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Deep neural network for Wannier function centers

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arXiv 2019年

作者： Zhang, Linfeng Chen, Mohan Wu, Xifan Wang, Han Weinan, E. Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States CAPT HEDPS College of Engineering Peking University Beijing100871 China Department of Physics Temple University PhiladelphiaPA19122 United States Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Huayuan Road 6 Beijing100088 China Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University PrincetonNJ08544 United States

We introduce a deep neural network (DNN) model that assigns the position of the centers of the electronic charge in the snapshots of a molecular dynamics trajectory. The electronic centers are uniquely specified by the unitary transformation that maps the occupied eigenstates onto maximally localized Wannier functions. In combination with deep potential molecular dynamics, a DNN approach to model the atomic potential energy surface at the ab initio density functional level of theory, the scheme makes possible to predict the dielectric properties of insulators for samples and trajectories inaccessible to direct ab initio simulation, without loss of accuracy. As an example, we report calculations of the infrared absorption spectra of light and heavy water at a dispersion inclusive hybrid functional level of theory, finding good agreement with experiment. Extensions to other spectroscopies, like Raman and sum frequency generation, are discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum chemistry

Local and global strong solutions for SQG in bounded domains

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arXiv 2017年

作者： Constantin, Peter Nguyen, Huy Quang DEPARTMENT OF MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States

We prove local well-posedness for the inviscid surface quasigeostrophic (SQG) equation in bounded domains of R2. When fractional Dirichlet Laplacian dissipation is added, global existence of strong solutions is obtained for small data for critical and supercritical cases. Global existence of strong solutions with arbitrary data is obtained in the subcritical *** Codes 35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

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MODELING A SIMPLE CHOICE TASK: STOCHASTIC DYNAMICS OF MUTUALLY INHIBITORY NEURAL GROUPS

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Stochastics and Dynamics 2001年第2期1卷 159-191页

作者： ERIC BROWN PHILIP HOLMES Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA Also at Department of Mechanical and Aerospace Engineering Princeton University Princeton NJ 08544 USA.

We describe the dynamical and bifurcational behavior of two mutually inhibitory, leaky, neural units subject to external stimulus, random noise, and "priming biases". The model describes a simple forced choice experiment and accounts for varying levels of expectation and control. By projecting the model's dynamics onto slow manifolds, using judicious linear approximations, and solving for one-dimensional (reduced) probability densities, analytical estimates are developed for reaction time distributions and shown to compare satisfactorily with "full" numerical data. A sensitivity analysis is performed and the effects of parameters assessed. The predictions are also compared with behavioral data. These results may help correlate low-dimensional models of stochastic neural networks with cognitive test data, and hence assist in parameter choices and model building.

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On quasisymmetric plasma equilibria sustained by small force

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arXiv 2020年

作者： Constantin, Peter Drivas, Theodore D. Ginsberg, Daniel Department of Mathematics Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the Magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction ξ, to change the smooth structure on space so that the vector field ξ is Killing for the new metric and construct ξ–symmetric solutions of the Magnetohydrostatic equations on that background by solving a generalized Grad-Shafranov equation. If ξ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing. Copyright © 2020, The Authors. All rights reserved.

关键词： Magnetoplasma

MONGE-AMPÈRE FLOW FOR GENERATIVE MODELING

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arXiv 2018年

作者： Zhang, Linfeng Weinan, E. Wang, Lei Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University United States Center for Data Science Peking University Beijing Institute of Big Data Research Beijing100871 China Institute of Physics Chinese Academy of Sciences Beijing100190 China

We present a deep generative model, named Monge-Ampère flow, which builds on continuous-time gradient flow arising from the Monge-Ampère equation in optimal transport theory. The generative map from the latent space to the data space follows a dynamical system, where a learnable potential function guides a compressible fluid to flow towards the target density distribution. Training of the model amounts to solving an optimal control problem. The Monge-Ampère flow has tractable likelihoods and supports efficient sampling and inference. One can easily impose symmetry constraints in the generative model by designing suitable scalar potential functions. We apply the approach to unsupervised density estimation of the MNIST dataset and variational calculation of the two-dimensional Ising model at the critical point. This approach brings insights and techniques from Monge-Ampère equation, optimal transport, and fluid dynamics into reversible flow-based generative models. Copyright © 2018, The Authors. All rights reserved.

