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Toward an orbital-free density functional theory of transition metals based on an electron density decomposition

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Physical Review B 2012年第4期85卷 045126-045126页

作者： Chen Huang Emily A. Carter Departments 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.

To properly apply approximate kinetic energy density functionals (KEDFs) used in orbital-free density functional theory (OF-DFT) to the highly inhomogeneous electron density in transition metals, we introduce a decomposition scheme in which the electron density is partitioned into delocalized and localized parts, which makes it possible to apply a different physically justifiable KEDF to each density. Using this decomposition scheme, we demonstrate that OF-DFT is able to quantitatively reproduce Kohn-Sham- (KS-)DFT predictions of the basic bulk properties of many phases of Ag, as well as surface energies, vacancy formation energies, and elastic moduli of face-centered-cubic Ag. However, OF-DFT Ag-Al alloy properties differ substantially from those predicted by KS-DFT using nonlocal pseudopotentials. These errors are due to remaining inaccuracies in the Ag and Al local electron-ion pseudopotentials, as well as the oversimplified model KEDF used for treating the interaction KE term between the localized and the delocalized electron densities. Future extensions to improve the treatment of transition metals are briefly outlined.

关键词： DENSITY functionals MOLECULAR orbitals TRANSITION metals ELECTRON distribution DECOMPOSITION (Chemistry) QUANTITATIVE research

Multiple time scale numerical methods for the inverted pendulum problem

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Lecture Notes in computational Science and Engineering 2005年 44卷 241-261页

作者： Sharp, Richard Tsai, Yen-Hsi Engquist, Björn Program in Applied and Computational Mathematics Princeton University NJ 08544 United States Department of Mathematics University of Texas at Austin 1 University Station C1200 Austin TX 78712 United States

ISBN: (纸本)9783540253358

In this article, we study a class of numerical ODE schemes that use a time filtering strategy and operate in two time scales. The algorithms follow the framework of the heterogeneous multiscale methods (HMM) [1]. We apply the methods to compute the averaged path of the inverted pendulum under a highly oscillatory vertical forcing on the pivot. The averaged equation for related problems has been studied analytically in [9]. We prove and show numerically that the proposed methods approximate the averaged equation and thus compute the average path of the inverted pendulum.

关键词： Inverted pendulum

Artificial neural network approach for turbulence models: A local framework

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Physical Review Fluids 2021年第8期6卷 084612-084612页

作者： Chenyue Xie Xiangming Xiong Jianchun Wang Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Department of Mechanics and Aerospace Engineering Southern University of Science and Technology Shenzhen 518055 People's Republic of China

A local artificial neural network (LANN) framework is developed for turbulence modeling. The Reynolds-averaged Navier-Stokes (RANS) unclosed terms are reconstructed by the artificial neural network based on the local coordinate system which is orthogonal to the curved walls. We verify the proposed model in the flows over periodic hills. The correlation coefficients of the RANS unclosed terms predicted by the LANN model can be made larger than 0.96 in an a priori analysis, and the relative error of the unclosed terms can be made smaller than 18%. In an a posteriori analysis, detailed comparisons are made on the results of RANS simulations using the LANN, global artificial neural network (GANN), Spalart-Allmaras (SA), and shear stress transport (SST) k−ω models. It is shown that the LANN model performs better than the GANN, SA, and SST k−ω models in the prediction of the average velocity, wall-shear stress, and average pressure, which gives the results that are essentially indistinguishable from the direct numerical simulation data. The LANN model trained at low Reynolds number, Re=2800, can be directly applied to the cases of high Reynolds numbers, Re=5600, 10 595, 19 000, and 37 000, with accurate predictions. Furthermore, the LANN model is verified for flows over periodic hills with varying slopes. These results suggest that the LANN framework has a great potential to be applied to complex turbulent flows with curved walls.

