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Coarse graining the dynamics of heterogeneous oscillators in networks with spectral gaps

Physical Review E 2011年第3期84卷 036708-036708页

作者： Karthikeyan Rajendran Ioannis G. Kevrekidis Department of Chemical and Biological Engineering Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA

We present a computer-assisted approach to coarse graining the evolutionary dynamics of a system of nonidentical oscillators coupled through a (fixed) network structure. The existence of a spectral gap for the coupling network graph Laplacian suggests that the graph dynamics may quickly become low dimensional. Our first choice of coarse variables consists of the components of the oscillator states—their (complex) phase angles—along the leading eigenvectors of this Laplacian. We then use the equation-free framework, circumventing the derivation of explicit coarse-grained equations, to perform computational tasks such as coarse projective integration, coarse fixed-point, and coarse limit-cycle computations. In a second step, we explore an approach to incorporating oscillator heterogeneity in the coarse-graining process. The approach is based on the observation of fast-developing correlations between oscillator state and oscillator intrinsic properties and establishes a connection with tools developed in the context of uncertainty quantification.

关键词： GRAINING COMPUTER-aided design LAPLACIAN operator EIGENVECTORS HARMONIC oscillators

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Deuteration removes quantum dipolar defects from KDP crystals

arXiv

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arXiv 2024年

作者： Yang, Bingjia Xie, Pinchen Car, Roberto Department of Chemistry Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Materials Institute Princeton University PrincetonNJ08544 United States

The structural, dielectric, and thermodynamic properties of the hydrogen-bonded ferroelectric crystal potassium dihydrogen phosphate (KH2PO4), KDP for short, differ significantly from those of DKDP (KD2PO4) [1, 2]. It is well established that deuteration affects the interplay of hydrogen-bond switches and heavy ion displacements that underlie the emergence of macroscopic polarization, but a detailed microscopic model is missing [3, 4]. Here we show that all-atom path integral molecular dynamics simulations can predict the isotope effects, revealing the microscopic mechanism that differentiates KDP and DKDP. Proton tunneling in the hydrogen bonds generates phosphate configurations that do not contribute to the polarization. These dipolar defects are always present in KDP, but disappear at low temperatures in DKDP, which behaves more classically. Quantum disorder confers residual entropy to ferroelectric KDP, explaining its lower spontaneous polarization and transition entropy relative to DKDP. Tunneling should also contribute to the anomalous heat capacity observed in KDP near absolute zero [5, 6]. The prominent role of quantum fluctuations in KDP is related to the unusual strength of the hydrogen bonds in this system and should be equally important in the other crystals of the KDP family, which exhibit similar isotope effects [7]. © 2024, CC BY.

关键词： Hydrogen bonds

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Applications of mean field games in financial engineering and economic theory

arXiv

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arXiv 2020年

作者： Carmona, René DEPARTMENT OF OPERATIONS RESEARCH & FINANCIAL ENGINEERING BENDHEIM CENTER FOR FINANCE PROGRAM IN APPLIED & COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY United States

This is an expanded version of the lecture given at the AMS Short Course on Mean Field Games, on January 13, 2020 in Denver CO. The assignment was to discuss applications of Mean Field Games in finance and economics. I need to admit upfront that several of the examples reviewed in this chapter were already discussed in book form. Still, they are here accompanied with discussions of, and references to, works which appeared over the last three years. Moreover, several completely new sections are added to show how recent developments in financial engineering and economics can benefit from being viewed through the lens of the Mean Field Game paradigm. The new financial engineering applications deal with bitcoin mining and the energy markets, while the new economic applications concern models offering a smooth transition between macro-economics and finance, and contract *** Codes 60, 91 © 2020, CC BY.

关键词： Finance

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NOMAD: nonlinear manifold decoders for operator learning 22

NOMAD: nonlinear manifold decoders for operator learning

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Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Jacob H. Seidman Georgios Kissas Paris Perdikaris George J. Pappas Graduate Program in Applied Mathematics and Computational Science University of Pennsylvania Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Department of Electrical and Systems Engineering University of Pennsylvania

ISBN: (纸本)9781713871088

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost.

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NUMERICAL ANALYSIS OF ELLIPTIC HEMIVARIATIONAL INEQUALITIES FOR SEMIPERMEABLE MEDIA

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Journal of computational mathematics 2019年第4期37卷 506-523页

作者： Weimin Han Ziping Huang Cheng Wang Wei Xu Program in Applied Mathematical and Computational Sciences (A MCS)& Department of Mathematics University of Iowa Iowa City IA 522/2 USA School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College J taxing 314051 China School of Mathematical Sciences Tojigji University Shanghai 200092 China School of Mathematical Sciences Tongji University Shanghai 200092 China Tongji Zhejiang College Jiaxing 314051 China

In this paper, we consider elliptic hemivariational inequalities arising in applications in semipermeable media. In its general form, the model includes both interior and boundary semipermeability terms. Detailed study is given on the hemivariational inequality in the case of isotropic and homogeneous semipermeable media. Solution existence and uniqueness of the problem are explored. Convergence of the Galerkin method is shown under the basic solution regularity available from the existence result. An optimal order error estimate is derived for the linear finite element solution under suitable solution regularity assumptions. The results can be readily extended to the study of more general hemivariational inequalities for non-isotropic and heterogeneous semipermeable media with interior semipermeability and/or boundary semiperrneability. Numerical examples are presented to show the performance of the finite element approximations;in particular, the theoretically predicted optimal first order convergence in H' norm of the linear element solutions is clearly observed.

关键词： Hemivariational inequality Interior semipermeability Boundary semipermeability Finite element method Error estimate

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How SGD selects the global minima in over-parameterized learning: a dynamical stability perspective 18

How SGD selects the global minima in over-parameterized lear...

