PINA/tutorials/tutorial10/tutorial.py

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

# # Tutorial: Averaging Neural Operator for solving Kuramoto Sivashinsky equation
# 
# [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial10/tutorial.ipynb)
# 
# In this tutorial we will build a Neural Operator using the
# `AveragingNeuralOperator` model and the `SupervisedSolver`. At the end of the
# tutorial you will be able to train a Neural Operator for learning
# the operator of time dependent PDEs.
# 
# 
# First of all, some useful imports. Note we use `scipy` for i/o operations.
# 

# In[1]:


## 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"')
  # get the data
  get_ipython().system('mkdir "data"')
  get_ipython().system('wget "https://github.com/mathLab/PINA/raw/refs/heads/master/tutorials/tutorial10/data/Data_KS.mat" -O "data/Data_KS.mat"')
  get_ipython().system('wget "https://github.com/mathLab/PINA/raw/refs/heads/master/tutorials/tutorial10/data/Data_KS2.mat" -O "data/Data_KS2.mat"')

import torch
import matplotlib.pyplot as plt
plt.style.use('tableau-colorblind10')
from scipy import io
from pina import Condition, LabelTensor
from pina.problem import AbstractProblem
from pina.model import AveragingNeuralOperator
from pina.solvers import SupervisedSolver
from pina.trainer import Trainer


# ## Data Generation
# 
# We will focus on solving a specific PDE, the **Kuramoto Sivashinsky** (KS) equation.
# The KS PDE is a fourth-order nonlinear PDE with the following form:
# 
# $$
# \frac{\partial u}{\partial t}(x,t) = -u(x,t)\frac{\partial u}{\partial x}(x,t)- \frac{\partial^{4}u}{\partial x^{4}}(x,t) - \frac{\partial^{2}u}{\partial x^{2}}(x,t).
# $$
# 
# In the above $x\in \Omega=[0, 64]$ represents a spatial location, $t\in\mathbb{T}=[0,50]$ the time and $u(x, t)$ is the value of the function $u:\Omega \times\mathbb{T}\in\mathbb{R}$. We indicate with $\mathbb{U}$ a suitable space for $u$, i.e. we have that the solution $u\in\mathbb{U}$.
# 
# 
# We impose Dirichlet boundary conditions on the derivative of $u$ on the border of the domain $\partial \Omega$
# $$
# \frac{\partial u}{\partial x}(x,t)=0 \quad \forall (x,t)\in \partial \Omega\times\mathbb{T}.
#  $$
# 
# Initial conditions are sampled from a distribution over truncated Fourier series with random coefficients 
# $\{A_k, \ell_k, \phi_k\}_k$ as
# $$
#     u(x,0) = \sum_{k=1}^N A_k \sin(2 \pi \ell_k x / L + \phi_k) \ ,
# $$
# 
# where $A_k \in [-0.4, -0.3]$, $\ell_k = 2$, $\phi_k  = 2\pi \quad \forall k=1,\dots,N$. 
# 
# 
# We have already generated some data for differenti initial conditions, and our objective will
# be to build a Neural Operator that, given $u(x, t)$ will output $u(x, t+\delta)$, where
# $\delta$ is a fixed time step. We will come back on the Neural Operator architecture, for now
# we first need to import the data.
# 
# **Note:**
# *The numerical integration is obtained by using pseudospectral method for spatial derivative discratization and
# implicit Runge Kutta 5 for temporal dynamics.*
# 

# In[2]:


# load data
data=io.loadmat("dat/Data_KS.mat")

# converting to label tensor
initial_cond_train = LabelTensor(torch.tensor(data['initial_cond_train'], dtype=torch.float), ['t','x','u0'])
initial_cond_test = LabelTensor(torch.tensor(data['initial_cond_test'], dtype=torch.float), ['t','x','u0'])
sol_train = LabelTensor(torch.tensor(data['sol_train'], dtype=torch.float), ['u'])
sol_test = LabelTensor(torch.tensor(data['sol_test'], dtype=torch.float), ['u'])

print('Data Loaded')
print(f' shape initial condition: {initial_cond_train.shape}')
print(f' shape solution: {sol_train.shape}')


# The data are saved in the form `B \times N \times D`, where `B` is the batch_size
# (basically how many initial conditions we sample), `N` the number of points in the mesh
# (which is the product of the discretization in `x` timese the one in `t`), and 
# `D` the dimension of the problem (in this case we have three variables `[u, t, x]`).
# 
# We are now going to plot some trajectories!

