PINNs文章框架和简单思路介绍

作者: Neural_PDE | 来源:发表于2021-11-09 16:22 被阅读0次

1 inference数值解

1.1连续时间模型 Shrödinger Equation

$\begin{aligned} &i h_{t}+0.5 h_{x x}+|h|^{2} h=0, \quad x \in[-5,5], \quad t \in[0, \pi / 2] \\ &h(0, x)=2 \operatorname{sech}(x) \\ &h(t,-5)=h(t, 5) \\ &h_{x}(t,-5)=h_{x}(t, 5) \end{aligned}$
上式分为三个部分，原方程f，初始条件0 和边界条件b
$M S E=M S E_{0}+M S E_{b}+M S E_{f}$

1.2离散时间模型 Allen-Cahn Equation

2 identification发现原方程

2.1连续时间模型 Navier-Stokes Equation

2.2离散时间模型 Korteweg–de Vries Equation

3 附录

Burgers’ Equation:

$\begin{aligned} &u_{t}+u u_{x}-(0.01 / \pi) u_{x x}=0, \quad x \in[-1,1], \quad t \in[0,1] \\ &u(0, x)=-\sin (\pi x) \\ &u(t,-1)=u(t, 1)=0 \end{aligned}$
上式分为三个部分，原方程f，初边值条件u (初始条件0 和边界条件b)
$M S E=M S E_{u}+M S E_{f}\\ M S E_{u}=\frac{1}{N_{u}} \sum_{i=1}^{N_{u}}\left|u\left(t_{u}^{i}, x_{u}^{i}\right)-u^{i}\right|^{2}\\ M S E_{f}=\frac{1}{N_{f}} \sum_{i=1}^{N_{f}}\left|f\left(t_{f}^{i}, x_{f}^{i}\right)\right|^{2}$
这里作者将初边值写在一起，原文这样叙述：Here, $\left\{t_{u}^{i}, x_{u}^{i}, u^{i}\right\}_{i=1}^{N_{u}}$ denote the initial and boundary training data on $u(t, x)$ .
这里作者将初边值写在一起我认为是由于
$M S E_{u}=M S E_{0}+M S E_{b}\\M S E_{0}=\frac{1}{N_{0}} \sum_{i=1}^{N_{0}}\left|u\left(0, x_{0}^{i}\right)-u_{0}^{i}\right|^{2}\\ M S E_{b}=\frac{1}{N_{b}} \sum_{i=1}^{N_{b}}(\left|u\left(t_{b}^{i}, 1\right)-u_{b}^{i}\right|^{2}+\left|u\left(t_{b}^{i}, -1\right)-u_{b}^{i}\right|^{2})$
所以
$M S E_{u}=M S E_{0}+M S E_{b}\\ =\frac{1}{N_{u}} \sum_{i=1}^{N_{u}}(\left| u\left(0, x_{0}^{i}\right)-u_{0}^{i}+ \right|^{2}+\left|u\left(t_{b}^{i}, 1\right)-u_{b}^{i}\right|^{2}+\left|u\left(t_{b}^{i}, -1\right)-u_{b}^{i}\right|^{2})\\ =\frac{1}{N_{u}} \sum_{i=1}^{N_{u}}\left|u\left(t_{u}^{i}, x_{u}^{i}\right)-u^{i}\right|^{2}$