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Proceedings of the 32nd International Conference on Neural Information Processing Systems

作者： Lei Wu Chao Ma Weinan E. School of Mathematical Sciences Peking University Beijing P.R. China Program in Applied and Computational Mathematics Princeton University Princeton NJ Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton NJ and Beijing Institute of Big Data Research Beijing P.R. China

The question of which global minima are accessible by a stochastic gradient decent (SGD) algorithm with specific learning rate and batch size is studied from the perspective of dynamical stability. The concept of non-uniformity is introduced, which, together with sharpness, characterizes the stability property of a global minimum and hence the accessibility of a particular SGD algorithm to that global minimum. In particular, this analysis shows that learning rate and batch size play different roles in minima selection. Extensive empirical results seem to correlate well with the theoretical findings and provide further support to these claims.

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Coarse Graining the Dynamics of Coupled Oscillators

引用

Physical Review Letters 2006年第14期96卷 144101-144101页

作者： Sung Joon Moon R. Ghanem I. G. Kevrekidis Department of Chemical Engineering & Program in Applied and Computational Mathematics (PACM) Princeton University Princeton New Jersey 08544 USA Department of Civil Engineering University of Southern California Los Angeles California 90089 USA

We present an equation-free computational approach to the study of the coarse-grained dynamics of finite assemblies of nonidentical coupled oscillators at and near full synchronization. We use coarse-grained observables which account for the (rapidly developing) correlations between phase angles and natural frequencies. Exploiting short bursts of appropriately initialized detailed simulations, we circumvent the derivation of closures for the long-term dynamics of the assembly statistics.

关键词： Oscillators (electronic)

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Direct minimization of the optimized effective problem based on efficient finite differences

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Physical Review B 2011年第16期84卷 165122-165122页

作者： Chen Huang Emily A. Carter Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and the Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

To avoid the difficult-to-solve optimized effective potential (OEP) integral equation, we introduce an efficient direct minimization scheme for performing OEP calculations within Kohn–Sham density functional theory (KS-DFT). We reformulated the functional derivative of the total energy with respect to the KS effective potential in terms of efficient finite differences. Our method only uses the orbitals involved in the construction of the KS exchange-correlation functionals. We demonstrate our scheme by performing exact-exchange OEP for sodium clusters, in which only occupied KS orbitals are needed to obtain the OEP. Our efficient direct minimization scheme should aid future development of orbital-dependent density functionals and render OEP to be a practical choice for various applications.

关键词： INTEGRAL equations DENSITY functionals TOTAL energy systems (On-site electric power production) FINITE differences PHYSICS

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Minuscule corrections to near-surface solar internal rotation using mode-coupling

arXiv

引用

arXiv 2023年

作者： Das, Srijan Bharati Kashyap, Samarth G. Oktay, Deniz Hanasoge, Shravan M. Tromp, Jeroen Department of Geosciences Princeton University PrincetonNJ United States Department of Astronomy and Astrophysics Tata Institute of Fundamental Research Mumbai India Department of Computer Science Princeton University PrincetonNJ United States Department of Geosciences and Program in Applied & Computational Mathematics Princeton University PrincetonNJ United States

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow — differential rotation — is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency-splittings do not consider the effect of mode-coupling. This approximation worsens for high-angular-degree modes, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ` ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a non-linear inversion using an eigenvalue solver. Differences in inverted profiles for frequency splitting measurements from MDI and HMI are compared and discussed. We find that corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields. © 2023, CC BY.

关键词： Rotation

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Angular momentum dependent orbital-free density functional theory: Formulation and implementation

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Physical Review B 2014年第15期89卷 155112-155112页

作者： Youqi Ke Florian Libisch Junchao Xia Emily A. Carter []Department of Mechanical and Aerospace Engineering Program in Applied and Computational Mathematics and Andlinger Center for Energy and the Environment Princeton University Princeton New Jersey 08544-5263 USA

Orbital-free density functional theory (OFDFT) directly solves for the ground-state electron density. It scales linearly with respect to system size, providing a promising tool for large-scale material simulations. Removal of the orbitals requires use of approximate noninteracting kinetic energy density functionals. If replacing ionic cores with pseudopotentials, removal of the orbitals also requires these pseudopotentials to be local. These are two severe challenges to the capabilities of conventional OFDFT. While main group elements are often well described within conventional OFDFT, transition metals remain intractable due to their localized d electrons. To advance the accuracy and general applicability of OFDFT, we have recently reported a general angular momentum dependent formulation as a next-generation OFDFT. In this formalism, we incorporate the angular momenta of electrons by devising a hybrid scheme based on a muffin tin geometry: inside spheres centered at the ionic cores, the electron density is expanded in a set of atom-centered basis functions combined with an onsite density matrix. The explicit treatment of the angular momenta of electrons provides an important basis for accurately describing the important ionic core region, which is not possible in conventional OFDFT. In addition to the conventional OFDFT total energy functional, we introduce a nonlocal energy term containing a set of angular momentum dependent energies to correct the errors due to the approximate kinetic energy density functional and local pseudopotentials. Our approach greatly increases the accuracy of OFDFT while largely preserving its numerical simplicity. Here, we provide details of the theoretical formulation and practical implementation, including the hybrid scheme, the derivation of the nonlocal energy term, the choice of basis functions, the direct minimization of the total energy, the procedure to determine the angular momentum dependent energies, the force formula with Pu

关键词： ANGULAR momentum (Nuclear physics) DENSITY functional theory GROUND state (Quantum mechanics) ELECTRON density SIMULATION methods & models DENSITY functionals

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