# In[3]:


# helper function
def plot_trajectory(coords, real, no_sol=None):
    # find the x-t shapes
    dim_x = len(torch.unique(coords.extract('x')))
    dim_t = len(torch.unique(coords.extract('t')))
    # if we don't have the Neural Operator solution we simply plot the real one
    if no_sol is None:
        fig, axs = plt.subplots(1, 1, figsize=(15, 5), sharex=True, sharey=True)
        c = axs.imshow(real.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto')
        axs.set_title('Real solution')
        fig.colorbar(c, ax=axs)
        axs.set_xlabel('t')
        axs.set_ylabel('x')
    # otherwise we plot the real one, the Neural Operator one, and their difference
    else:
        fig, axs = plt.subplots(1, 3, figsize=(15, 5), sharex=True, sharey=True)
        axs[0].imshow(real.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto')
        axs[0].set_title('Real solution')
        axs[1].imshow(no_sol.reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto')
        axs[1].set_title('NO solution')
        c = axs[2].imshow((real - no_sol).abs().reshape(dim_t, dim_x).T.detach(),extent=[0, 50, 0, 64], cmap='PuOr_r', aspect='auto')
        axs[2].set_title('Absolute difference')
        fig.colorbar(c, ax=axs.ravel().tolist())
        for ax in axs:
            ax.set_xlabel('t')
            ax.set_ylabel('x')
    plt.show()

# a sample trajectory (we use the sample 5, feel free to change)
sample_number = 20
plot_trajectory(coords=initial_cond_train[sample_number].extract(['x', 't']),
                real=sol_train[sample_number].extract('u'))


# As we can see, as the time progresses the solution becomes chaotic, which makes
# it really hard to learn! We will now focus on building a Neural Operator using the
# `SupervisedSolver` class to tackle the problem.
# 
# ## Averaging Neural Operator
# 
# We will build a neural operator $\texttt{NO}$ which takes the solution at time $t=0$ for any $x\in\Omega$,
# the time $(t)$ at which we want to compute the solution, and gives back the solution to the KS equation $u(x, t)$, mathematically:
# $$
# \texttt{NO}_\theta : \mathbb{U} \rightarrow  \mathbb{U},
# $$
# such that
# $$
# \texttt{NO}_\theta[u(t=0)](x, t) \rightarrow  u(x, t).
# $$
# 
# There are many ways on approximating the following operator, e.g. by 2D [FNO](https://mathlab.github.io/PINA/_rst/models/fno.html) (for regular meshes),
# a [DeepOnet](https://mathlab.github.io/PINA/_rst/models/deeponet.html), [Continuous Convolutional Neural Operator](https://mathlab.github.io/PINA/_rst/layers/convolution.html),
# [MIONet](https://mathlab.github.io/PINA/_rst/models/mionet.html). 
# In this tutorial we will use the *Averaging Neural Operator* presented in [*The Nonlocal Neural Operator: Universal Approximation*](https://arxiv.org/abs/2304.13221)
# which is a [Kernel Neural Operator](https://mathlab.github.io/PINA/_rst/models/base_no.html) with integral kernel:
# 
# $$
# K(v) = \sigma\left(Wv(x) + b + \frac{1}{|\Omega|}\int_\Omega v(y)dy\right)
# $$
# 
# where:
# 
# *   $v(x)\in\mathbb{R}^{\rm{emb}}$ is the update for a function $v$ with $\mathbb{R}^{\rm{emb}}$ the embedding (hidden) size
# *   $\sigma$ is a non-linear activation
# *   $W\in\mathbb{R}^{\rm{emb}\times\rm{emb}}$ is a tunable matrix.
# *   $b\in\mathbb{R}^{\rm{emb}}$ is a tunable bias.
# 
# If PINA many Kernel Neural Operators are already implemented, and the modular componets of the [Kernel Neural Operator](https://mathlab.github.io/PINA/_rst/models/base_no.html) class permits to create new ones by composing base kernel layers.
# 
# **Note:*** We will use the already built class* `AveragingNeuralOperator`, *as constructive excercise try to use the* [KernelNeuralOperator](https://mathlab.github.io/PINA/_rst/models/base_no.html) *class for building a kernel neural operator from scratch. You might employ the different layers that we have in pina, e.g.* [FeedForward](https://mathlab.github.io/PINA/_rst/models/fnn.html), *and* [AveragingNeuralOperator](https://mathlab.github.io/PINA/_rst/layers/avno_layer.html) *layers*.

