基于物理信息的神经网络的高阶KdV方程家族的数据驱动解决方案和参数估计
Jiajun Chen1, Jianping Shi2, Ao He1
1Department of Mathematics, Kunming University of Science and Technology, Kunming, 650500, Yunnan, People's Republic of China.
Scientific reports
|October 12, 2024
概括
与tanh相比,具有正弦激活函数的物理信息神经网络 (PINNs) 为解决更高阶KdV方程提供了更高的精度. 这种深度学习方法增强了非线性部分微分方程的解决方案和参数估计.
科学领域:
- 计算物理 计算物理
- 应用数学 应用数学 应用数学
- 机器学习 机器学习
背景情况:
- 非线性局部微分方程 (NLPDEs) 在建模复杂的物理现象中至关重要.
- 基于物理学的神经网络 (PINNs) 将物理定律与数据驱动的方法结合起来,用于解决PDEs.
- 在PINNs中选择激活函数可以显著影响学习PDE解决方案的性能.
研究的目的:
- 调查使用tanh和正弦激活函数的两个PINN方法的有效性.
- 为了比较它们在解决高阶KdV方程家族的前向和反向问题的表现.
- 评估它们在数据驱动解决方案和参数估计方面的能力.
主要方法:
- 实施两个物理信息神经网络 (PINN) 框架.
- 使用过度的触角 (tanh) 和正弦作为不同的激活函数.
- 应用于一个高阶的科尔特韦格-德弗里斯 (KdV) 方程家族,用于解决和参数估计.
主要成果:
- 与基于tanh的PINN相比,具有正弦激活功能的PINN框架在学习各种解决方案 (solitons,周期波,扭曲) 中取得了更高的精度.
- 弦激活函数在参数估计任务中表现出卓越的性能.
- 发现方程的复杂性会影响PINN方法的准确性和计算效率.
结论:
- 使用正弦激活函数的PINN在解决高阶KdV方程方面比使用tanh的PINN更有效.
- 正弦激活函数提高了解决方案的准确性和参数估计能力.
- 这项研究促进了对复杂的NLPDEs的深度学习的应用,特别是在科学建模和模拟中.
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