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docs: Fix a typo in the inverse wave problem

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weipengOO98 2021-08-04 17:00:17 +08:00
parent 3493e5068f
commit 0cc817b15b
1 changed files with 3 additions and 3 deletions
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6
docs/user/get_started/5_inverse_wave_equation.md
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@ -3,7 +3,7 @@ Consider the 1d wave equation:
$$ $$
\begin{equation} \begin{equation}
\frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2}, \frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2},
\end{equation} \end{equation}
$$ $$
where $c>0$ is unknown and is to be estimated. A group of data pairs $\{x_i, t_i, u_i\}_{i=1,2,\cdot,N}$ is observed. where $c>0$ is unknown and is to be estimated. A group of data pairs $\{x_i, t_i, u_i\}_{i=1,2,\cdot,N}$ is observed.
@ -11,7 +11,7 @@ Then the problem is formulated as:
$$ $$
\min_{u,c} \sum_{i=1,2,\cdots,N} \|u(x_i, t_i)-u_i\|^2\\ \min_{u,c} \sum_{i=1,2,\cdots,N} \|u(x_i, t_i)-u_i\|^2\\
s.t. \frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2} s.t. \frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}
$$ $$
In the context of PINN, $u$ is parameterized to $u_\theta$. In the context of PINN, $u$ is parameterized to $u_\theta$.
@ -20,7 +20,7 @@ The problem above is transformed to the discrete form:
$$ $$
\min_{\theta,c} \min_{\theta,c}
w_1\sum_{i=1,2,\cdots,N} \|u_\theta(x_i, t_i)-u_i\|^2 w_1\sum_{i=1,2,\cdots,N} \|u_\theta(x_i, t_i)-u_i\|^2
+w_2\sum_{i=1,2,\cdots,M}\left|\frac{\partial^2u_\theta(x_i,t_i)}{\partial t^2}-c\frac{\partial^2u_\theta(x_i,t_i)}{\partial x^2}\right|^2. +w_2\sum_{i=1,2,\cdots,M}\left|\frac{\partial^2u_\theta(x_i,t_i)}{\partial t^2}-c^2\frac{\partial^2u_\theta(x_i,t_i)}{\partial x^2}\right|^2.
$$ $$
## Importing External Data ## Importing External Data