Dario Coscia
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tutorials/tutorial5/tutorial.py
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209
tutorials/tutorial5/tutorial.py
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#!/usr/bin/env python
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# coding: utf-8
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# # Tutorial: Two dimensional Darcy flow using the Fourier Neural Operator
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#
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# [](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial5/tutorial.ipynb)
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#
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# In this tutorial we are going to solve the Darcy flow problem in two dimensions, presented in [*Fourier Neural Operator for
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# 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 operations.
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# In[ ]:
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## routine needed to run the notebook on Google Colab
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try:
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import google.colab
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IN_COLAB = True
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except:
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IN_COLAB = False
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if IN_COLAB:
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get_ipython().system('pip install "pina-mathlab"')
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get_ipython().system('pip install scipy')
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# get the data
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get_ipython().system('wget https://github.com/mathLab/PINA/raw/refs/heads/master/tutorials/tutorial5/Data_Darcy.mat')
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import torch
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import matplotlib.pyplot as plt
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import warnings
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# !pip install scipy # install scipy
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from scipy import io
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from pina.model import FNO, FeedForward # let's import some models
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from pina import Condition, Trainer
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from pina.solver import SupervisedSolver
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from pina.problem.zoo import SupervisedProblem
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warnings.filterwarnings("ignore")
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# ## Data Generation
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#
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# We will focus on solving a specific PDE, the **Darcy Flow** equation. The Darcy PDE is a second-order elliptic PDE with the following form:
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#
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# $$
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# -\nabla\cdot(k(x, y)\nabla u(x, y)) = f(x) \quad (x, y) \in D.
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# $$
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#
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# 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.
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#
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# In[2]:
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# download the dataset
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data = io.loadmat("Data_Darcy.mat")
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# extract data (we use only 100 data for train)
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k_train = torch.tensor(data["k_train"], dtype=torch.float)
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u_train = torch.tensor(data["u_train"], dtype=torch.float)
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k_test = torch.tensor(data["k_test"], dtype=torch.float)
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u_test = torch.tensor(data["u_test"], dtype=torch.float)
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x = torch.tensor(data["x"], dtype=torch.float)[0]
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y = torch.tensor(data["y"], dtype=torch.float)[0]
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# Let's visualize some data
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# In[3]:
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plt.subplot(1, 2, 1)
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plt.title("permeability")
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plt.imshow(k_train[0])
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plt.subplot(1, 2, 2)
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plt.title("field solution")
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plt.imshow(u_train[0])
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plt.show()
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# We now create the Neural Operators problem class. Learning Neural Operators is similar as learning in a supervised manner, therefore we will use `SupervisedProblem`.
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# In[4]:
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# make problem
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problem = SupervisedProblem(
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input_=k_train.unsqueeze(-1), output_=u_train.unsqueeze(-1)
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)
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# ## Solving the problem with a FeedForward Neural Network
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#
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# 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.
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# In[5]:
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# make model
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model = FeedForward(input_dimensions=1, output_dimensions=1)
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# make solver
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solver = SupervisedSolver(problem=problem, model=model, use_lt=False)
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# make the trainer and train
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trainer = Trainer(
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solver=solver,
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max_epochs=10,
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accelerator="cpu",
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enable_model_summary=False,
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batch_size=10,
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train_size=1.0,
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val_size=0.0,
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test_size=0.0,
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)
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trainer.train()
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# The final loss is pretty high... We can calculate the error by importing `LpLoss`.
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# In[6]:
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from pina.loss import LpLoss
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# make the metric
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metric_err = LpLoss(relative=False)
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model = solver.model
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err = (
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float(
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metric_err(u_train.unsqueeze(-1), model(k_train.unsqueeze(-1))).mean()
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)
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* 100
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)
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print(f"Final error training {err:.2f}%")
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err = (
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float(metric_err(u_test.unsqueeze(-1), model(k_test.unsqueeze(-1))).mean())
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* 100
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)
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print(f"Final error testing {err:.2f}%")
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# ## Solving the problem with a Fourier Neural Operator (FNO)
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#
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# 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.
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# In[7]:
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# make model
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lifting_net = torch.nn.Linear(1, 24)
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projecting_net = torch.nn.Linear(24, 1)
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model = FNO(
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lifting_net=lifting_net,
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projecting_net=projecting_net,
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n_modes=8,
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dimensions=2,
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inner_size=24,
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padding=8,
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)
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# make solver
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solver = SupervisedSolver(problem=problem, model=model, use_lt=False)
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# make the trainer and train
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trainer = Trainer(
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solver=solver,
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max_epochs=10,
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accelerator="cpu",
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enable_model_summary=False,
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batch_size=10,
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train_size=1.0,
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val_size=0.0,
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test_size=0.0,
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)
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trainer.train()
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# We can clearly see that the final 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, when many data samples are used.
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# In[8]:
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model = solver.model
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err = (
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float(
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metric_err(u_train.unsqueeze(-1), model(k_train.unsqueeze(-1))).mean()
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)
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* 100
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)
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print(f"Final error training {err:.2f}%")
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err = (
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float(metric_err(u_test.unsqueeze(-1), model(k_test.unsqueeze(-1))).mean())
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* 100
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)
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print(f"Final error testing {err:.2f}%")
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# As we can see the loss is way lower!
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# ## What's next?
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#
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# 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.
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