# In[4]:


class SIREN(torch.nn.Module):
    def forward(self, x):
        return torch.sin(x)
    
embedding_dimesion = 40     # hyperparameter embedding dimension
input_dimension = 3         # ['u', 'x', 't']
number_of_coordinates = 2   # ['x', 't']
lifting_net = torch.nn.Linear(input_dimension, embedding_dimesion)   # simple linear layers for lifting and projecting nets
projecting_net = torch.nn.Linear(embedding_dimesion + number_of_coordinates, 1)
model = AveragingNeuralOperator(lifting_net=lifting_net,
                                projecting_net=projecting_net,
                                coordinates_indices=['x', 't'],
                                field_indices=['u0'],
                                n_layers=4,
                                func=SIREN
                                ) 


# Super easy! Notice that we use the `SIREN` activation function, more on [Implicit Neural Representations with Periodic Activation Functions](https://arxiv.org/abs/2006.09661).
# 
# ## Solving the KS problem
# 
# We will now focus on solving the KS equation using the `SupervisedSolver` class
# and the `AveragingNeuralOperator` model. As done in the [FNO tutorial](https://github.com/mathLab/PINA/blob/master/tutorials/tutorial5/tutorial.ipynb) we now create the `NeuralOperatorProblem` class with `AbstractProblem`.

# In[6]:


# expected running time ~ 1 minute

class NeuralOperatorProblem(AbstractProblem):
    input_variables = initial_cond_train.labels
    output_variables = sol_train.labels
    conditions = {'data' : Condition(input_points=initial_cond_train, 
                                     output_points=sol_train)}


# initialize problem
problem = NeuralOperatorProblem()
# initialize solver
solver = SupervisedSolver(problem=problem, model=model,optimizer_kwargs={"lr":0.001})
# train, only CPU and avoid model summary at beginning of training (optional)
trainer = Trainer(solver=solver, max_epochs=40, accelerator='cpu', enable_model_summary=False, log_every_n_steps=-1, batch_size=5) # we train on CPU and avoid model summary at beginning of training (optional)
trainer.train()


# We can now see some plots for the solutions

# In[7]:


sample_number = 2
no_sol = solver(initial_cond_test)
plot_trajectory(coords=initial_cond_test[sample_number].extract(['x', 't']),
                real=sol_test[sample_number].extract('u'),
                no_sol=no_sol[5])


# As we can see we can obtain nice result considering the small trainint time and the difficulty of the problem!
# Let's see how the training and testing error:

# In[8]:


from pina.loss import PowerLoss

error_metric = PowerLoss(p=2)                                                   # we use the MSE loss

with torch.no_grad():
    no_sol_train = solver(initial_cond_train)
    err_train = error_metric(sol_train.extract('u'), no_sol_train).mean()       # we average the error over trajectories
    no_sol_test = solver(initial_cond_test)
    err_test = error_metric(sol_test.extract('u'),no_sol_test).mean()           # we average the error over trajectories
    print(f'Training error: {float(err_train):.3f}')
    print(f'Testing error: {float(err_test):.3f}')


# as we can see the error is pretty small, which agrees with what we can see from the previous plots.

# ## What's next?
# 
# Now you know how to solve a time dependent neural operator problem in **PINA**! There are multiple directions you can go now:
# 
# 1. Train the network for longer or with different layer sizes and assert the finaly accuracy
# 
# 2. We left a more challenging dataset [Data_KS2.mat](dat/Data_KS2.mat) where $A_k \in [-0.5, 0.5]$, $\ell_k \in [1, 2, 3]$, $\phi_k \in [0, 2\pi]$ for loger training
# 
# 3. Compare the performance between the different neural operators (you can even try to implement your favourite one!)