export tutorials changed in db9df8b

tutorials/tutorial3/tutorial.py

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 # coding: utf-8
 
 # # Tutorial: Applying Hard Constraints in PINNs to solve the Wave Problem
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+#
 # [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial3/tutorial.ipynb)
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 # In this tutorial, we will present how to solve the wave equation using **hard constraint Physics-Informed Neural Networks (PINNs)**. To achieve this, we will build a custom `torch` model and pass it to the **PINN solver**.
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 # First of all, some useful imports.
 
 # In[ ]:
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 warnings.filterwarnings("ignore")
 
 
-# ## The problem definition 
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+# ## The problem definition
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 # The problem is described by the following system of partial differential equations (PDEs):
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 # \begin{equation}
 # \begin{cases}
 # \Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
@@ -48,9 +48,9 @@ warnings.filterwarnings("ignore")
 # u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
 # \end{cases}
 # \end{equation}
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 # Where:
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 # - $D$ is a square domain $[0, 1]^2$.
 # - $\Gamma_i$, where $i = 1, \dots, 4$, are the boundaries of the square where Dirichlet conditions are applied.
 # - The velocity in the standard wave equation is fixed to $1$.
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 # ## Hard Constraint Model
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 # Once the problem is defined, a **torch** model is needed to solve the PINN. While **PINA** provides several pre-implemented models, users have the option to build their own custom model using **torch**. The hard constraint we impose is on the boundary of the spatial domain. Specifically, the solution is written as:
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 # $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t), $$
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 # where $NN$ represents the neural network output. This neural network takes the spatial coordinates $x$, $y$, and time $t$ as input and provides the unknown field $u$. By construction, the solution is zero at the boundaries.
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 # The residuals of the equations are evaluated at several sampling points (which the user can manipulate using the `discretise_domain` method). The loss function minimized by the neural network is the sum of the residuals.
 
 # In[3]:
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 # The results are not ideal, and we can clearly see that as time progresses, the solution deteriorates. Can we do better?
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 # One valid approach is to impose the initial condition as a hard constraint as well. Specifically, we modify the solution to:
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 # $$
 # u_{\rm{pinn}} = xy(1-x)(1-y) \cdot NN(x, y, t) \cdot t + \cos(\sqrt{2}\pi t)\sin(\pi x)\sin(\pi y),
 # $$
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 # Now, let us start by building the neural network.
 
 # In[8]:
@@ -329,19 +329,19 @@ plot_solution(solver=pinn, time=1)
 # We can now see that the results are much better! This improvement is due to the fact that, previously, the network was not correctly learning the initial condition, which led to a poor solution as time evolved. By imposing the initial condition as a hard constraint, the network is now able to correctly solve the problem.
 
 # ## What's Next?
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 # Congratulations on completing the two-dimensional Wave tutorial of **PINA**! Now that you’ve got the basics down, there are several directions you can explore:
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 # 1. **Train the Network for Longer**: Train the network for a longer duration or experiment with different layer sizes to assess the final accuracy.
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 # 2. **Propose New Types of Hard Constraints in Time**: Experiment with new time-dependent hard constraints, for example:
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 #    $$
 #    u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t)(1-\exp(-t)) + \cos(\sqrt{2}\pi t)\sin(\pi x)\sin(\pi y)
 #    $$
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 # 3. **Exploit Extrafeature Training**: Apply extrafeature training techniques to improve models from 1 and 2.
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 # 4. **...and many more!**: The possibilities are endless! Keep experimenting and pushing the boundaries.
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 # For more resources and tutorials, check out the [PINA Documentation](https://mathlab.github.io/PINA/).