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Better approximations of high dimensional smooth functions by deep neural networks with rectified power units

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arXiv 2019年

作者： Li, Bo Tang, Shanshan Yu, Haijun NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China China Justice Big Data Institute Beijing100043 China

MSC Codes 65D15, 65M12, 65M15Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance Ε, by our constructive proofs, are in general O(log1Ε ) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function. Copyright © 2019, The Authors. All rights reserved.

关键词： Deep neural networks

An enhanced SDR based global algorithm for nonconvex complex quadratic programs with signal processing applications

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arXiv 2019年

作者： Lu, Cheng Liu, Ya-Feng Zhou, Jing School of Economics and Management North China Electric Power University Beijing102206 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics College of Science Zhejiang University of Technology Hangzhou310023 China

In this paper, we consider a class of nonconvex complex quadratic programming (CQP) problems, which find a broad spectrum of signal processing applications. By using the polar coordinate representations of the complex variables, we first derive a new enhanced semidefinite relaxation (SDR) for problem (CQP). Based on the newly derived SDR, we further propose an efficient branch-and-bound algorithm for solving problem (CQP). Key features of our proposed algorithm are: (1) it is guaranteed to find the global solution of the problem (within any given error tolerance);(2) it is computationally efficient because it carefully utilizes the special structure of the problem. We apply our proposed algorithm to solve the multiinput multi-output (MIMO) detection problem, the unimodular radar code design problem, and the virtual beamforming design problem. Simulation results show that our proposed enhanced SDR, when applied to the above problems, is generally much tighter than the conventional SDR and our proposed global algorithm can efficiently solve these problems. In particular, our proposed algorithm significantly outperforms the state-of-the-art sphere decode algorithm for solving the MIMO detection problem in the hard cases (where the number of inputs and outputs is equal or the SNR is low) and a state-of-the-art general-purpose global optimization solver called Baron for solving the virtual beamforming design problem. Copyright © 2019, The Authors. All rights reserved.

关键词： Beamforming

PowerNet: Efficient representations of polynomials and smooth functions by deep neural networks with rectified power units

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arXiv 2019年

作者： Li, Bo Tang, Shanshan Yu, Haijun Lsec Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 School OfMathematical Sciences University of Chinese Academy of Sciences Beijing100049 National Center ForMathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing100190 China

MSC Codes 65M12, 65M15, 65P40Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct corresponding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our constructive algorithms show clearly a close connection between spectral methods and deep neural networks: a PowerNet with n layers can exactly represent polynomials up to degree sn, where s is the power of RePUs. The proposed PowerNets have potential applications in the situations where high-accuracy is desired or smoothness is required. Copyright © 2019, The Authors. All rights reserved.

关键词： Deep neural networks

Conductivity in the Square Lattice Hubbard Model at High Temperatures: Importance of Vertex Corrections

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Physical Review Letters 2019年第3期123卷 036601-036601页

作者： J. Vučičević J. Kokalj R. Žitko N. Wentzell D. Tanasković J. Mravlje Scientific Computing Laboratory Center for the Study of Complex Systems Institute of Physics Belgrade University of Belgrade Pregrevica 118 11080 Belgrade Serbia University of Ljubljana Faculty of Civil and Geodetic Engineering Jamova 2 Ljubljana Slovenia Jozef Stefan Institute Jamova 39 SI-1000 Ljubljana Slovenia University of Ljubljana Faculty of Mathematics and Physics Jadranska 19 Ljubljana Slovenia Center for Computational Quantum Physics Simons Foundation Flatiron Institute New York New York 10010 USA

Recent experiments on cold atoms in optical lattices allow for a quantitative comparison of the measurements to the conductivity calculations in the square lattice Hubbard model. However, the available calculations do not give consistent results, and the question of the exact solution for the conductivity in the Hubbard model remained open. In this Letter, we employ several complementary state-of-the-art numerical methods to disentangle various contributions to conductivity and identify the best available result to be compared to experiment. We find that, at relevant (high) temperatures, the self-energy is practically local, yet the vertex corrections remain rather important, contrary to expectations. The finite-size effects are small even at the lattice size 4×4, and the corresponding Lanczos diagonalization result is, therefore, close to the exact result in the thermodynamic limit.

关键词： Electrical conductivity 2-dimensional systems Mott insulators Strongly correlated systems Dynamical mean field theory Exact diagonalization Hubbard model Lattice models in condensed matter Monte Carlo methods Numerical Renormalization Group

Double parameters full-waveform inversion for velocity and density in the time domain

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Double parameters full-waveform inversion for velocity and d...

