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Structured time-delay models for dynamical systems with connections to Frenet-Serret frame

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arXiv 2021年

作者： Hirsh, Seth M. Ichinaga, Sara M. Brunton, Steven L. Kutz, J. Nathan Brunton, Bingni W. Department of Physics University of Washington SeattleWA United States Applied and Computational Mathematical Sciences Program University of Washington SeattleWA United States Department of Mechanical Engineering University of Washington SeattleWA United States Department of Applied Mathematics University of Washington SeattleWA United States Department of Biology University of Washington SeattleWA United States

Time-delay embeddings and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition (DMD) on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal with a zero diagonal;for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modeling procedure on data from several nonlinear synthetic and real-world examples. © 2021, CC BY.

关键词： Time delay

Manifold learning with arbitrary norms

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arXiv 2020年

作者： Kileel, Joe Moscovich, Amit Zelesko, Nathan Singer, Amit Department of Mathematics Oden Institute University of Texas at Austin United States Department of Statistics and Operations Research Tel-Aviv University Israel Department of Mathematics Brown University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using any norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry. Copyright © 2020, The Authors. All rights reserved.

关键词： Mathematical operators

Isotope effects in molecular structures and electronic properties of liquid water via deep potential molecular dynamics based on the SCAN functional

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Physical Review B 2020年第21期102卷 214113-214113页

作者： Jianhang Xu Chunyi Zhang Linfeng Zhang Mohan Chen Biswajit Santra Xifan Wu Department of Physics Temple University Philadelphia Pennsylvania 19122 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA CAPT HEDPS College of Engineering Peking University Beijing 100871 China Institute for Computational Molecular Science Temple University Philadelphia Pennsylvania 19122 USA

Feynman path-integral deep potential molecular dynamics (PI-DPMD) calculations have been employed to study both light (H2O) and heavy water (D2O) within the isothermalisobaric ensemble. In particular, the deep neural network is trained based on ab initio data obtained from the strongly constrained and appropriately normed (SCAN) exchange-correlation functional. Because of the lighter mass of hydrogen than deuteron, the properties of light water are more influenced by nuclear quantum effect than those of heavy water. Clear isotope effects are observed and analyzed in terms of hydrogen-bond structure and electronic properties of water that are closely associated with experimental observables. The molecular structures of both liquid H2O and D2O agree well with the data extracted from scattering experiments. The delicate isotope effects on radial distribution functions and angular distribution functions are well reproduced as well. Our approach demonstrates that deep neural network combined with SCAN functional based ab initio molecular dynamics provides an accurate theoretical tool for modeling water and its isotope effects.

关键词： Atomic & molecular structure Electronic structure Isotope effect Water Ab initio calculations Density functional theory Machine learning Molecular dynamics Path-integral methods

Interior electroneutrality in nernst-planck-navier-stokes systems

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arXiv 2020年

作者： Constantin, Peter Ignatova, Mihaela Lee, Fizay-Noah DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States DEPARTMENT OF MATHEMATICS TEMPLE UNIVERSITY PHILADELPHIAPA19122 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider the limit of vanishing Debye length for ionic diffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. In the asymptotically stable cases of blocking (vanishing normal flux) and uniform selective (special Dirichlet) boundary conditions for the ionic concentrations, we prove that the ionic charge density ρ converges in time to zero in the interior of the domain, in the limit of vanishing Debye length (ǫ → 0). For the unstable regime of Dirichlet boundary conditions for the ionic concentrations, we prove bounds that are uniform in time and ǫ. We also consider electroneutral boundary conditions, for which we prove that electroneutrality ρ → 0 is achieved at any fixed ǫ > 0, exponentially fast in time in Lp, for all 1 ≤ p MSC Codes 35Q30, 35Q35, 35Q92 Copyright © 2020, The Authors. All rights reserved.

关键词： Boundary conditions

Threshold and decay properties of transient isolated turbulent band in plane Couette flow

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applied mathematics and Mechanics(English Edition) 2019年第10期40卷 1449-1456页

作者： Jianzhou LU Jianjun TAO Weitao ZHOU Xiangming XIONG CAPT-HEDPS SKLTCS Collaborative Innovation Center of IFSA Department of Mechanics and Engineering Science College of Engineering Peking University Beijing 100871 China The Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 U. S.A.

