PINA/tutorials/tutorial5/tutorial.py

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

# # Tutorial 5: Fourier Neural Operator Learning

# In this tutorial we are going to solve the Darcy flow 2d problem, presented in [Fourier Neural Operator for
# Parametric Partial Differential Equation](https://openreview.net/pdf?id=c8P9NQVtmnO). First of all we import the modules needed for the tutorial. Importing `scipy` is needed for input output operation, run `pip install scipy` for installing it.

# In[29]:



from scipy import io
import torch
from pina.model import FNO, FeedForward  # let's import some models
from pina import Condition
from pina import LabelTensor
from pina.solvers import SupervisedSolver
from pina.trainer import Trainer
from pina.problem import AbstractProblem
import matplotlib.pyplot as plt


# ## Data Generation
# 
# We will focus on solving the a specfic PDE, the **Darcy Flow** equation. The Darcy PDE is a second order, elliptic PDE with the following form:
# 
# $$
# -\nabla\cdot(k(x, y)\nabla u(x, y)) = f(x) \quad (x, y) \in D.
# $$
# 
# Specifically, $u$ is the flow pressure, $k$ is the permeability field and $f$ is the forcing function. The Darcy flow can parameterize a variety of systems including flow through porous media, elastic materials and heat conduction. Here you will define the domain as a 2D unit square Dirichlet boundary conditions. The dataset is taken from the authors original reference.
# 

# In[36]:


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

# extract data
k_train = torch.tensor(data['k_train'], dtype=torch.float).unsqueeze(-1)
u_train = torch.tensor(data['u_train'], dtype=torch.float).unsqueeze(-1)
k_test = torch.tensor(data['k_test'], dtype=torch.float).unsqueeze(-1)
u_test= torch.tensor(data['u_test'], dtype=torch.float).unsqueeze(-1)
x = torch.tensor(data['x'], dtype=torch.float)[0]
y = torch.tensor(data['y'], dtype=torch.float)[0]


# Let's visualize some data

# In[88]:


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


# We now create the neural operator class. It is a very simple class, inheriting from `AbstractProblem`.

# In[69]:


class NeuralOperatorSolver(AbstractProblem):
    input_variables = ['u_0']
    output_variables = ['u']
    conditions = {'data' : Condition(input_points=LabelTensor(k_train, input_variables), 
                                     output_points=LabelTensor(u_train, input_variables))}

# make problem
problem = NeuralOperatorSolver()


# ## Solving the problem with a FeedForward Neural Network
# 
# We will first solve the problem using a Feedforward neural network. We will use the `SupervisedSolver` for solving the problem, since we are training using supervised learning.

# In[78]:


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


# make solver
solver = SupervisedSolver(problem=problem, model=model)

# make the trainer and train
trainer = Trainer(solver=solver, max_epochs=100)
trainer.train()


# The final loss is pretty high... We can calculate the error by importing `LpLoss`.

# In[79]:


from pina.loss import LpLoss

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


err = float(metric_err(u_train.squeeze(-1), solver.models[0](k_train).squeeze(-1)).mean())*100
print(f'Final error training {err:.2f}%')

err = float(metric_err(u_test.squeeze(-1), solver.models[0](k_test).squeeze(-1)).mean())*100
print(f'Final error testing {err:.2f}%')


# ## Solving the problem with a Fuorier Neural Operator (FNO)
# 
# We will now move to solve the problem using a FNO. Since we are learning operator this approach is better suited, as we shall see.

# In[70]:


# 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=16,
            dimensions=2,
            inner_size=24,
            padding=11)


# make solver
solver = SupervisedSolver(problem=problem, model=model)

# make the trainer and train
trainer = Trainer(solver=solver, max_epochs=20)
trainer.train()


# We can clearly see that with 1/3 of the total epochs the loss is lower. Let's see in testing.. Notice that the number of parameters is way higher than a `FeedForward` network. We suggest to use GPU or TPU for a speed up in training.

# In[77]:


err = float(metric_err(u_train.squeeze(-1), solver.models[0](k_train).squeeze(-1)).mean())*100
print(f'Final error training {err:.2f}%')

err = float(metric_err(u_test.squeeze(-1), solver.models[0](k_test).squeeze(-1)).mean())*100
print(f'Final error testing {err:.2f}%')


# As we can see the loss is way lower!

# ## What's next?
# 
# We have made a very simple example on how to use the `FNO` for learning neural operator. Currently in **PINA** we implement 1D/2D/3D cases. We suggest to extend the tutorial using more complex problems and train for longer, to see the full potential of neural operators.