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A predictor-corrector deep learning algorithm for high dimensional stochastic partial differential equations

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arXiv 2022年

作者： Zhang, He Zhang, Ran Zhou, Tao School of Mathematics Jilin University Jilin Changchun130012 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely, we decompose the original SPDE into a degenerate SPDE and a deterministic PDE. Then in the prediction step, we solve the degenerate SPDE with the Euler scheme, while in the correction step we solve the second-order deterministic PDE by deep neural networks via its equivalent backward stochastic differential equation (BSDE). Under standard assumptions, error estimates and the rate of convergence of the proposed algorithm are presented. The efficiency and accuracy of the proposed algorithm are illustrated by numerical examples. Copyright © 2022, The Authors. All rights reserved.

关键词： Numerical methods

An extended plane wave framework for the electronic structure calculations of twisted bilayer material systems

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arXiv 2023年

作者： Dai, Xiaoying Zhou, Aihui Zhou, Yuzhi 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 CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

In this paper, we propose an extended plane wave framework to make the electronic structure calculations of the twisted bilayer 2D material systems practically feasible. Based on the foundation in [Y. Zhou, H. Chen, A. Zhou, J. Comput. Phys. 384, 99 (2019)], following extensions take place: (1) an tensor-producted basis set, which adopts PWs in the incommensurate dimensions, and localized basis in the interlayer dimension, (2) a practical application of a novel cutoff techniques we have recently developed, and (3) a quasi-band structure picture under the small twisted angles and weak interlayer coupling limits. With (1) and (2) now the dimensions of Hamiltonian matrix are reduced by about 2 orders of magnitude compared with the original framework. And (3) enables us to better organize the calculations and understand the results. For numerical examples, we study the electronic structures of the linear bilayer graphene lattice system with the magic twisted angle (∼ 1.05◦). The famous flat bands have been reproduced with their features in quantitative agreement with those from experiments and other theoretical calculations. Moreover, the extended framework has much less computational cost compared to the commensurate cell approximations, and is more extendable compared to the traditional model hamiltonians and tight binding models. Finally this framework can readily accommodate nonlinear models thus will laid the foundations for more effective yet accurate Density Functional Theory (DFT) calculations. © 2023, CC BY.

关键词： Electronic structure

Localization in the Incommensurate Systems: A Plane Wave Study via Effective Potentials

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arXiv 2023年

作者： Wang, Ting Zhou, Yuzhi Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100094 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper, we apply the effective potentials in the localization landscape theory (Filoche et al., 2012, Arnold et al., 2016) to study the spectral properties of the incommensurate systems. We uniquely develop a plane wave method for the effective potentials of the incommensurate systems and utilize that, the localization of the electron density can be inferred from the effective potentials. Moreover, we show that the spectrum distribution can also be obtained from the effective potential version of Weyl’s law. We perform some numerical experiments on some typical incommensurate systems, showing that the effective potential provides an alternative tool for investigating the localization and spectrum distribution of the systems. Copyright © 2023, The Authors. All rights reserved.

关键词： Elastic waves

Evidence for validity and reliability of a research-based assessment instrument on measurement uncertainty

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Physical Review Physics Education Research 2024年第2期20卷 020125页

作者： Gayle Geschwind Michael Vignal Marcos D. Caballero H. J. Lewandowski National Institute of Standards and Technology and the University of Colorado Boulder Colorado 80309 USA Department of Physics Department of Physics and Astronomy and CREATE for STEM Institute Department of Computational Mathematics Science and Engineering Department of Physics and Center for Computing in Science Education

The Survey of Physics Reasoning on Uncertainty Concepts in Experiments (SPRUCE) was designed to measure students’ proficiency with measurement uncertainty concepts and practices across ten different assessment objectives to help facilitate the improvement of laboratory instruction focused on this important topic. To ensure the reliability and validity of this assessment, we conducted a comprehensive statistical analysis using classical test theory. This analysis includes an evaluation of the test as a whole, as well as an in-depth examination of individual items and assessment objectives. We make use of a previously reported on scoring scheme involving pairing items with assessment objectives, creating a new unit for statistical analysis referred to as a “couplet.” The findings from our analysis provide evidence for the reliability and validity of SPRUCE as an assessment tool for undergraduate physics labs. This increases both instructors’ and researchers’ confidence in using SPRUCE for measuring students’ proficiency with measurement uncertainty concepts and practices to ultimately improve laboratory instruction. Additionally, our results using couplets and assessment objectives demonstrate how these can be used with traditional classic test theory analysis.

关键词： Assessment Research methodology

A RIEMANNIAN EXPONENTIAL AUGMENTED LAGRANGIAN METHOD FOR computing THE PROJECTION ROBUST WASSERSTEIN DISTANCE

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arXiv 2022年

作者： Jiang, Bo Liu, Ya-Feng Key Laboratory for NSLSCS of Jiangsu Province School of Mathematical Sciences Nanjing Normal University Nanjing 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 Beijing China

