利用机器学习对非线性薛定谔方程进行粗网格模拟

IF 2.3 3区 数学 Q1 MATHEMATICS Mathematics Pub Date : 2024-09-09 DOI:10.3390/math12172784
Benjamin F. Akers, Kristina O. F. Williams
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引用次数: 0

摘要

本研究开发了一种在粗空间网格上演化非线性薛定谔方程的数值方法。该方法训练神经网络生成最佳模版权重,以离散化非线性薛定谔方程解的二次导数。神经网络被嵌入到一个对称矩阵中,以控制该方案的特征值,从而确保稳定性。机器学习方法的性能优于其母有限差分法和傅立叶谱法。训练后的方案与其母有限差分法具有相同的渐近运算成本。与传统方法不同的是,其性能取决于初始数据与训练集的接近程度。
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Coarse-Gridded Simulation of the Nonlinear Schrödinger Equation with Machine Learning
A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schrödinger equation. The neural network is embedded in a symmetric matrix to control the scheme’s eigenvalues, ensuring stability. The machine-learned method can outperform both its parent finite difference method and a Fourier spectral method. The trained scheme has the same asymptotic operation cost as its parent finite difference method after training. Unlike traditional methods, the performance depends on how close the initial data are to the training set.
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来源期刊
Mathematics
Mathematics Mathematics-General Mathematics
CiteScore
4.00
自引率
16.70%
发文量
4032
审稿时长
21.9 days
期刊介绍: Mathematics (ISSN 2227-7390) is an international, open access journal which provides an advanced forum for studies related to mathematical sciences. It devotes exclusively to the publication of high-quality reviews, regular research papers and short communications in all areas of pure and applied mathematics. Mathematics also publishes timely and thorough survey articles on current trends, new theoretical techniques, novel ideas and new mathematical tools in different branches of mathematics.
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