idrlnet/docs/user/get_started/3_burgers_equation.md

# Burgers' Equation
Burgers' equation is formulated as following:

$$
\begin{equation}
\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\nu \frac{\partial^{2} u}{\partial x^{2}}
\end{equation}
$$
We have added the template of the equation into `idrlnet.pde_op.equations`.
In this example, we take $\nu=-0.01/\pi$, and the problem is

$$
\begin{equation}
\begin{array}{l}
u_t+u u_{x}-(0.01 / \pi) u_{x x}=0, \quad x \in[-1,1], \quad t \in[0,1] \\
u(0, x)=-\sin (\pi x) \\
u(t,-1)=u(t, 1)=0
\end{array}
\end{equation}.
$$

## Time-dependent Domain
The equation is time-dependent. In addition, we define a time symbol `t` and its range.
```python
t_symbol = Symbol('t')
time_range = {t_symbol: (0, 1)}
```
The parameter range `time_range` will be passed to methods `geo.Geometry.sample_interior()` and `geo.Geometry.sample_boundary()`.
The sampling methods generate samples containing the additional dims provided in `param_ranges.keys()`.
```python
# Interior domain
points = geo.sample_interior(10000, bounds={x: (-1., 1.)}, param_ranges=time_range)

# Initial value condition
points = geo.sample_interior(100, param_ranges={t_symbol: 0.0})

# Boundary condition
points = geo.sample_boundary(100, param_ranges=time_range)
```

The result is shown as follows:

![burgers](https://raw.githubusercontent.com/weipeng0098/picture/master/20210617081844.png)

## Use TensorBoard
To monitor the training process, we employ [TensorBoard](https://www.tensorflow.org/tensorboard). 
The learning rate, losses on different domains, and the total loss will be recorded automatically.
Users can call `Solver.summary_receiver()` to get the instance of `SummaryWriter`.
As default, one starts TensorBoard at `./network_idr`:
```bash
tensorboard --logdir ./network_dir
```
Users can monitor the status of training:

![tensorboard](https://raw.githubusercontent.com/weipeng0098/picture/master/20210617081853.png)


See `examples/burgers_equation`.