based correction. The method utilizes a greedy index selection (gIS) algorithm for compressing Lagrange multipliers and Proper Orthogonal Decomposition (POD) applied to velocity snapshots. Additionally, we explore hyper-reduction techniques, comparing the Empirical Interpolation Method (EIM) and the Empirical Quadrature (EQ) procedure from both computational complexity and accuracy perspectives. Finally, we demonstrate the applicability of the methodology in a complex scenario involving many agents in a highly congested geometric configuration. This work represents the first attempt to apply model order reduction to a discrete contact problem of the type introduced in [Maury et al., 2011] and paves the way for future advancements in nonlinear MOR specifically for this class of problems.
2. 用于随机微分方程的递归多项式混沌演化方法
A Recursive Polynomial Chaos Evolution Method for Stochastic Differential Equations
作者: Guillaume Bal, Shengbo Ma, Su Zhang, Zhiwen Zhang
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03853v1
链接: http://arxiv.org/abs/2605.03853v1
摘要: Numerical simulation of stochastic differential equations over long time intervals poses significant computational challenges. In this paper, we propose a novel recursive polynomial chaos evolution method that achieves model reduction without sampling by exploiting the Markov property to maintain a fixed low-dimensional representation throughout the time evolution. At each time step, we construct orthogonal polynomial bases adapted to the current probability measure, and project the one-step-ahead solution onto this new basis together with the new Brownian increments. This dynamic updating strategy effectively reduces the dimension of the random variables during long-time evolution. Under appropriate assumptions, we prove the convergence of the method, specifically that the distributions generated by the method preserve convergence in the Wasserstein-1 distance. We present numerical results demonstrating that the method can accurately capture complex dynamical behaviors with high accuracy and low computational cost.
3. 基于几何守恒律的高阶直线拉格朗日方法
A high-order rectilinear Lagrangian method based on the geometric conservation law
作者: Xun Wang, Chengdi Ma
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03739v1
链接: http://arxiv.org/abs/2605.03739v1
摘要: This paper presents a mesh moving strategy for high-order Lagrangian method on quadrilateral meshes. The primary evidence of this method stems from principle of area conservative linearization and the asymptotic properties of the velocity. The former strictly adheres to the requirements of geometric conservation laws, while the latter provides a high-order accuracy guarantee. Two smooth vortex test cases verify the feasibility of the proposed scheme.
4. 基于凸半无限规划与切割平面法的一维非负样条光滑
One-Dimensional Nonnegative Spline Smoothing via Convex Semi-Infinite Programming with a Cutting-Plane Method
作者: Hiroki Arai, Daichi Kitahara
类别: math.OC, eess.SP, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03711v1
链接: http://arxiv.org/abs/2605.03711v1
摘要: Spline functions are smooth piecewise polynomials widely used for interpolation and smoothing, and nonnegative spline smoothing is also studied for nonnegative data. Previous research used sufficient conditions for the nonnegativity of spline functions because necessary and sufficient conditions for the nonnegativity are infinitely many linear inequalities, which are difficult to handle in optimization algorithms. This conventional method quickly computes a nonnegative spline function via quadratic programming (QP), but the optimal solution may be slightly degraded by using the sufficient condition. In this paper, we express 1D nonnegative spline smoothing as a convex semi-infinite programming (CSIP) problem that directly deals with infinite inequality constraints. As optimization algorithms for general SIP problems, local-reduction-based sequential quadratic programming (LRSQP) methods are used, but their convergence performance deteriorates for certain problems due to multiple approximations during updates. To quickly solve the CSIP problem, we propose a cutting-plane (CP) method. In the proposed method, after giving an initial solution by the standard spline smoothing, we find the minimizer of each polynomial piece by using the closed-form solution for a low-degree polynomial or a numerical solution for a high-degree polynomial. If the minimum value is negative, then such minimizer is added into the constraint of the problem to guarantee the nonnegativity. This constrained problem is quickly solved via QP, and we find the minimizer of each polynomial piece again. We repeat these procedures until there are no negative minimum values. The proposed method guarantees convergence to the original CSIP solution, and its effectiveness is demonstrated in numerical experiments by comparison to the conventional methods, QP under the sufficient condition and CSIP using the MATLAB LRSQP algorithm.
