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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Department of Mathematics Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solutions can be represented efficiently using artificial neural networks, even in high dimension. Conversely, we present examples in which the solution fails to lie in the function space associated to a neural network under *** Codes 68T07, 35C15, 65M80 Copyright © 2020, The Authors. All rights reserved.

关键词： Neural networks

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Visualized Low-Energy Interference Structure with Attosecond Temporal Resolution

Visualized Low-Energy Interference Structure with Attosecond...

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第六届全国计算原子与分子物理学术会议

作者： Weifeng Yang Xianhuan Yu Xiaohong Song Cheng Lin Jing Chen Wei Quan Xiaojun Liu Department of Physics College of Science Shantou University Institute of Applied Physics and Computational Mathematics

＜正＞Accessing precisely to the phase variation of electronic wavepacket（EWP）provides unprece dented s patiotemporal information of microworld.A radial interference pattern at near zero energy has been widely observed in experiments of strong field ***,the underlying physical picture of this in terference pattern is still debated and remains *** we report an experimental and theoretical ver

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A priori estimates for classification problems using neural networks

arXiv

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arXiv 2020年

作者： Weinan, E. Wojtowytsch, Stephan Program for Applied and Computational Mathematics Department of Mathematics Princeton University PrincetonNJ08544 United States Program for Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss *** Codes 68T07, 60-08 Copyright © 2020, The Authors. All rights reserved.

关键词： Classification (of information)

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Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

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Journal of Fluid Mechanics 1992年第1期238卷 1-30页

作者： Karniadakis, George Em Triantafyllou, George S. Department of Mechanical and Aerospace Engineering and Program of Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States The Levich Institute and Department of Mechanical Engineering The City College of New York New York NY 10031 United States

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast ’ transition, from a laminar two-dimensional state a t Reynolds number 200 to a turbulent state a t Reynolds number 400. The process has been documented in several eXperimental mvestigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the NavierStokes equations at representative Reynolds numbers, up to 500. A high-order timeaccurate, miXed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vorteX street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vorteX filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vorteX filaments. Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance intermittent phenomena. It is concluded that the wake undergoes transit

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Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

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Communications in computational physics 2014年第3期15卷 692-711页

作者： Xu Qian Yaming Chen Songhe Song Department of Mathematics and Systems Science and State Key Laboratory of High Performance ComputingNational University of Defense TechnologyChangsha 410073P.R.China Program in Applied and Computational Mathematics Princeton UniversityPrincetonNJ 08544USA School of Mathematical Sciences Queen Mary University of LondonLondon E14NSUK

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum *** two methods can preserve the intrinsic properties of original problems as much as *** splitting technique increases the computational ***,the error estimation and some conservative properties are *** is proved to preserve the charge conservation *** global energy and momentum conservation laws can be preserved under several *** experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

关键词： Schrödinger equation wavelet collocation method splitting technique conservative property

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Discovery of Partial Differential Equations from Highly Noisy and Sparse Data with physics-Informed Information Criterion

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Research 2024年第1期2023卷 247-259页

作者： Hao Xu Junsheng Zeng Dongxiao Zhang BIC-ESAT EREand SKLTCSCollege of EngineeringPeking UniversityBeijing 100871P.R.China Institute of Applied Physics and Computational Mathematics Beijing 100088P.R.China Eastern Institute for Advanced Study Eastern Institute of TechnologyNingbo 315200ZhejiangP.R.China National Center for Applied Mathematics Shenzhen(NCAMS) Southern University of Science and TechnologyShenzhen 518055GuangdongP.R.China Department of Mathematics and Theories Peng Cheng LaboratoryShenzhen 518000GuangdongP.R.China

Data-driven discovery of partial differential equations(PDEs)has recently made tremendous progress,and many canonical PDEs have been discovered successfully for proof of ***,determining the most proper PDE without prior references remains challenging in terms of practical *** this work,a physics-informed information criterion(PIC)is proposed to measure the parsimony and precision of the discovered PDE *** proposed PIC achieves satisfactory robustness to highly noisy and sparse data on 7 canonical PDEs from different physical scenes,which confirms its ability to handle difficult *** PIC is also employed to discover unrevealed macroscale governing equations from microscopic simulation data in an actual physical *** results show that the discovered macroscale PDE is precise and parsimonious and satisfies underlying symmetries,which facilitates understanding and simulation of the physical *** proposition of the PIC enables practical applications of PDE discovery in discovering unrevealed governing equations in broader physical scenes.

