export tutorials changed in 85b9edc (#634)

Co-authored-by: dario-coscia <dario-coscia@users.noreply.github.com>

tutorials/tutorial9/tutorial.py

@@ -2,14 +2,14 @@
 # coding: utf-8
 
 # # Tutorial: Applying Periodic Boundary Conditions in PINNs to solve the Helmholtz 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/tutorial9/tutorial.ipynb)
-# 
-# This tutorial demonstrates how to solve a one-dimensional Helmholtz equation with periodic boundary conditions (PBC) using Physics-Informed Neural Networks (PINNs).  
+#
+# This tutorial demonstrates how to solve a one-dimensional Helmholtz equation with periodic boundary conditions (PBC) using Physics-Informed Neural Networks (PINNs).
 # We will use standard PINN training, augmented with a periodic input expansion as introduced in [*An Expert’s Guide to Training Physics-Informed Neural Networks*](https://arxiv.org/abs/2308.08468).
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+#
 # Let's start with some useful imports:
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+#
 
 # In[1]:
 
@@ -42,33 +42,33 @@ warnings.filterwarnings("ignore")
 
 
 # ## Problem Definition
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+#
 # The one-dimensional Helmholtz problem is mathematically expressed as:
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+#
 # $$
 # \begin{cases}
 # \frac{d^2}{dx^2}u(x) - \lambda u(x) - f(x) &= 0 \quad \text{for } x \in (0, 2) \\
 # u^{(m)}(x = 0) - u^{(m)}(x = 2) &= 0 \quad \text{for } m \in \{0, 1, \dots\}
 # \end{cases}
 # $$
-# 
-# In this case, we seek a solution that is $C^{\infty}$ (infinitely differentiable) and periodic with period 2, over the infinite domain $x \in (-\infty, \infty)$. 
-# 
+#
+# In this case, we seek a solution that is $C^{\infty}$ (infinitely differentiable) and periodic with period 2, over the infinite domain $x \in (-\infty, \infty)$.
+#
 # A classical PINN approach would require enforcing periodic boundary conditions (PBC) for all derivatives—an infinite set of constraints—which is clearly infeasible.
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+#
 # To address this, we adopt a strategy known as *coordinate augmentation*. In this approach, we apply a coordinate transformation $v(x)$ such that the transformed inputs naturally satisfy the periodicity condition:
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+#
 # $$
 # u^{(m)}(x = 0) - u^{(m)}(x = 2) = 0 \quad \text{for } m \in \{0, 1, \dots\}
 # $$
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+#
 # For demonstration purposes, we choose the specific parameters:
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+#
 # - $\lambda = -10\pi^2$
 # - $f(x) = -6\pi^2 \sin(3\pi x) \cos(\pi x)$
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+#
 # These yield an analytical solution:
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+#
 # $$
 # u(x) = \sin(\pi x) \cos(3\pi x)
 # $$
@@ -111,39 +111,39 @@ problem.discretise_domain(200, "grid", domains=["phys_cond"])
 
 
 # As usual, the Helmholtz problem is implemented in **PINA** as a class. The governing equations are defined as `conditions`, which must be satisfied within their respective domains. The `solution` represents the exact analytical solution, which will be used to evaluate the accuracy of the predicted solution.
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-# For selecting collocation points, we use Latin Hypercube Sampling (LHS), a common strategy for efficient space-filling in high-dimensional domains 
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+#
+# For selecting collocation points, we use Latin Hypercube Sampling (LHS), a common strategy for efficient space-filling in high-dimensional domains
+#
 # ## Solving the Problem with a Periodic Network
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-# Any $\mathcal{C}^{\infty}$ periodic function  $u : \mathbb{R} \rightarrow \mathbb{R}$ with period $L \in \mathbb{N}$  
+#
+# Any $\mathcal{C}^{\infty}$ periodic function  $u : \mathbb{R} \rightarrow \mathbb{R}$ with period $L \in \mathbb{N}$
 # can be constructed by composing an arbitrary smooth function  $f : \mathbb{R}^n \rightarrow \mathbb{R}$ with a smooth, periodic mapping$v : \mathbb{R} \rightarrow \mathbb{R}^n$ of the same period $L$. That is,
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+#
 # $$
 # u(x) = f(v(x)).
 # $$
-# 
-# This formulation is general and can be extended to arbitrary dimensions.  
+#
+# This formulation is general and can be extended to arbitrary dimensions.
 # For more details, see [*A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks*](https://arxiv.org/pdf/2007.07442).
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 # In our specific case, we define the periodic embedding as:
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+#
 # $$
 # v(x) = \left[1, \cos\left(\frac{2\pi}{L} x\right), \sin\left(\frac{2\pi}{L} x\right)\right],
 # $$
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+#
 # which constitutes the coordinate augmentation. The function $f(\cdot)$ is approximated by a neural network $NN_{\theta}(\cdot)$, resulting in the approximate PINN solution:
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+#
 # $$
 # u(x) \approx u_{\theta}(x) = NN_{\theta}(v(x)).
 # $$
-# 
-# In **PINA**, this is implemented using the `PeriodicBoundaryEmbedding` layer for $v(x)$,  
-# paired with any `pina.model` to define the neural network $NN_{\theta}$.  
-# 
+#
+# In **PINA**, this is implemented using the `PeriodicBoundaryEmbedding` layer for $v(x)$,
+# paired with any `pina.model` to define the neural network $NN_{\theta}$.
+#
 # Let’s see how this is put into practice!
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-# 
+#
+#
 
 # In[18]:
 
@@ -160,11 +160,11 @@ model = torch.nn.Sequential(
 
 
 # As simple as that!
-# 
-# In higher dimensions, you can specify different periods for each coordinate using a dictionary.  
-# For example, `periods = {'x': 2, 'y': 3, ...}` indicates a periodicity of 2 in the $x$ direction,  
+#
+# In higher dimensions, you can specify different periods for each coordinate using a dictionary.
+# For example, `periods = {'x': 2, 'y': 3, ...}` indicates a periodicity of 2 in the $x$ direction,
 # 3 in the $y$ direction, and so on.
-# 
+#
 # We will now solve the problem using the usual `PINN` and `Trainer` classes. After training, we'll examine the losses using the `MetricTracker` callback from `pina.callback`.
 
 # In[ ]:
@@ -240,15 +240,15 @@ with torch.no_grad():
 
 # It's clear that the network successfully captures the periodicity of the solution, with the error also exhibiting a periodic pattern. Naturally, training for a longer duration or using a more expressive neural network could further improve the results.
 # ## What's next?
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 # Congratulations on completing the one-dimensional Helmholtz tutorial with **PINA**! Here are a few directions you can explore next:
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 # 1. **Train longer or with different architectures**: Experiment with extended training or modify the network's depth and width to evaluate improvements in accuracy.
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 # 2. **Apply `PeriodicBoundaryEmbedding` to time-dependent problems**: Explore more complex scenarios such as spatiotemporal PDEs (see the official documentation for examples).
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 # 3. **Try extra feature training**: Integrate additional physical or domain-specific features to guide the learning process more effectively.
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 # 4. **...and many more!**: Extend to higher dimensions, test on other PDEs, or even develop custom embeddings tailored to your problem.
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 # For more resources and tutorials, check out the [PINA Documentation](https://mathlab.github.io/PINA/).