CPS/SEG北京2018国际地球物理会议暨展览

作者： Wensheng Zhang UCAS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences

In this paper, the double parameters full-waveform inversion for velocity and density simultaneously in the time domain is investigated. Comparing with the single parameter inversion, the double parameters inversion is more difficult because of the interactive influences between velocity and density. The forward is solved by the efficient staggered-grid difference scheme with the perfectly matched absorbing layer. The inversion is an optimization iterative process based on LBFGS algorithm. The combination of L2 regularization and TV regularization and the stepwise inversion strategy from low frequency to high frequency are applied to overcome the ill-posedness. Moreover, the adaptive regularization parameter is adopted to improve inversion precision. The double parameters inversion for Marmousi model is completed. The numerical results show the double parameters inversion can yield good inversion results of velocity and density even in the case of noisy data. The numerical computations show the correctness and effectiveness of the method in this paper.

关键词： FWI Double parameters full-waveform inversion for velocity and density in the time domain

Electron spin mediated distortion in metallic systems

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arXiv 2019年

作者： Anand, G. Eisenbach, Markus Goodall, Russell Freeman, Colin L. Department of Metallurgy and Materials Engineering Indian Institute of Engineering Science and Technology-Shibpur Howrah WB India Department of Materials Science and Engineering University of Sheffield United Kingdom Warwick Centre for Predictive Modelling University of Warwick United Kingdom Scientific Computing Group Centre for Computational Sciences Oak Ridge National Laboratory Oak RidgeTN United States

The deviation of positions of atoms from their ideal lattice sites in crystalline solid state systems causes distortion and can lead to variation in structural [1] and functional properties [2]. Distortion in molecular systems has been traditionally understood in the term of Jahn-Teller distortion [3], while for the one-dimensional chain of metals, the Peierls distortion mechanism has been proposed [4]. In real three dimensional metallic systems, a fundamental description of the distortion is missing, which we need to design alloys with attractive structural properties. The present investigation presents the evolution of distortion in metallic systems in terms of magnetovolume effects arising due to magnetic ground-state of the system. Particularly the significant distortion in the case of Cr due to presence of other transition metals is seen. Cr with transition metals as a candidate system to study the correlation between charge disproportion, spin-flip, magnetovolume effects and Fermi surface nesting. This study provides a novel explanation of strengthening effect of Cr observed in alloys due to its unique magnetic properties. Copyright © 2019, The Authors. All rights reserved.

关键词： Magnetic moments

Two-grid based adaptive proper orthogonal decomposition algorithm for time dependent partial differential equations

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arXiv 2019年

作者： Dai, Xiaoying Kuang, Xiong Xin, Jack Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Department of Mathematics University of California at Irvine IrvineCA92697 United States

In this article, we propose a two-grid based adaptive proper orthogonal decomposition(POD) algorithm to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new algorithm is cheap and easy to implement. We implement our new method to the solution of time-dependent advection-diffusion equations with Kolmogorov flow and ABC flow. The numerical results show that our method is more efficient than the existing POD algorithms. Copyright © 2019, The Authors. All rights reserved.

关键词： Principal component analysis

On optimal finite element schemes for biharmonic equation

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arXiv 2018年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China

In this paper, two nonconforming finite element schemes that use piecewise cubic and piecewise quartic polynomials respectively are constructed for the planar biharmonic equation with optimal convergence rates on general shape-regular triangulations. Therefore, it is proved that optimal finite element schemes of arbitrary order for planar biharmonic equation can be constructed on general shape-regular triangulations. Copyright © 2018, The Authors. All rights reserved.

关键词： Finite element method

Dimensional scaling of flame propagation in discrete particulate clouds

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arXiv 2019年

作者： Lam, Fredric Y.K. Mi, XiaoCheng Higgins, Andrew J. Institute for Computational and Mathematical Engineering Stanford University StanfordCA94305 United States Laboratory for Scientific Computing Cavendish Laboratory Department of Physics University of Cambridge CambridgeCB3 0HE United Kingdom Department of Mechanical Engineering McGill University MontrealQCH3A 0C3 Canada

The critical dimension necessary for a flame to propagate in suspensions of fuel particles in oxidizer is studied analytically and numerically. Two types of models are considered: First, a continuum model, wherein the individual particulate sources are not resolved and the heat release is assumed spatially uniform, is solved via conventional finite difference techniques. Second, a discrete source model, wherein the heat diffusion from individual sources is modeled via superposition of the Green’s function of each source, is employed to examine the influence of the random, discrete nature of the media. Heat transfer to cold, isothermal walls and to a layer of inert gas surrounding the reactive medium are considered as the loss mechanisms. Both cylindrical and rectangular (slab) geometries of the reactive medium are considered, and the flame speed is measured as a function of the diameter and thickness of the domains, respectively. In the continuum model with inert gas confinement, a universal scaling of critical diameter to critical thickness near 2:1 is found. In the discrete source model, as the time scale of heat release of the sources is made small compared to the interparticle diffusion time, the geometric scaling between cylinders and slabs exhibits values greater than 2:1. The ability of the flame in the discrete regime to propagate in thinner slabs than predicted by continuum scaling is attributed to the flame being able to exploit local fluctuations in concentration across the slab to sustain propagation. As the heat release time of the sources is increased, the discrete source model reverts back to results consistent with the continuum model. Implications of these results for experiments are discussed. Copyright © 2019, The Authors. All rights reserved.

关键词： Solvents