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Coarse-grained spectral projection: A deep learning assisted approach to quantum unitary dynamics

Physical Review B 2021年第2期103卷 024304-024304页

作者： Pinchen Xie Weinan E Department of Mathematics and Program in Applied and Computational Mathematics Princeton University New Jersey 08544 USA

We propose the coarse-grained spectral projection method (CGSP), a deep learning assisted approach for tackling quantum unitary dynamic problems with an emphasis on quench dynamics. We show that CGSP can extract spectral components of many-body quantum states systematically with a sophisticated neural network quantum ansatz. CGSP fully exploits the linear unitary nature of the quantum dynamics and is potentially superior to other quantum Monte Carlo methods for ergodic dynamics. Preliminary numerical results on one-dimensional XXZ models with periodic boundary conditions are carried out to demonstrate the practicality of CGSP.

关键词： Quantum quench Quantum statistical mechanics Nonequilibrium lattice models Machine learning Monte Carlo methods

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Factor Analysis for Spectral Estimation

Factor Analysis for Spectral Estimation

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International Conference on Sampling Theory and Applications

作者： Joakim Anden Amit Singer Program in Applied and Computational Mathematics Princeton University Department of Mathematics and Program in Applied and Computational Mathematics Princeton University

ISBN: (纸本)9781538615669

Power spectrum estimation is an important tool in many applications, such as the whitening of noise. The popular multitaper method enjoys significant success, but fails for short signals with few samples. We propose a statistical model where a signal is given by a random linear combination of fixed, yet unknown, stochastic sources. Given multiple such signals, we estimate the subspace spanned by the power spectra of these fixed sources. Projecting individual power spectrum estimates onto this subspace increases estimation accuracy. We provide accuracy guarantees for this method and demonstrate it on simulated and experimental data from cryo-electron microscopy.

关键词： Spectrum Analysis power spectra Subspace STATIONARY SOURCES Factor analysis Cryoelectron Microscopy Linear Combination statistical models (nuclear) signals Individual Power simulating

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Renormalization theory for eddy diffusivity in turbulent transport

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Physical Review Letters 1992年第20期68卷 3028-3028页

作者： Marco Avellaneda Andrew J. Majda Department of Mathematics Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544

We examine the derivation of eddy-diffusivity equations for transport of passive scalars in a turbulent velocity field. Our main contention is that, in the long-time–large-distance limit, the eddy-diffusivity equations can take very different forms according to the statistical properties of the subgrid velocity, and that these equations depend very sensitively on the interplay between spatial and temporal velocity fluctuations. Such crossovers can be represented in a ‘‘phase diagram’’ involving two relevant statistical parameters. Strikingly, the Kolmogorov-Obukhov statistical theory is shown to lie on a phase-transition boundary.

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Coupled MRT Lattice Boltzmann Study of Electrokinetic Mixing of Power-Law Fluids in Microchannels with Heterogeneous Surface Potential

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Advances in applied mathematics and Mechanics 2018年第6期10卷 1478-1496页

作者： Xuguang Yang Lei Wang Baochang Shi School of Mathematics and Computational Science Hunan First Normal UniversityChangsha 410205HunanChina Department of Applied Mathematics The Hong Kong Polytechnic UniversityHung HomHong Kong School of Mathematics and Physics China University of GeosciencesWuhan430074HubeiChina School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074HubeiChina

The electrokinetic mixing,as a powerful technique in microfluidic devices,is widely used in many *** this study,a more general dynamic model,which consists of Poisson equation,Nernst-Planck equation and Navier-Stokes equations,is used to describe the electrokinetic mixing of non-Newtonian fluids in ***,a coupled multiple-relaxation-time(MRT)lattice Boltzmann(LB)framework is developed to solve this complicated multi-physics transport *** numerical simulations,we mainly consider the effects of the arrangement of nonuniform surface potentials and the power-law index on the mixing efficiency and the volumetric flow *** results show that the mixing efficiency and the volumetric flow rate of shear-thinning fluids are higher than that of shear-thickening fluids under the same *** is also shown that for both types of fluids,there should be a balance between the mixing and liquid transport in electrokinetic microfluidics.

关键词： Lattice Boltzmann method electrokinetic mixing non-Newtonian fluids

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Modelling the effect of changes in defect geometry on chemically active surfaces by the boundary element technique

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Surface Science 1988年第1-2期202卷 32-57页

作者： Peirce, Anthony P. Rabitz, Herschel Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Chemistry Princeton University Princeton NJ 08544 United States

The boundary element (BE) technique is used to analyze the effect of defect structures upon desorption processes on two-dimensional chemically active surfaces. The standard BE algorithm for diffusion is modified to incorporate the effects of bulk desorption, and an explicit scheme is proposed for the treatment of the non-linear equations associated with localized defect structures. The BE algorithm proposed here provides an elegant representation of the effects of localized non-linear reactions which allows arbitrarily oriented defect structures to be modelled without having to perform mesh deformation. A class of trapping reactions is assumed to occur along defect structures, and the effect of changes in defect geometry on the balance between the desorptive processes is explored. A number of interesting competitive/cooperative phenomena are observed to occur for the various shapes of defect geometry, including strong intrinsic competition in circular defect structures that form islands of nearly constant concentration, a redistribution of material along V-shaped defect structures in a way that reflects relative competitiveness of defects on opposite sides of the defect structure, and a reduction of competitiveness for defect distributions that are less regular in shape. The proposed BE algorithm is shown to provide a useful technique for modelling the effect of defect structures on chemically active surfaces. © 1988.

