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An Inexact Augmented Lagrangian Algorithm for Training Leaky ReLU Neural Network with Group Sparsity

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arXiv 2022年

作者： Liu, Wei Liu, Xin Chen, Xiaojun Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The leaky ReLU network with a group sparse regularization term has been widely used in the recent years. However, training such network yields a nonsmooth nonconvex optimization problem and there exists a lack of approaches to compute a stationary point deterministically. In this paper, we first resolve the multi-layer composite term in the original optimization problem by introducing auxiliary variables and additional constraints. We show the new model has a nonempty and bounded solution set and its feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Moreover, we show the relationship between the new model and the original problem. Remarkably, we propose an inexact augmented Lagrangian algorithm for solving the new model, and show the convergence of the algorithm to a KKT point. Numerical experiments demonstrate that our algorithm is more efficient for training sparse leaky ReLU neural networks than some well-known algorithms. Copyright © 2022, The Authors. All rights reserved.

关键词： Constrained optimization

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

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arXiv 2022年

作者： Bruno, Oscar P. Yin, Tao Department of Computing & Mathematical Sciences California Institute of Technology 1200 East California Blvd. CA91125 United States LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems;2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1);3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation;and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed *** Codes 35L05, 65M80, 65T99, 65R20 Copyright © 2022, The Authors. All rights reserved.

关键词： Wave equations

A New Mixed Finite Element Method with Exponential Coefficients for Solving Steady-State Poisson-Nernst-Planck Equations

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SSRN

SSRN 2024年

作者： Shen, Ruigang Yang, Ying Wang, Qin Lu, Benzhuo Guangxi Colleges Universities Key Laboratory of Data Analysis and Computation Guilin University of Electronic Technology Guangxi Guilin541004 China 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

In this work, a new mixed finite element method with exponential coefficients is produced for solving the Poisson-Nernst-Planck (PNP) equations. Based on the exponential properties of the PNP equation, the current (flux) density function is directly introduced through the Slotboom transformation to transform the original PNP equation into a new PNP system with exponential coefficients. The BDM mixed finite element method is used to discretize the Nernst-Planck (NP) equation based on the new variables. The electrostatic potential Poisson equation is still discretized using the standard finite element method. The well-posedness and error estimates of the new mixed finite element solutions are derived. Numerical experiments are presented to verify the optimal order estimation, fluxes conservation, and the effectiveness and reliability of the algorithm for hybrid discrete systems. © 2024, The Authors. All rights reserved.

关键词： Poisson equation

Novel structure-preserving schemes for stochastic Klein–Gordon–Schrödinger equations with additive noise

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arXiv 2023年

作者： Hong, Jialin Hou, Baohui Sun, Liying Zhang, Xiaojing 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 Shanghai University Shanghai200444 China Capital Normal University Beijing100048 China

Stochastic Klein–Gordon–Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete scheme for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings. © 2023, CC BY.

关键词： Runge Kutta methods

A Deep Learning Method for computing Eigenvalues of the Fractional Schrödinger Operator

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arXiv 2023年

作者： Yixiao, Guo Pingbing, Ming LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator. Our approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem. These improvements enable our method to handle both high-dimensional problems and problems posed on irregular bounded domains. We successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators. As an application, we share a conjecture to the fractional order isospectral problem that has not yet been studied. Copyright © 2023, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

STRONG ERROR ANALYSIS OF EULER METHODS FOR OVERDAMPED GENERALIZED LANGEVIN EQUATIONS WITH FRACTIONAL NOISE: NONLINEAR CASE

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arXiv 2022年

作者： Dai, Xinjie Hong, Jialin Sheng, Derui Zhou, Tau 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

This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the fractional Brownian motion and the singular kernel, which is overcome by means of the Malliavin calculus and fine estimates of several multiple singular integrals. Consequently, these two methods are proved to be strongly convergent with order nearly min{2(H + α − 1), α}, where H ∈ (1/2, 1) and α ∈ (1 − H, 1) respectively characterize the singularity levels of fractional noises and singular kernels in the underlying equation. This result improves the existing convergence order H + α − 1 of Euler methods for the nonlinear case, and gives a positive answer to the open problem raised in [4]. As an application of the theoretical findings, we further investigate the complexity of the multilevel Monte Carlo simulation based on the fast Euler method, which turns out to behave better performance than the standard Monte Carlo simulation when computing the expectation of functionals of the considered *** Codes 65C20, 65C30, 65C05, 60H07 Copyright © 2022, The Authors. All rights reserved.

关键词： Monte Carlo methods

SOLVING TIME DEPENDENT FOKKER-PLANCK EQUATIONS VIA TEMPORAL NORMALIZING FLOW

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arXiv 2021年

作者： Feng, Xiaodong Zeng, Li Zhou, Tao LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach. Copyright © 2021, The Authors. All rights reserved.

关键词： Fokker Planck equation

Bridging Fairness Gaps: A (Conditional) Distance Covariance Perspective in Fairness Learning

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arXiv 2024年

作者： Huang, Ruifan Liu, Haixia School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan Hubei China School of Mathematics and Statistics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan China

We bridge fairness gaps from a statistical perspective by selectively utilizing either conditional distance covariance or distance covariance statistics as measures to assess the independence between predictions and sensitive attributes. We enhance fairness by incorporating sample (conditional) distance covariance as a manageable penalty term into the machine learning process. Additionally, we present the matrix form of empirical (conditional) distance covariance for parallel calculations to enhance computational efficiency. Theoretically, we provide a proof for the convergence between empirical and population (conditional) distance covariance, establishing necessary guarantees for batch computations. Through experiments conducted on a range of real-world datasets, we have demonstrated that our method effectively bridges the fairness gap in machine learning. Copyright © 2024, The Authors. All rights reserved.

关键词： Contrastive Learning

Three Kinds of Novel Multi-Symplectic Methods for Stochastic Hamiltonian Partial Differential Equations

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SSRN

SSRN 2022年

作者： Hong, Jialin Hou, Baohui Li, Qiang Sun, Liying 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

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods. © 2022, The Authors. All rights reserved.

关键词： Stochastic systems