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arXiv 2025年

作者： Dong, Suchuan Ni, Naxian Center for Computational and Applied Mathematics Department of Mathematics Purdue University United States

We present a method leveraging extreme learning machine (ELM) type randomized neural networks (NNs) for learning the exact time integration algorithm for initial value problems. The exact time integration algorithm for non-autonomous (including autonomous) systems can be represented by an algorithmic function in higher dimensions, which satisfies an associated system of partial differential equations with corresponding boundary conditions. Our method learns the algorithmic function by solving this associated system using ELM with a physics informed approach. The trained ELM network serves as the learned algorithm and can be used to solve the initial value problem with arbitrary initial data or arbitrary step sizes from some domain. When the right hand side of the non-autonomous system exhibits a periodicity with respect to any of its arguments, while the solution itself to the problem is not periodic, we show that in this case the algorithmic function is either periodic, or when it is not, satisfies a well-defined relation for different periods. This property can greatly simplify the network training for algorithm learning in many problems. We consider explicit and implicit NN formulations, leading to explicit and implicit time integration algorithms, and discuss how to train the ELM network by the nonlinear least squares method. Extensive numerical experiments with benchmark problems, including non-stiff, stiff and chaotic systems, show that the learned NN algorithm produces highly accurate solutions in long-time simulations, with its time-marching errors decreasing nearly exponentially with increasing degrees of freedom in the neural network. We compare extensively the computational performance (time-marching accuracy vs. time-marching cost) between the current NN algorithm and the leading traditional time integration algorithms. The learned NN algorithm is computationally competitive, markedly outperforming the traditional algorithms in many problems. Copyright ©

关键词： Least squares approximations

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A charged hadronic string model within the R.P. Feynman proper time paradigm and vacuum field theory approach 14

A charged hadronic string model within the R.P. Feynman prop...

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14th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields, IARD 2024

作者： Bogolubov, Nikolai N. Prykarpatski, Anatolij K. ICTP Trieste Italy Department of Applied Mathematics The Cracov University of Technology Kraków Poland Department of Computational Mathematics and Programming The Lviv Polytechnic National University Lviv Ukraine

We study a novel charged hadronic string model within the least action principle and the vacuum field theory approach based on the classical R.P. Feynman’s proper time paradigm. It is stated that the hadronic string model allows the conformal local coordinates, with respect to which the resulting string dynamics is described by means of the linear second order elliptic equation under the Dirac type constraint, suitable for the canonical quantization and demonstrating a related gauge type invariance of the model. The related Lagrangian and Hamiltonian aspects of the string model, interacting with ambient both electrical and electromagnetic potential fields, are analyzed in detail. © 2025 Institute of Physics Publishing. All rights reserved.

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Traveling antiferromagnetic domain walls in a magnetic field

arXiv

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arXiv 2025年

作者： Theodorou, George Komineas, Stavros Department of Mathematics and Applied Mathematics University of Crete Crete Heraklion70013 Greece Institute of Applied and Computational Mathematics FORTH Crete Heraklion Greece

We consider an antiferromagnet in one space dimension with easy-axis anisotropy in a perpendicular magnetic field. We study propagating domain wall solutions that can have a velocity up to a maximum vc. The width of the domain wall is a non-monotonic function of the velocity and it diverges to infinity at vc. Both features are in contrast to the case of the Lorentz invariant model in the absence of the field. We further study the modification of the wall profile when a Dzyaloshinskii-Moriya interaction is added. Finally, we present a propagating spiral expected to form when the system is forced at a velocity higher than the maximum velocity for domain walls and we give numerical results for the effect of the Dzyaloshinskii-Moriya interaction. © 2025, CC BY.

关键词： Magnetic fields

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A flow-kick model of dryland vegetation patterns: the impact of rainfall variability on resilience

arXiv

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arXiv 2025年

作者： Gandhi, Punit Oline, Matthew Silber, Mary Department of Mathematics and Applied Mathematics Virginia Commonwealth University RichmondVA23284 United States Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

In many drylands around the globe, vegetation self-organizes into regular spatial patterns in response to aridity stress. We consider the regularly-spaced vegetation bands, on gentle hill-slopes, that survive low rainfall conditions by harvesting additional stormwater from upslope low-infiltration bare zones. We are interested in the robustness of this pattern formation survival mechanism to changes in rainfall variability. For this, we use a flow-kick modeling framework that treats storms as instantaneous kicks to the soil water. The positive feedbacks in the storm-level hydrology, that act to concentrate water within the vegetation bands, are captured through the spatial profiles of the soil water kicks. Between storms, the soil water and vegetation, modeled by a two-component reaction-diffusion system, evolve together. We use a combination of linear stability analysis and numerical simulation to compare predictions of idealized periodic rainfall, with no variability, to predictions when there is randomness in the timing and magnitude of water input from storms. We show that including these random elements leads to a decrease in the parameter range over which patterns appear. This suggests that an increase in storm variability, even with the same mean annual rainfall, may negatively impact the resilience of these pattern-forming dryland ecosystems. Copyright © 2025, The Authors. All rights reserved.