Projecting the distance measures onto a low-dimensional space is an efficient way of mitigating the curse of dimensionality in the classical Wasserstein distance using optimal transport. The obtained maximized distance is referred to as projection robust Wasserstein (PRW) distance. In this paper, we equivalently reformulate the computation of the PRW distance as an optimization problem over the Cartesian product of the Stiefel manifold and the Euclidean space with additional nonlinear inequality constraints. We propose a Riemannian exponential augmented Lagrangian method (ReALM) with a global convergence guarantee to solve this problem. Compared with the existing approaches, ReALM can potentially avoid too small penalty parameters. Moreover, we propose a framework of inexact Riemannian gradient descent methods to solve the subproblems in ReALM efficiently. In particular, by using the special structure of the subproblem, we give a practical algorithm named as the inexact Riemannian Barzilai-Borwein method with Sinkhorn iteration (iRBBS). The remarkable features of iRBBS lie in that it performs a flexible number of Sinkhorn iterations to compute an inexact gradient with respect to the projection matrix of the problem and adopts the Barzilai-Borwein stepsize based on the inexact gradient information to improve the performance. We show that iRBBS can return an ǫ-stationary point of the original PRW distance problem within O(ǫ−3) iterations. Extensive numerical results on synthetic and real datasets demonstrate that our proposed ReALM as well as iRBBS outperform the state-of-the-art solvers for computing the PRW *** Codes 65K10, 90C26, 90C47 Copyright © 2022, The Authors. All rights reserved.

关键词： Constrained optimization

A Non-Isothermal Phase-Field Crystal Model with Lattice Expansion: Analysis and Benchmarks

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arXiv 2024年

作者： Punke, Maik Salvalaglio, Marco Voigt, Axel Wise, Steven M. Institute of Scientific Computing TU Dresden Dresden01062 Germany Dresden Center for Computational Materials Science TU Dresden Dresden01062 Germany Department of Mathematics The University of Tennessee KnoxvilleTN37996 United States

We introduce a non-isothermal phase-field crystal model including heat flux and thermal expansion of the crystal lattice. The thermal compatibility condition, as well as a positive entropy-production property, is derived analytically and further verified by numerical benchmark simulations. Furthermore, we examine how the different model parameters control density and temperature evolution during dendritic solidification through extensive parameter studies. Finally, we extend our framework to the modeling of open systems considering external mass and heat fluxes. This work sets the ground for a comprehensive mesoscale model of non-isothermal solidification including thermal expansion within a positive entropy-producing framework, and provides a benchmark for further meso- to macroscopic modeling of solidification. Copyright © 2024, The Authors. All rights reserved.

关键词： Crystal lattices

A Finite Element Framework for Solving the Density-Gradient Model

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A Finite Element Framework for Solving the Density-Gradient ...

International Conference on Simulation of Semiconductor Processes and Devices (SISPAD)

作者： Pengcong Mu Tao Cui Lijun Xu Kun Luo Zhiqiang Li Zhenhua Wu Institute of Microelectronics Chinese Academy of Sciences Beijing China University of Chinese Academy of Sciences Beijing China LSEC NCMIS Institute of Computational Mathematics and Scientific Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China School of Pyhsics Zhejiang University Hangzhou China

ISBN: (数字)9798331516352

ISBN: (纸本)9798331516369

This paper presents a positive-preserving, stable finite element scheme for the density-gradient (DG) model. An ef-ficient Newton-Krylov solver is designed to address the nonlinear coupled discrete system. Simulation results for gate-all-around (GAA) devices demonstrate the robustness and effectiveness of the proposed scheme. Notably, the simulation time using our method is only half that of the Sentaurus Device, highlighting its superior computational efficiency.

关键词： Semiconductor device modeling computational modeling Simulation Gallium arsenide Field effect transistors Logic gates Robustness Semiconductor process modeling Finite element analysis computational efficiency

Smooth approximation of feedback laws for infinite horizon control problems with non-smooth value functions

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arXiv 2023年

作者： Kunisch, Karl Vásquez-Varas, Donato Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstraße 69 Upper Austria Linz4040 Austria Institute of Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Styria Graz8010 Austria

In this work the synthesis of approximate optimal and smooth feedback laws for infinite horizon optimal control problems is addressed. In this regards, Lp type error bounds of the approximating smooth feedback laws are derived, depending on either the C1 norm of the value function or its semi-concavity. These error bounds combined with the existence of a Lyapunov type function are used to prove the existence of an approximate optimal sequence of smooth feedback laws. Moreover, we extend this result to the Hölder continuous case by a diagonalization argument combined with the Moreau envelope. It is foreseen that these error bounds could be applied to study the convergence of synthesis of feedback laws via data driven machine learning methods. Additionally, we provide an example of an infinite horizon optimal control problem for which the value functions is non-differentiable but Lipschitz continuous. We point out that in this example no restrictions on either the controls or the trajectories are *** Codes 49L12, 49L25, 49N35, 93B52, 35D40 © 2023, CC BY.

关键词： Learning systems

Novel semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems

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arXiv 2024年

作者： Hong, Jialin Hou, Baohui 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 Department of Mathematics Shanghai University Shanghai200444 China Academy for Multidisciplinary Studies Capital Normal University Beijing100048 China

In this manuscript, we propose efficient stochastic semi-explicit symplectic schemes tailored for nonseparable stochastic Hamiltonian systems (SHSs). These semi-explicit symplectic schemes are constructed by introducing augmented Hamiltonians and using symmetric projection. In the case of the artificial restraint in augmented Hamiltonians being zero, the proposed schemes also preserve quadratic invariants, making them suitable for developing semi-explicit charge-preserved multi-symplectic schemes for stochastic cubic Schrödinger equations with multiplicative noise. Through numerical experiments that validate theoretical results, we demonstrate that the proposed stochastic semi-explicit symplectic scheme, which features a straightforward Newton iteration solver, outperforms the traditional stochastic midpoint scheme in terms of effectiveness and accuracy. Copyright © 2024, The Authors. All rights reserved.

关键词： Hamiltonians