关键词： Statistical mechanics

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)

Offering Supplementary Wireless Technologies: Adoption Behavior and Offloading Benefits

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Offering Supplementary Wireless Technologies: Adoption Behav...

IEEE INFOCOM

作者： Carlee Joe-Wong Soumya Sen Sangtae Ha Program in Applied and Computational Mathematics Princeton University Princeton NJ USA Department of Electrical Engineering Princeton University Princeton NJ USA

ISBN: (纸本)9781467359443

To alleviate the congestion caused by rapid growth in demand for mobile data, ISPs have begun encouraging users to offload some of their traffic onto a supplementary, better quality network technology, e.g., offloading from 3G or 4G to WiFi and femtocells. With the growing popularity of such offerings, a deeper understanding of the underlying economic principles and their impact on technology adoption is necessary. To this end, we develop a model for user adoption of a base wireless technology and a bundle of the base plus a supplementary technology. In our model, individual users make their adoption decisions based on several factors, including the technologies' intrinsic qualities, throughput degradation due to congestion externalities from other subscribers, and the flat access rates that an ISP charges. We study the adoption dynamics and show that they converge to a unique equilibrium for a given set of exogenously determined system parameters. In particular, we characterize the occurrence of interesting adoption behaviors, including a possible decrease in the adoption of the supplementary technology as its coverage increases. Similar behaviors occur at an ISP's profit-maximizing prices and the optimal coverage area for the supplementary technology. To account for the potential benefits from offloading in practice, we collect 3G and WiFi usage and location data from twenty mobile users. We then use this data to numerically investigate the profit-maximizing adoption levels when an ISP accounts for its cost of deploying the supplemental technology and savings from offloading traffic onto this technology.

关键词： UNLOADING internet service providers third generation Adoption Wireless Technology Wi-Fi Femtocells 4G mobile communication

DeePCG: Constructing coarse-grained models via deep neural networks

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arXiv 2018年

作者： Weinan, E. Zhang, Linfeng Han, Jiequn Wang, Han Car, Roberto Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Institute of Applied Physics and Computational Mathematics Fenghao East Road 2 Beijing100094 China Caep Software Center for High Performance Numerical Simulation Huayuan Road 6 Beijing100088 China Program in Applied and Computational Mathematics Princeton Institute for Science and Technology of Materials Princeton University PrincetonNJ08544 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States Beijing Institute of Big Data Research Beijing100871 China

We introduce a general framework for constructing coarse-grained potential models without ad hoc approximations such as limiting the potential to two- A nd/or three-body contributions. The scheme, called Deep Coarse-Grained Potential (abbreviated DeePCG), exploits a carefully crafted neural network to construct a many-body coarse-grained potential. The network is trained with full atomistic data in a way that preserves the natural symmetries of the system. The resulting model is very accurate and can be used to sample the configurations of the coarse-grained variables in a much faster way than with the original atomistic model. As an application we consider liquid water and use the oxygen coordinates as the coarse-grained variables, starting from a full atomistic simulation of this system at the ab initio molecular dynamics level. We find that the two-body, three-body, and higher-order oxygen correlation functions produced by the coarse-grained and full atomistic models agree very well with each other, illustrating the effectiveness of the DeePCG model on a rather challenging task. Copyright © 2018, The Authors. All rights reserved.

关键词： Deep neural networks

STEFAN PROBLEM WITH SURFACE TENSION: GLOBAL EXISTENCE OF PHYSICAL SOLUTIONS UNDER RADIAL SYMMETRY

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arXiv 2022年

作者： Nadtochiy, Sergey Shkolnikov, Mykhaylo Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States ORFE Department Bendheim Center for Finance Program in Applied & Computational Mathematics Princeton University Princeton NJ08544 United States

We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of solution, which can accommodate the discontinuities in time (of the radius) of the evolving aggregate. Our main result establishes the global existence of a probabilistic solution satisfying the natural upper bound on the sizes of the discontinuities. Moreover, we prove that the upper bound is sharp in dimensions d ⩾ 3, in the sense that none of the discontinuities in the solution can be decreased in magnitude. The detailed analysis of the discontinuities, via appropriate stochastic representations, differentiates this work from the previous literature on weak solutions to the Stefan problem with surface *** Codes 80A22, 60H30, 35B44 Copyright © 2022, The Authors. All rights reserved.

关键词： Surface tension