关键词： Vegetation

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Axisymmetric type II blowup solutions to the three dimensional Keller-Segel system

arXiv

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arXiv 2025年

作者： Hou, Thomas Y. Nguyen, Van Tien Song, Peicong Applied and Computational Mathematics Caltech PasadenaCA United States Department of Mathematics National Taiwan University Taiwan

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blows up in finite time. In particular, the singularity is of type II, which admits locally a leading order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate. © 2025, CC BY.

关键词： Eigenvalues and eigenfunctions

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Structure stability of steady supersonic shear flow with inflow boundary conditions

arXiv

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arXiv 2025年

作者： Jiang, Song Zhou, Chunhui Department of Mathematics Southeast University Nanjing210096 China Institute of Applied Physics and Computational Mathematics Beijing China

We study the existence and zero viscous limit of smooth solutions to steady compressible Navier-Stokes equations near plane shear flow between two moving parallel walls. Under the assumption 0 0 = (µ(x2), 0), there exist smooth solutions near U0 to steady compressible Navier-Stokes equations in a 2-dimension domain Ω = (0, L) × (0, 2). Moreover, based on the uniform-in-Ε estimates, we establish the zero viscosity limit of the solutions obtained above to the solutions of the steady Euler equations. © 2025, CC BY-NC-ND.

关键词： Navier Stokes equations

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Splitting algorithms for paraxial and Itô-Schrödinger models of wave propagation in random media

arXiv

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arXiv 2025年

作者： Bal, Guillaume Nair, Anjali Departments of Statistics and Mathematics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper introduces a full discretization procedure to solve wave beam propagation in random media modeled by a paraxial wave equation or an Itô-Schrödinger stochastic partial differential equation. This method bears similarities with the phase screen method used routinely to solve such problems. The main axis of propagation is discretized by a centered splitting scheme with step ∆z while the transverse variables are treated by a spectral method after appropriate spatial truncation. The originality of our approach is its theoretical validity even when the typical wavelength θ of the propagating signal satisfies θ ∆z. More precisely, we obtain a convergence of order ∆z in mean-square sense while the errors on statistical moments are of order (∆z)2 as expected for standard centered splitting schemes. This is a surprising result as splitting schemes typically do not converge when ∆z is not the smallest scale of the problem. The analysis is based on equations satisfied by statistical moments in the Itô-Schrödinger case and on integral (Duhamel) expansions for the paraxial model. Several numerical simulations illustrate and confirm the theoretical findings. Copyright © 2025, The Authors. All rights reserved.

关键词： Schrodinger equation

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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

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应用数学与计算数学学报 2025年第1期7卷 3-39页

作者： Hendrik Ranocha Andrew R.Winters Hugo Guillermo Castro Lisandro Dalcin Michael Schlottke-Lakemper Gregor J.Gassner Matteo Parsani Applied Mathematics University of HamburgBundesstr.5520146 HamburgGermany Department of Mathematics Applied Mathematics Linköping UniversityLinköpingSweden Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Applied and Computational Mathematics RWTH Aachen UniversityAachenGermany High-Performance Computing Center Stuttgart University of StuttgartStuttgartGermany Department of Mathematics and Computer Science University of CologneCologneGermany Center for Data and Simulation Science University of CologneCologneGermany Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Physical Science and Engineering Division(PSE) King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia

We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fluid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler *** demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)*** numerical examples show that the error-based step size control is easy to use,robust,and efficient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy *** demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages *** with *** and the C/Fortran code SSDC based on PETSc.

关键词： Explicit Runge-Kutta(RK)methods Step size control Compressible fluid dynamics Adaptivity in space and time Shock capturing

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Lawvere’s Frobenius Reciprocity, the Modular Connections of Grandis and Dilworth’s Abstract Principal Ideals

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applied Categorical Structures 2025年第3期33卷 1-8页

作者： Goswami, Amartya Janelidze, Zurab Manuell, Graham Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg South Africa National Institute for Theoretical and Computational Sciences (NITheCS) Stellenbosch South Africa Mathematics Division Department of Mathematical Sciences Stellenbosch University Stellenbosch South Africa

The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.

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GEOMETRIC HARDY INEQUALITIES ON THE HEISENBERG GROUPS VIA CONVEXITY

arXiv

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arXiv 2025年

作者： Barbatis, Gerassimos Chatzakou, Marianna Tertikas, Achilles Department of Mathematics National and Kapodistrian University of Athens Athens15784 Greece Department of Mathematics: Analysis Logic and Discrete Mathematics Ghent University Belgium Department of Mathematics & Applied Mathematics University of Crete Heraklion70013 Greece Institute of Applied and Computational Mathematics FORTH Heraklion71110 Greece

We prove Lp-Hardy inequalities with distance to the boundary for domains in the Heisenberg group Hn, n ≥ 1. Our results are based on a geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is also implemented on half-spaces and convex polytopes for the distance defined by the gauge quasi-norm on Hn related to the fundamental solution of the horizontal Laplacian. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak H-concavity of the Euclidean distance to the boundary, thus obtaining an alternative proof for the L2-Hardy inequality on convex domains. In all cases the constants are shown to be *** Codes 35R03, 26D10 (Primary) 35A23, 35H20, 35J75 (Secondary) Copyright © 2025, The Authors. All rights reserved.

关键词： Lie groups

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