5. 基于特征伽辽金方法的半拉格朗日间断有限元格式中时间依赖检验函数的误差分析
Error Analysis of Time-Dependent Test Functions in the Semi-Lagrangian Discontinuous Finite Element Scheme Based on the Characteristic Galerkin Method
作者: Zhengrong Xie
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03695v1
链接: http://arxiv.org/abs/2605.03695v1
摘要: This paper studies the time-dependent test-function error in the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method caused by numerical integration errors of the characteristic ODE solver, and its effect on convergence. Unlike classical finite element methods and standard DG methods, the test functions in CSLDG are constructed by characteristic backtracking. As a result, ODE errors affect not only the upstream integration region but also enter directly into the discrete weak formulation through perturbations of the test function support. Neglecting mesh geometric errors, we introduce an ideal auxiliary solution and an auxiliary solution retaining the test-function error, and derive the corresponding error equation and recursive relations. First, based on a global time-dependent test-function error analysis, we show that the optimal $L^2$ convergence order of the $P^K$-CSLDG method is preserved provided that the ODE solver order satisfies $D\ge 2K+3+d$. We then propose a new analysis based on the propagation of local modal coefficient errors, which controls the error cell by cell through the actual modal update formulas and recovers the global $L^2$ estimate. This leads to an improved sufficient condition $D\ge K+1+\frac{d}{2}$. The results show that the required ODE accuracy depends strongly on the analytical route, and that the modal-coefficient-based analysis is both closer to the actual implementation and more consistent with numerical observations.
6. 基于代数谱曲线的自由解压
Free Decompression with Algebraic Spectral Curves
作者: Siavash Ameli, Chris van der Heide, Liam Hodgkinson, Michael W. Mahoney
类别: stat.ML, cs.LG, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03634v1
链接: http://arxiv.org/abs/2605.03634v1
摘要: Tools from random matrix theory have become central to deep learning theory, using spectral information to provide mechanisms for modeling generalization, robustness, scaling, and failure modes. While often capable of modeling empirical behavior, practical computations are limited by matrix size, often imposing a restriction to models that are too small to be realistic. This motivates the inference of properties of larger models from the behavior of smaller ones. Free decompression (FD) is a recently proposed method for extrapolating spectral information across matrix sizes, but its utility is currently limited by strong assumptions that preclude its implementation on more realistic machine learning (ML) models. We use algebraic spectral curve theory to provide a general FD methodology for spectral densities whose Stieltjes transform satisfies an algebraic relation, a modeling assumption that is more likely to hold in practice. This recasts FD as an evolution along spectral curves which can be readily integrated. Our framework enables the expansion of spectral densities that have multiple or multi-modal bulks, that exist at multiple scales, and that contain atoms, all characteristic of real-world data and popular ML models. We demonstrate the efficacy of our framework on models of interest in modern ML, including Hessian and activation matrices associated with neural networks and large-scale diffusion models.
7. 真实与热完全气体可压缩流动方程的压力平衡保持与完全守恒离散格式
Pressure-equilibrium-preserving and fully conservative discretization of compressible flow equations for real and thermally perfect gases
作者: Gennaro Coppola, Alessandro Aiello, Carlo De Michele
类别: physics.flu-dyn, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03617v1
链接: http://arxiv.org/abs/2605.03617v1
摘要: Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure equilibrium condition for the compressible Euler equations, typically at the cost of spoiling the correct conservation of total energy or by overspecifying the thermodynamical variables. This study proposes for the first time a numerical discretization procedure which is able to discretely preserve the full conservation of the linear invariants (mass, momentum and total energy) and to exactly enforce the pressure equilibrium condition. The method also preserves the conservation of kinetic energy by convection, and is based on the specification of nonlinear numerical fluxes for mass and internal energy which depend on the details of the equation of state. Both thermally perfect and real gases with an arbitrary equation of state are considered, and a simplified approximate pressure equilibrium preserving formulation with excellent performances is also proposed. The effectiveness of the novel formulations is assessed through a series of numerical simulations in supercritical and transcritical conditions with some of the most popular cubic equations of state.
8. 傅里叶残差网络对不连续函数实现谱精度
Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions
作者: Owen Davis, Mohammad Motamed, Olof Runborg
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03549v1
链接: http://arxiv.org/abs/2605.03549v1
摘要: We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions – including those with jump discontinuities in the function and its derivatives – and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work.
