forked from p94628173/idrlnet
docs: Fix a typo in the inverse wave problem
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@ -3,7 +3,7 @@ Consider the 1d wave equation:
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$$
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\begin{equation}
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\frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2},
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\frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2},
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\end{equation}
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$$
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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.
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@ -11,7 +11,7 @@ Then the problem is formulated as:
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$$
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\min_{u,c} \sum_{i=1,2,\cdots,N} \|u(x_i, t_i)-u_i\|^2\\
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s.t. \frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2}
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s.t. \frac{\partial^2u}{\partial t^2}=c^2\frac{\partial^2u}{\partial x^2}
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$$
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In the context of PINN, $u$ is parameterized to $u_\theta$.
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@ -20,7 +20,7 @@ The problem above is transformed to the discrete form:
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$$
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\min_{\theta,c}
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w_1\sum_{i=1,2,\cdots,N} \|u_\theta(x_i, t_i)-u_i\|^2
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+w_2\sum_{i=1,2,\cdots,M}\left|\frac{\partial^2u_\theta(x_i,t_i)}{\partial t^2}-c\frac{\partial^2u_\theta(x_i,t_i)}{\partial x^2}\right|^2.
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+w_2\sum_{i=1,2,\cdots,M}\left|\frac{\partial^2u_\theta(x_i,t_i)}{\partial t^2}-c^2\frac{\partial^2u_\theta(x_i,t_i)}{\partial x^2}\right|^2.
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$$
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## Importing External Data
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