关键词： Compressible boundary layers Turbulence Artificial neural networks

APPLE picker: Automatic particle picking, a low-effort cryo-EM framework

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arXiv 2018年

作者： Heimowitz, Ayelet Andén, Joakim Singer, Amit Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Center for Computational Biology Flatiron Institute New YorkNY United States Department of Mathematics Princeton University PrincetonNJ United States

Particle picking is a crucial first step in the computational pipeline of single-particle cryo-electron microscopy (cryo-EM). Selecting particles from the micrographs is difficult especially for small particles with low contrast. As high-resolution reconstruction typically requires hundreds of thousands of particles, manually picking that many particles is often too time-consuming. While semi-automated particle picking is currently a popular approach, it may suffer from introducing manual bias into the selection process. In addition, semi-automated particle picking is still somewhat time-consuming. This paper presents the APPLE (Automatic Particle Picking with Low user Effort) picker, a simple and novel approach for fast, accurate, and fully automatic particle picking. While our approach was inspired by template matching, it is completely template-free. This approach is evaluated on publicly available datasets containing micrographs of β-Galactosidase, T20S proteasome, 70S ribosome and keyhole limpet hemocyanin projections. Copyright © 2018, The Authors. All rights reserved.

关键词： Support vector machines

Warm dense matter simulation via electron temperature dependent deep potential molecular dynamics

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arXiv 2019年

作者： Zhang, Yuzhi Gao, Chang Zhang, Linfeng Wang, Han Chen, Mohan Center for Applied Physics and Technology HEDPS College of Engineering Peking University Beijing100871 China Beijing Institute of Big Data Research Beijing100871 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China

Simulating warm dense matter that undergoes a wide range of temperatures and densities is challenging. Predictive theoretical models, such as quantum-mechanics-based first-principles molecular dynamics (FPMD), require a huge amount of computational resources. Herein, we propose a deep learning based scheme, called electron temperature dependent deep potential molecular dynamics (TDDPMD), for efficiently simulating warm dense matter with the accuracy of FPMD. The TDDPMD simulation is several orders of magnitudes faster than FPMD, and, unlike FPMD, its efficiency is not affected by the electron temperature. We apply the TDDPMD scheme to beryllium (Be) in a wide range of temperatures (0.4 to 2500 eV) and densities (3.50 to 8.25 g/cm3). Our results demonstrate that the TDDPMD method not only accurately reproduces the structural properties of Be along the principal Hugoniot curve at the FPMD level, but also yields even more reliable diffusion coefficients than typical FPMD simulations due to its ability to simulate larger systems with longer time. Copyright © 2019, The Authors. All rights reserved.

关键词： Molecular dynamics

Gait transitions in a phase oscillator model of an insect central pattern generator

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arXiv 2017年

作者： Aminzare, Zahra Srivastava, Vaibhav Holmes, Philip Program in Applied and Computational Mathematics Department of Mechanical and Aerospace Engineering Princeton University NJ United States Department of Electrical and Computer Engineering Michigan State University LansingMI United States Program in Applied and Computational Mathematics Department of Mechanical and Aerospace Engineering Princeton Neuroscience Institute Princeton University NJ United States

Legged locomotion involves various gaits. It has been observed that fast running insects (cockroaches) employ a tripod gait with three legs lifted off the ground simultaneously in swing, while slow walking insects (stick insects) use a tetrapod gait with two legs lifted off the ground simultaneously. Fruit flies use both gaits and exhibit a transition from tetrapod to tripod at intermediate speeds. Here we study the effect of stepping frequency on gait transition in an ion-channel bursting neuron model in which each cell represents a hemi-segmental thoracic circuit of the central pattern generator. Employing phase reduction, we collapse the network of bursting neurons represented by 24 ordinary differential equations to 6 coupled nonlinear phase oscillators, each corresponding to a sub-network of neurons controlling one leg. Assuming that the left and right legs maintain constant phase differences (contralateral symmetry), we reduce from 6 equations to 3, allowing analysis of a dynamical system with 2 phase differences defined on a torus. We show that bifurcations occur from multiple stable tetrapod gaits to a unique stable tripod gait as speed increases. Finally, we consider gait transitions in two sets of data fitted to freely walking fruit flies.34C16, 34C60, 37G10, 92B20, 92C2034C16, 34C60, 37G10, 92B20, 92C20 Copyright © 2017, The Authors. All rights reserved.