By direct numerical simulations of the plane Couette flow (PCF) in a large computational domain, it is shown that an isolated turbulent band decays monotonically at low Reynolds numbers but experiences transient growth before the eventual relaminarization at moderate Reynolds numbers. The lower bound Reynolds number of the transient-growth regime is determined as 286. The width, length, and tilt angle of the iso- lated band structure are defined based on the disturbance kinetic energy in the mid-plane, and the geometric characteristics of the band can be described with a tilted rectangle. It is illustrated that before its eventual fragmentation, the isolated turbulent band decays in a style of longitudinal contraction, where the center, width, and tilt angle of the band keep almost constant but the band length contracts with a statistically constant velocity.

关键词： plane Couette flow (PCF) subcritical transition turbulent band

Nernst-Planck-Navier-stokes systems far from equilibrium

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arXiv 2020年

作者： Constantin, Peter Ignatova, Mihaela Lee, Fizay-Noah Department of Mathematics Princeton University PrincetonNJ08544 United States Department of Mathematics Temple University PhiladelphiaPA19122 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We consider ionic electrodiffusion in fluids, described by the Nernst-Planck-Navier-Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data: aribitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions for the potential, arbitrary positive initial concentrations, and arbitrary regular divergence-free initial velocities. The result holds for any positive diffusivities of ions, in bounded domains with smooth boundary in three space dimensions, in the case of two ionic species, coupled to Stokes equations for the fluid. The result also holds in the case of Navier-Stokes coupling, if the velocity is regular. The global smoothness of solutions is also true for arbitrarily many ionic species, if all their diffusivities are the *** Codes 35Q30, 35Q35, 35Q92 Copyright © 2020, The Authors. All rights reserved.

关键词： Navier Stokes equations

computational Design of Anisotropic Stealthy Hyperuniform Composites with Engineered Directional Scattering Properties

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arXiv 2023年

作者： Shi, Wenlong Keeney, David Chen, Duyu Jiao, Yang Torquato, Salvatore Materials Science and Engineering Arizona State University TempeAZ85287 United States Materials Research Laboratory University of California Santa BarbaraCA93106 United States Department of Physics 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

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here, we generalize the Fourier-space based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ˜V (k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform (SHU) microstructures with χ˜V (k) = 0 for k ∈ Ω, i.e., complete suppression of scattering in an "exclusion" region Ω around the origin in the Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the and phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions, and are non-hyperuniform along others. We find that, while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotr

关键词： Volume fraction

NERNST-PLANCK-NAVIER-STOKES SYSTEMS NEAR EQUILIBRIUM

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arXiv 2020年

作者： Constantin, Peter Ignatova, Mihaela Lee, Fizay-Noah DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States DEPARTMENT OF MATHEMATICS TEMPLE UNIVERSITY PHILADELPHIAPA19122 United States PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States

The Nernst-Planck-Navier-Stokes system models electrodiffusion of ions in a fluid. We prove global existence of solutions in bounded domains in three dimensions with either blocking (no-flux) or uniform selective (special Dirichlet) boundary conditions for ion concentrations. The global existence of strong solutions is established for initial conditions that are sufficiently small perturbations of steady state solutions. The solutions remain close to equilbrium in strong norms. The main two steps of the proof are (1) the decay of the sum of relative entropies (Kullback-Leibler divergences) and (2) the control of L2 norms of deviations by the sum of relative *** Codes 35Q30, 35Q35, 35Q92 Copyright © 2020, The Authors. All rights reserved.

关键词： Navier Stokes equations

A hybrid semi-Lagrangian cut cell method for advection-diffusion problems with robin boundary conditions in moving domains

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arXiv 2021年

作者： Barrett, Aaron Fogelson, Aaron L. Griffith, Boyce E. Department of Mathematics University of Utah Salt Lake CityUT United States Departments of Mathematics and Bioengineering University of Utah Salt Lake CityUT United States Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

We present a new discretization approach to advection-diffusion problems with Robin boundary conditions on complex, time-dependent domains. The method is based on second order cut cell finite volume methods introduced by Bochkov et al. [8] to discretize the Laplace operator and Robin boundary condition. To overcome the small cell problem, we use a splitting scheme along with a semi-Lagrangian method to treat advection. We demonstrate second order accuracy in the L1, L2, and L∞norms for both analytic test problems and numerical convergence studies. We also demonstrate the ability of the scheme to convert one chemical species to another across a moving boundary. © 2021, CC BY-SA.

关键词： Advection