Docs: Add Example for Boussinesq Equation

docs/src/examples/pde/boussinesq.md

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+## Discovering Unknown Closure Term for Approximated Boussinesq Equation using Universal Partial Differential Equations
+
+## Introduction
+The Boussinesq equations, which are derived from simplifying incompressible Navier-Stokes equations, are often used in climate modelling. In this documentation, we solve the **Universal Partial Differential Equation (UPDE)** by training a neural network with generated data to discover the unknown function in the UPDE, instead of a conventional approach, which is to manually approximate the function by physical laws.
+
+
+## The Approximated Boussinesq Equation without Closure
+By an approximation of Boussinesq equations, we obtain a local advection-diffusion equation describing the evolution of the horizontally-averaged temperature $\overline{T}$:
+
+$$\frac{\partial \overline{T}}{\partial t} + \frac{\partial \overline{wT}}{\partial z} = \kappa \frac{\partial^2 \overline{T}}{\partial z^2}$$
+
+where $\overline{T}(z, t)$ is the horizontally-averaged temperature, $\kappa$ is the thermal diffusivity, and $\overline{wT}$ is the horizontal average temperature flux in the vertical direction.
+
+
+Since $\overline{wT}$ is unknown, this one-dimensional approximating system is not closed. Instead of closing the system manually by determining an approximating $\overline{wT}$ from ad-hoc models, physical reasoning and scaling laws, we can use an UDE-automated approach to approximate $\overline{wT}$ from data. We let 
+
+$$\overline{wT} = {U}_\theta \left( \mathbf{P}, \overline{T}, \frac{\partial \overline{T}}{\partial z} \right)$$
+
+where $P$ are the physical parameters, $\overline{T}$ is the averaged temperature, and $\frac{\partial \overline{T}}{\partial z}$ is its gradient.
+
+
+## Generating Data for Training
+
+To train the neural network, we can generate data using the function $\overline{wT} = cos(sin(T^3)) + sin(cos(T^2))$ with $N$ spatial points discretized by a finite difference method, with the time domain $t \in [0,1.5]$ and Neumann zero-flux boundary conditions, meaning $\frac{\partial \overline{T}}{\partial z} = 0$ at the edges.
+
+
+## Training Neural Network
+We train a neural network with two hidden layers, each of size 8, and with tanh activation functions against 30 data points sampled from the true PDE. 
+
+The ADAM optimizer is used to fit the UPDE. Learning rate $10^{−2}$ for 200 iterations and then ADAM with a learning rate of $10^{−3}$ for 1000 iterations.
+
+
+## Results and Conclusion