9. 随机检验函数、$H^{-1}$范数等价性与随机变分物理信息神经网络
Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
作者: Diego Marcondes
类别: math.NA, cs.LG
发布日期: 2026-05-05
arXiv ID: 2605.03542v1
链接: http://arxiv.org/abs/2605.03542v1
摘要: The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of a residual requires a supremum over an infinite-dimensional function space. We prove that the $H^{-1}$ norm of any functional is equivalent to its expected squared evaluation against a random test function whose distribution depends only on the domain. Crucially, realisations of this random test function have negative Sobolev regularity for $d \geq 2$, yet this roughness is not an obstacle: averaging over the distribution exactly recovers the correct weak topology, independently of the differential operator. This equivalence introduces the notion of stochastically weak solutions, which coincide with classical weak solutions, and motivates stochastic variational physics-informed neural networks (SV-PINNs): neural networks trained by minimising an empirical approximation of the stochastic norm of the PDE residual. Although instantiated here with neural networks as trial spaces, the underlying principle is independent of the approximation architecture and suggests a broader paradigm for numerical methods based on stochastic rather than deterministic test spaces. The framework extends naturally to higher-order elliptic, parabolic and hyperbolic equations and to abstract operator equations on Hilbert spaces. As a proof of concept, we present numerical experiments on eight challenging second-order linear elliptic problems spanning high-frequency and multi-scale solutions, indefinite operators, variable coefficients, and non-standard domains, in which SV-PINNs consistently and significantly outperform standard PINNs, recovering solutions to within one percent relative error in hundreds of L-BFGS steps.
10. 脑动脉瘤中器械诱导的血栓形成：将患者特异性血栓建模与功能性闭塞关联至虚拟血管造影评估
Device-Induced Thrombus Formation in Cerebral Aneurysms: Linking Patient-Specific Clot Modeling and Functional Occlusion to Virtual Angiographic Assessment
作者: Fabian Holzberger, Struan Hume, Markus Muhr, Malebogo Ngoepe, Barbara Wohlmuth
类别: cs.CE, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03536v1
链接: http://arxiv.org/abs/2605.03536v1
摘要: Endovascular treatment of cerebral aneurysms aims to achieve functional occlusion and isolation of the aneurysm sac from bloodflow. In clinical practice, treatment success is assessed primarily through digital subtraction angiography (DSA), which visualizes contrast-agent inflow and washout but does not directly resolve thrombus formation driving early occlusion. We present a computational framework that couples acute fibrin thrombus formation with virtual angiography, enabling early thrombus growth to be interpreted through clinically familiar DSA-like imaging. Three common treatment strategies: endovascular coiling, flow diversion, and stent-assisted coiling, are modeled under pulsatile hemodynamics and linked to simulated contrast transport. Across three representative aneurysm morphologies, the simulations demonstrate that while devices reduce inflow, residual contrast access and trapping may persist, with early thrombus formation contributing substantially to perfusion suppression and altered washout patterns. These effects are clearly reflected in the virtual angiographic imaging. The importance of vortical structures in device-induced thrombosis is highligthed in one of the cases. By seeking to align modelling and simulation tools with clinically-relevant metrics, with a particular focus on occlusion outcome, this work presents a good starting point for bridging the gap between these two paradigms.
11. 用次线性神经网络参数化凸集
Parametrizing Convex Sets Using Sublinear Neural Networks
作者: Eloi Martinet
类别: math.OC, cs.AI, cs.LG, math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03520v1
链接: http://arxiv.org/abs/2605.03520v1
摘要: We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.
