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一种基于神经网络的方法,用于解决时间分数扩散波浪方程
Z Soori1, H Azin2, A Habibirad2
1Faculty of Mathematics, K. N Toosi University of Technology, Iran.
本研究应用神经网络 (NN) 来解决时间微分扩散波方程 (TFDWE). 深度神经网络 (DNN) 方法,使用亚当优化,有效地处理复杂的分数计算问题.
科学领域:
- 计算数学 计算数学 计算数学
- 应用数学 应用数学 应用数学
- 数字分析 数字分析
背景情况:
- 时间微分扩散波方程 (TFDWE) 模拟了各种科学学科的复杂现象.
- 解决TFDWE,特别是使用α (1 < α < 2) 顺序的卡普托衍生品,带来了重大的数值挑战.
- 现有的数值方法可能会在不规则的域和高维问题上扎.
研究的目的:
- 开发和应用一个新的深度神经网络 (DNN) 框架来解决TFDWE.
- 实现一个数值方案,以 O ((t3-α) 精度近似卡普托导数.
- 用亚当优化算法对二维不规则域验证DNN方法.
主要方法:
- 设计了一个结构化的多层神经网络 (NN) 模型,具有输入,隐藏和输出层.
- 一个试验解决方案是条件满足项和一个NN输出项的和.
- 反向传播算法和梯度下降被用来通过最小化损失函数来训练NN.
主要成果:
- 拟议的DNN框架成功解决了不规则域名的TFDWE.
- 讨论了TFDWE的梯度下降算法的对汇率分析.
- 数字实验证明了该方法的有效性,比较了Sigmoid和ReLU激活函数.
结论:
- 开发的DNN方法提供了一种有效和准确的方法来解决时间微分扩散波方程.
- 使用Adam优化可以提高神经网络的性能.
- 该研究验证了DNN在处理二维复杂的分数微分方程中的能力.
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