关键词： Bifurcation (mathematics)

Physics-informed machine learning with smoothed particle hydrodynamics: Hierarchy of reduced Lagrangian models of turbulence

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Physical Review Fluids 2023年第5期8卷 054602-054602页

作者： Michael Woodward Yifeng Tian Criston Hyett Chris Fryer Mikhail Stepanov Daniel Livescu Michael Chertkov Graduate Interdisciplinary Program in Applied Mathematics University of Arizona Tucson Arizona 85721 USA Department of Mathematics University of Arizona Tucson Arizona 85721 USA Computer Computational and Statistical Sciences Division LANL Los Alamos New Mexico 87544 USA

Building efficient, accurate, and generalizable reduced-order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and it investigates the effects of enforcing physical structure through smoothed particle hydrodynamics (SPH) versus relying on neural networks (NNs) as universal function approximators. Starting from NN parametrizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational, and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using automatic differentiation and sensitivity analysis. Each model within the hierarchy is trained on two data sets associated with weakly compressible homogeneous isotropic turbulence: (1) a validation set using weakly compressible SPH; and (2) a high-fidelity set from direct numerical simulations. Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

关键词： Sensitivity analysis

Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence

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arXiv 2022年

作者： Han, Jiequn Hu, Ruimeng Long, Jihao Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States Department of Mathematics Department of Statistics and Applied Probability University of California Santa BarbaraCA93106-3080 United States The Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States

One of the core problems in mean-field control and mean-field games is to solve the corresponding McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs). Most existing methods are tailored to special cases in which the mean-field interaction only depends on expectation or other moments and thus is inadequate to solve problems when the mean-field interaction has full distribution dependence. In this paper, we propose a novel deep learning method for computing MV-FBSDEs with a general form of mean-field interactions. Specifically, built on fictitious play, we recast the problem into repeatedly solving standard FBSDEs with explicit coefficient functions. These coefficient functions are used to approximate the MV-FBSDEs’ model coefficients with full distribution dependence, and are updated by solving another supervising learning problem using training data simulated from the last iteration’s FBSDE solutions. We use deep neural networks to solve standard BSDEs and approximate coefficient functions in order to solve high-dimensional MV-FBSDEs. Under proper assumptions on the learned functions, we prove that the convergence of the proposed method is free of the curse of dimensionality (CoD) by using a class of integral probability metrics (IPMs) previously developed in [Han, Hu and Long, arXiv:2104.12036]. The proved theorem shows the advantage of the method in high dimensions. We present the numerical performance in high-dimensional MV-FBSDE problems, including a mean-field game example of the well-known Cucker-Smale model whose cost depends on the full distribution of the forward *** Codes 65C30, 68T07, 49N80, 68Q25 © 2022, CC BY.

关键词： Stochastic systems

ON A LINEARIZATION OF QUADRATIC WASSERSTEIN DISTANCE

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arXiv 2022年

作者： Greengard, Philip Hoskins, Jeremy G. Marshall, Nicholas F. Singer, Amit Department of Statistics Columbia University United States Department of Statistics University of Chicago United States Program in Applied and Computational Mathematics The Department of Mathematics Princeton University United States

This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance W2. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein distance follows from a classic linearization of a general Monge-Ampére equation. Our contribution is threefold. First, we provide expository material on this classic linearization of Wasserstein distance including a quantitative error estimate. Second, we reduce the computational problem to solving a elliptic boundary value problem involving the Witten Laplacian, which is a Schrödinger operator of the form H = -∆ + V , and describe an associated embedding. Third, for the case of probability distributions on the unit square [0, 1]2 represented by n × n arrays we present a fast code demonstrating our approach. Several numerical examples are presented. Copyright © 2022, The Authors. All rights reserved.

关键词： Laplace transforms