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Exponential convergence of the deep neural network approximation for analytic functions Dedicated to Professor Ta Tsien Li on the Occasion of His 80th Birthday

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Science China mathematics 2018年第10期61卷 1733-1740页

作者： Weinan E Qingcan Wang Department of Mathematics and PACM Princeton University Princeton NJ 08544 USA Center for Big Data Research Peking University Beijing 100871 China Beijing Institute of Big Data Research Beijing 100871 China Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 USA

We prove that for analytic functions in low dimension, the convergence rate of the deep neural network approximation is exponential.

关键词： neural networks approximation theory analytic functions

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A qualitative study of the dynamic behavior for adaptive gradient algorithms

arXiv

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Stanford University Program in Applied and Computational Mathematics Princeton University Program in Applied and Computational Mathematics Department of Mathematics Princeton University

The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods

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Noisy Hegselmann-Krause systems: Phase transition and the 2R-conjecture 55

Noisy Hegselmann-Krause systems: Phase transition and the 2R...

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55th IEEE Conference on Decision and Control, CDC 2016

作者： Wang, Chu Li, Qianxiao Weinan, E. Chazelle, Bernard Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States Nokia Bell Labs 600 Mountain Avenue Murray HillNJ07974 United States Department of Mathematics Program in Applied and Computational Mathematics Princeton University NJ08540 United States Department of Computer Science Princeton University PrincetonNJ08540 United States

ISBN: (纸本)9781509018376

The classic Hegselmann-Krause (HK) model for opinion dynamics consists of a set of agents on the real line, each one instructed to move, at every time step, to the mass center of the agents within a fixed distance R. In this work, we investigate the effects of noise in the continuous-time version of the model as described by its mean-field Fokker-Planck equation. In the presence of a finite number of agents, the system exhibits a phase transition from order to disorder as the noise increases. We introduce an order parameter to track the phase transition and resolve the corresponding phase diagram. The system undergoes a phase transition for small R but none for larger R. Based on the stability analysis of the mean-field equation, we derive the existence of a forbidden zone for the disordered phase to emerge. We also provide a theoretical explanation for the well-known 2R conjecture, which states that, for a random initial distribution in a fixed interval, the final configuration consists of clusters separated by a distance of roughly 2R. Our theoretical analysis confirms previous simulations and predicts properties of the noisy HK model in higher dimension. © 2016 IEEE.

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DeePKS + ABACUS as a Bridge between Expensive Quantum Mechanical Models and Machine Learning Potentials

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Journal of Physical Chemistry A 2022年第49期126卷 9154-9164页

作者： Li, Wenfei Ou, Qi Chen, Yixiao Cao, Yu Liu, Renxi Zhang, Chunyi Zheng, Daye Cai, Chun Wu, Xifan Wang, Han Chen, Mohan Zhang, Linfeng AI for Science Institute Beijing100080 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States HEDPS CAPT College of Engineering School of Physics 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 DP Technology Beijing100080 China

Recently, the development of machine learning (ML) potentials has made it possible to perform large-scale and long-time molecular simulations with the accuracy of quantum mechanical (QM) models. However, for different levels of QM methods, such as density functional theory (DFT) at the meta-GGA level and/or with exact exchange, quantum Monte Carlo, etc., generating a sufficient amount of data for training an ML potential has remained computationally challenging due to their high cost. In this work, we demonstrate that this issue can be largely alleviated with Deep Kohn-Sham (DeePKS), an ML-based DFT model. DeePKS employs a computationally efficient neural network-based functional model to construct a correction term added upon a cheap DFT model. Upon training, DeePKS offers closely matched energies and forces compared with high-level QM method, but the number of training data required is orders of magnitude less than that required for training a reliable ML potential. As such, DeePKS can serve as a bridge between expensive QM models and ML potentials: one can generate a decent amount of high-accuracy QM data to train a DeePKS model and then use the DeePKS model to label a much larger amount of configurations to train an ML potential. This scheme for periodic systems is implemented in a DFT package ABACUS, which is open source and ready for use in various applications. © 2022 The Journal of Physical Chemistry A. All right reserved.

关键词： Density functional theory

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Level sets of quantum control landscapes

Level sets of quantum control landscapes

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4th International Symposium on Communications, Control, and Signal Processing, ISCCSP-2010

作者： Beltrani, Vincent Dominy, Jason Ho, Tak-San Rabitz, Herschel Department of Chemistry Princeton University NJ 08544 United States Program in Applied and Computational Mathematics Princeton University NJ 08544 United States

ISBN: (纸本)9781424462858

A controlled quantum system possesses a search landscape defined by the observable value as a functional of the control field. Within the search landscape, there exist level sets of controls giving the same observable value. This paper focuses on level sets of the transition probability P i→f. For transition probabilities 0 i→fi→f = 1, a second order D-MORPH algorithm is presented that can explore the perfect control level set. D-MORPH is utilized to identify level set members that exhibit certain desirable secondary characteristics, e.g., minimal pulse ftuence. Numerical simulations for finite level systems are presented to illustrate the variety of control behavior found across level set members. ©2010 IEEE.

关键词： Quantum optics

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