12. 带高度奇异位势的非线性薛定谔方程指数波积分器的最优误差界
Optimal error bounds on the exponential wave integrator for nonlinear Schrödinger equations with highly singular potential
作者: Weizhu Bao, Chushan Wang, Yifei Wu
类别: math.NA
发布日期: 2026-05-05
arXiv ID: 2605.03355v1
链接: http://arxiv.org/abs/2605.03355v1
摘要: We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a highly singular potential in $\R^d$ with $1 \leq d \leq 3$. Our results deal with singular potentials in $L^p_\text{\rm loc}(\R^d)$ with $p > \frac{d}{2}$ and $p \geq 1$, which is (almost) the weakest regularity of the potential required by the well-posedness of the NLSE. First, for $L^p_\text{loc}$-potentials with $p>2$, we establish an optimal first-order $L^2$-norm convergence for the EWI, with the convergence order slightly reduced to $1^-$ when $p=2$. To the best of our knowledge, the optimal first-order convergence for the three-dimensional $L^2$-potential is for the first time in the literature. The optimality of such an error bound is two-fold: (i) the first-order $L^2$-norm convergence is optimal for the EWI (and its higher-order versions) under the given $L^2$-regularity assumption on the potential, and (ii) to achieve the first-order $L^2$-norm convergence for the EWI, such an assumption is optimally weak. For more singular potentials in $L^p_\text{\rm loc}(\R^d)$ with $\frac{d}{2} < p < 2$ and $p \geq 1$, we prove that the $L^2$-norm convergence is (almost) of $(1 - α)$-order when $d=1, 2$, and of $(1 - \frac{3}{2}α)$-order when $d=3$, where $α:= d(1/p - 1/2)$ when $d =1,2,3$, $p>1$ and $α:= \frac{1}{2}^+$ when $d=1$, $p=1$. Notably, this result pushes the error estimate to the threshold regularity of the potential that matches the threshold regularity for the well-posedness of the NLSE, which is also for the first time. Two main ingredients are adopted in the proof: (i) the use of discrete space-time Lebesgue spaces together with discrete Strichartz estimates to establish the stability of the numerical scheme, and (ii) the use of normal form transformation and frequency decompositions to obtain optimal error bounds.
13. 作为Boltzmann流与Landau流近似的连续提升-投影流
The consecutive lifting-projection flow as an approximation of Boltzmann and Landau flow
作者: Kun Huang
类别: math.NA, math-ph
发布日期: 2026-05-05
arXiv ID: 2605.03273v1
链接: http://arxiv.org/abs/2605.03273v1
摘要: We introduce the consecutive lifting-projection (LP) flow as a novel approximation framework for the spatially homogeneous Boltzmann and Landau equations. The key idea is to lift the nonlinear collision operator to a higher dimensional linear Kac master equation on spheres, evolve this lifted equation in time, and project the solution back to the lower dimensional velocity space. The resulting LP flow is a tangent flow to the original kinetic dynamics and admits a clear semigroup structure. We show that the consecutive LP flow preserves mass, momentum, and energy, satisfies an entropy dissipation property, and converges to the correct Maxwellian equilibrium. In addition, the lifting removes the intrinsic nonlinearity of the collision operator and enables explicit analytical representations of the solution. For Maxwell molecules, we provide an error estimate quantifying the accuracy over finite time intervals. The framework provides a concise and general methodology for constructing reliable numerical solvers in kinetic theory. It unifies existing explicit discretizations, which helps understanding numerical stability and clarifying the trade-off between conservation and positivity. More importantly, it enables the development of new schemes. In particular, we propose the Green’s function method, which is not only unconditionally stable, but also perfectly compatible with fast spectral discretizations.
14. 基于张量密度近似的正则化路径依赖McKean–Vlasov动力学的高维增强采样
High-Dimensional Enhanced Sampling via Regularized Path-Dependent McKean–Vlasov Dynamics using Tensor Density Approximation
作者: Liyao Lyu, Siyu Guo, Huan Lei
类别: math.NA, physics.comp-ph, stat.CO
发布日期: 2026-05-04
arXiv ID: 2605.03080v1
链接: http://arxiv.org/abs/2605.03080v1
摘要: Sampling from high-dimensional Gibbs measures poses a challenge when the energy landscape consists of multiple metastable states. Enhanced-sampling methods mitigate this difficulty by introducing adaptive biasing potentials to facilitate the exploration along prescribed collective variables (CVs), but their scalability is often limited by the dimension of the CV space. Motivated by the Wasserstein-gradient-flow interpretation of adaptive biasing, we propose a regularized path-dependent McKean–Vlasov formulation for high-dimensional enhanced sampling. The formulation replaces the variational regularization of the Wasserstein functional by a direct regularization of the CV marginal density in the McKean–Vlasov drift, avoiding the outer convolution over the CV domain. Furthermore, it replaces the instantaneous law by a weighted path-history measure to improve statistical stability in the small-replica regime. We establish well-posedness of the resulting regularized and path-dependent stochastic dynamics under suitable assumptions. For numerical realization, the history-averaged CV marginal density is approximated using an optimization-free functional hierarchical tensor representation, leading to a scalable density-based adaptive biasing scheme. Numerical experiments on benchmark potentials and molecular systems demonstrate the effectiveness of the proposed method for sampling problems with CV dimensions up to 64.