tutorial.py

#!/usr/bin/env python
# coding: utf-8

# # Tutorial: Modeling 2D Darcy Flow with the Fourier Neural Operator
# 
# [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/PINA-org/PINA-tutorials/blob/main/tutorials/tutorial5/tutorial.ipynb)
# 
# In this tutorial, we are going to solve the **Darcy flow problem** in two dimensions, as presented in the paper [*Fourier Neural Operator for Parametric Partial Differential Equations*](https://openreview.net/pdf?id=c8P9NQVtmnO).
# 
# We begin by importing the necessary modules for the tutorial:
# 

# In[ ]:


## routine needed to run the notebook on Google Colab
try:
    import google.colab

    IN_COLAB = True
except:
    IN_COLAB = False
if IN_COLAB:
    get_ipython().system('pip install "pina-mathlab[tutorial]"')
    get_ipython().system('pip install scipy')
    # get the data
    get_ipython().system('wget https://github.com/PINA-org/PINA-tutorials/raw/refs/heads/main/tutorials/tutorial5/Data_Darcy.mat')

import torch
import matplotlib.pyplot as plt
import warnings

from scipy import io
from pina.model import FNO, FeedForward
from pina import Trainer
from pina.solver import SupervisedSingleModelSolver
from pina.problem.zoo import SupervisedProblem

warnings.filterwarnings("ignore")


# ## Data Generation
# 
# We will focus on solving a specific PDE: the **Darcy Flow** equation. This is a second-order elliptic PDE given by:
# 
# $$
# -\nabla\cdot(k(x, y)\nabla u(x, y)) = f(x, y), \quad (x, y) \in D.
# $$
# 
# Here, $u$ represents the flow pressure, $k$ is the permeability field, and $f$ is the forcing function. The Darcy flow equation can be used to model various systems, including flow through porous media, elasticity in materials, and heat conduction.
# 
# In this tutorial, the domain $D$ is defined as a 2D unit square with Dirichlet boundary conditions. The dataset used is taken from the authors' original implementation in the referenced paper.

# In[ ]:


# download the dataset
data = io.loadmat("Data_Darcy.mat")

# extract data (we use only 100 data for train)
k_train = torch.tensor(data["k_train"], dtype=torch.float)
u_train = torch.tensor(data["u_train"], dtype=torch.float)
k_test = torch.tensor(data["k_test"], dtype=torch.float)
u_test = torch.tensor(data["u_test"], dtype=torch.float)
x = torch.tensor(data["x"], dtype=torch.float)[0]
y = torch.tensor(data["y"], dtype=torch.float)[0]


# Before diving into modeling, it's helpful to visualize some examples from the dataset. This will give us a better understanding of the input (permeability field) and the corresponding output (pressure field) that our model will learn to predict.

# In[ ]:


plt.subplot(1, 2, 1)
plt.title("permeability")
plt.imshow(k_train[0])
plt.subplot(1, 2, 2)
plt.title("field solution")
plt.imshow(u_train[0])
plt.show()


# We now define the problem class for learning the Neural Operator. Since this task is essentially a supervised learning problem—where the goal is to learn a mapping from input functions to output solutions—we will use the `SupervisedProblem` class provided by **PINA**.

# In[ ]:


# make problem
problem = SupervisedProblem(
    input_=k_train.unsqueeze(-1), output_=u_train.unsqueeze(-1)
)


# ## Solving the Problem with a Feedforward Neural Network
# 
# We begin by solving the Darcy flow problem using a standard Feedforward Neural Network (FNN). Since we are approaching this task with supervised learning, we will use the `SupervisedSingleModelSolver` provided by **PINA** to train the model.

# In[ ]:


# make model
model = FeedForward(input_dimensions=1, output_dimensions=1)


# make solver
solver = SupervisedSingleModelSolver(problem=problem, model=model, use_lt=False)

# make the trainer and train
trainer = Trainer(
    solver=solver,
    max_epochs=10,
    accelerator="cpu",
    enable_model_summary=False,
    batch_size=10,
    train_size=1.0,
    val_size=0.0,
    test_size=0.0,
)
trainer.train()


# The final loss is relatively high, indicating that the model might not be capturing the solution accurately. To better evaluate the model's performance, we can compute the error using the `LpLoss` metric.

# In[ ]:


from pina.loss import LpLoss

# make the metric
metric_err = LpLoss(relative=False)

model = solver.model
err = (
    float(
        metric_err(u_train.unsqueeze(-1), model(k_train.unsqueeze(-1))).mean()
    )
    * 100
)
print(f"Final error training {err:.2f}%")

err = (
    float(metric_err(u_test.unsqueeze(-1), model(k_test.unsqueeze(-1))).mean())
    * 100
)
print(f"Final error testing {err:.2f}%")


# ## Solving the Problem with a Fourier Neural Operator
# 
# We will now solve the Darcy flow problem using a Fourier Neural Operator (FNO). Since we are learning a mapping between functions—i.e., an operator—this approach is more suitable and often yields better performance, as we will see.

# In[ ]:


# make model
lifting_net = torch.nn.Linear(1, 24)
projecting_net = torch.nn.Linear(24, 1)
model = FNO(
    lifting_net=lifting_net,
    projecting_net=projecting_net,
    n_modes=8,
    dimensions=2,
    inner_size=24,
    padding=8,
)


# make solver
solver = SupervisedSingleModelSolver(problem=problem, model=model, use_lt=False)

# make the trainer and train
trainer = Trainer(
    solver=solver,
    max_epochs=10,
    accelerator="cpu",
    enable_model_summary=False,
    batch_size=10,
    train_size=1.0,
    val_size=0.0,
    test_size=0.0,
)
trainer.train()


# We can clearly observe that the final loss is significantly lower when using the FNO. Let's now evaluate its performance on the test set.
# 
# Note that the number of trainable parameters in the FNO is considerably higher compared to a `FeedForward` network. Therefore, we recommend using a GPU or TPU to accelerate training, especially when working with large datasets.

# In[ ]:


model = solver.model
err = (
    float(
        metric_err(u_train.unsqueeze(-1), model(k_train.unsqueeze(-1))).mean()
    )
    * 100
)
print(f"Final error training {err:.2f}%")

err = (
    float(metric_err(u_test.unsqueeze(-1), model(k_test.unsqueeze(-1))).mean())
    * 100
)
print(f"Final error testing {err:.2f}%")


# As we can see, the loss is significantly lower with the Fourier Neural Operator!

# ## What's Next?
# 
# Congratulations on completing the tutorial on solving the Darcy flow problem using **PINA**! There are many potential next steps you can explore:
# 
# 1. **Train the network longer or with different hyperparameters**: Experiment with different configurations of the neural network. You can try varying the number of layers, activation functions, or learning rates to improve accuracy.
# 
# 2. **Solve more complex problems**: The Darcy flow problem is just the beginning! Try solving other complex problems from the field of parametric PDEs. The original paper and **PINA** documentation offer many more examples to explore.
# 
# 3. **...and many more!**: There are countless directions to further explore. For instance, you could try to add physics informed learning!
# 
# For more resources and tutorials, check out the [PINA Documentation](https://PINA-org.github.io/PINA/).