关键词： process canonical proof

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From point patterns to networks: to what extent does the Delaunay triangulation reproduce key spatial and density information?

arXiv

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arXiv 2025年

作者： Newby, Eli Shi, Wenlong Jiao, Yang Torquato, Salvatore Albert, Réka Department of Physics Pennsylvania State University University ParkPA16802 United States Department of Cardiovascular and Metabolic Sciences Lerner Research Institute Cleveland Clinic ClevelandOH44195 United States Arizona State University TempeAZ85287 United States Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Biology Pennsylvania State University University ParkPA16802 United States

To be used as an analysis tool, it is important that a spatial network’s construction algorithm reproduces the structural properties of the original physical embedding. One method for converting a two-dimensional (2D) point pattern into a spatial network is the Delaunay triangulation. Here, we apply the Delaunay triangulation to seven different types of 2D point patterns, including hyperuniform systems. The latter are characterized by completely suppressed normalized infinite-wavelength density fluctuations. We demonstrate that the quartile coefficients of dispersion of multiple centrality measures are capable of rank-ordering hyperuniform and nonhyperuniform systems independently, but they cannot distinguish a system that is nearly hyperuniform from hyperuniform systems. Thus, in each system, we investigate the local densities of the point pattern ρP (ri;) and of the network ρG(ni;). We reveal that there is a strong correlation between ρP (ri;) and ρG(ni;) in nonhyperuniform systems, but there is no such correlation in hyperuniform systems. When calculating the pair-correlation function and local density covariance function on the point pattern and network, the point pattern and network functions are similar only in nonhyperuniform systems. In hyperuniform systems, the triangulation has a positive covariance of local network densities in pairs of nodes that are close together;such covariance is not present in the point patterns. Thus, we demonstrate that the Delaunay triangulation accurately captures the density fluctuations of the underlying point pattern only when the point pattern possesses a positive correlation between ρP (ri;) for points that are close together. Such positive correlation is seen in most real-world systems, so the Delaunay triangulation is generally an effective tool for building a spatial network from a 2D point pattern, but there are situations (i.e., disordered hyperuniform systems) where we caution that the Delaunay triangulation would not

关键词： Triangulation

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Reducing Stress Level Cases of Hypogalactia Using Electric and Massage Stimulation Models 4

Reducing Stress Level Cases of Hypogalactia Using Electric a...

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4th International Conference on Energy, Environment, Epidemiology and Information System, ICENIS 2019

作者： Maula, Sheyla Najwatul Widyawati, Melyana Nurul Suryono, Suryono Post Graduate Applied Science Midwifery Program Poltekkes Kemenkes Semarang Indonesia Physics Department Faculty of Mathematics and Natural Sciences Diponegoro University Semarang Indonesia

Postpartum stress is very likely to take place as there are fluctuations in terms of feelings, pressure, anxiety, and guilt that may result in hypogalactia without proper treatment. Hypogalactia itself is an issue breastfeeding mothers face that may stall toddlers' growth, and hence, the future of a nation. This research aims to lower stress levels in hypogalactia cases. It employs nor pharmacological methods of electric and massage stimulation models to activate the biomechanical, psychological, and neurological systems within the body. The electric stimulation model was performed for 10 minutes in acupoints SI, ST, and SP at 0.5 watts frequency. Meanwhile, the massage stimulation was performed for 30 minutes in acupoints GB, BL, ST, CV, and ST. Samples were divided into intervention and control groups, both characteristics and difference tests were performed. Paired t-test results show that there is a significant difference (p © The Authors, published by EDP Sciences, 2019.

关键词： Electrophysiology

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Densest local sphere-packing diversity. II. Application to three dimensions

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Physical Review E 2011年第1期83卷 011304-011304页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry and Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N=1054. In the predecessor to this paper [A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space Rd and presented those packings in R2 for values of N up to N=348. Here we analyze the properties and characteristics of the densest local packings in R3 and employ knowledge of the Rmin(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R3, we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12–6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R3 (the Barlow packings) and find that the densest local packings are almost always most similar as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the fcc Barlow packing. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance Rmin(N). In particular, we find two “maracas” packings at N=77 and N=93, each consisting of a few unjammed sphere

关键词： DENSITY functionals SPHERES MATHEMATICAL models EUCLIDEAN metric CORRELATION (Statistics) SYMMETRY

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Disordered strictly jammed binary sphere packings attain an anomalously large range of densities

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Physical Review E 2013年第2期88卷 022205-022205页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton Institute for the Science and Technology of Materials Department of Physics Princeton Center for Theoretical Science Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions 0.634≤ϕ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain α and small sphere relative number concentrations x approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about 0.624 over the majority of the α-x plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the α-x plane where the backbone is not roughly constant, we find that backbone packing fractions range from about 0.606 to 0.829, with the volume of rattler spheres comprising between 1.6% and 26.9% of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such hig

关键词： PACKING fractions SHEAR (Mechanics) ALGORITHMS SOLID propellants CONCRETE CERAMICS

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