tutorial.py

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

# # Tutorial: Physics Informed Neural Networks on PINA
# 
# [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial1/tutorial.ipynb)
# 

# In this tutorial, we will demonstrate a typical use case of **PINA** on a toy problem, following the standard API procedure. 
# 
# <p align="center">
#     <img src="../../readme/API_color.png" alt="PINA API" width="400"/>
# </p>
# 
# Specifically, the tutorial aims to introduce the following topics:
# 
# * Explaining how to build **PINA** Problems,
# * Showing how to generate data for `PINN` training
# 
# These are the two main steps needed **before** starting the modelling optimization (choose model and solver, and train). We will show each step in detail, and at the end, we will solve a simple Ordinary Differential Equation (ODE) problem using the `PINN` solver.

# ## Build a PINA problem

# Problem definition in the **PINA** framework is done by building a python `class`, which inherits from one or more problem classes (`SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`, ...) depending on the nature of the problem. Below is an example:
# ### Simple Ordinary Differential Equation
# Consider the following:
# 
# $$
# \begin{equation}
# \begin{cases}
# \frac{d}{dx}u(x) &=  u(x) \quad x\in(0,1)\\
# u(x=0) &= 1 \\
# \end{cases}
# \end{equation}
# $$
# 
# with the analytical solution $u(x) = e^x$. In this case, our ODE depends only on the spatial variable $x\in(0,1)$ , meaning that our `Problem` class is going to be inherited from the `SpatialProblem` class:
# 
# ```python
# from pina.problem import SpatialProblem
# from pina.domain import CartesianProblem
# 
# class SimpleODE(SpatialProblem):
#     
#     output_variables = ['u']
#     spatial_domain = CartesianProblem({'x': [0, 1]})
# 
#     # other stuff ...
# ```
# 
# Notice that we define `output_variables` as a list of symbols, indicating the output variables of our equation (in this case only $u$), this is done because in **PINA** the `torch.Tensor`s are labelled, allowing the user maximal flexibility for the manipulation of the tensor. The `spatial_domain` variable indicates where the sample points are going to be sampled in the domain, in this case $x\in[0,1]$.
# 
# What if our equation is also time-dependent? In this case, our `class` will inherit from both `SpatialProblem` and `TimeDependentProblem`:
# 

# 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"')

import warnings

from pina.problem import SpatialProblem, TimeDependentProblem
from pina.domain import CartesianDomain

warnings.filterwarnings("ignore")


class TimeSpaceODE(SpatialProblem, TimeDependentProblem):

    output_variables = ["u"]
    spatial_domain = CartesianDomain({"x": [0, 1]})
    temporal_domain = CartesianDomain({"t": [0, 1]})

    # other stuff ...


# where we have included the `temporal_domain` variable, indicating the time domain wanted for the solution.
# 
# In summary, using **PINA**, we can initialize a problem with a class which inherits from different base classes: `SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`, and so on depending on the type of problem we are considering. Here are some examples (more on the official documentation):
# * ``SpatialProblem`` $\rightarrow$ a differential equation with spatial variable(s) ``spatial_domain``
# * ``TimeDependentProblem`` $\rightarrow$ a time-dependent differential equation with temporal variable(s) ``temporal_domain``
# * ``ParametricProblem`` $\rightarrow$ a parametrized differential equation with parametric variable(s) ``parameter_domain``
# * ``AbstractProblem`` $\rightarrow$ any **PINA** problem inherits from here

# ### Write the problem class
# 
# Once the `Problem` class is initialized, we need to represent the differential equation in **PINA**. In order to do this, we need to load the **PINA** operators from `pina.operator` module. Again, we'll consider Equation (1) and represent it in **PINA**:

# In[ ]:


import torch
import matplotlib.pyplot as plt

from pina.problem import SpatialProblem
from pina.operator import grad
from pina import Condition
from pina.domain import CartesianDomain
from pina.equation import Equation, FixedValue


# defining the ode equation
def ode_equation(input_, output_):

    # computing the derivative
    u_x = grad(output_, input_, components=["u"], d=["x"])

    # extracting the u input variable
    u = output_.extract(["u"])

    # calculate the residual and return it
    return u_x - u


class SimpleODE(SpatialProblem):

    output_variables = ["u"]
    spatial_domain = CartesianDomain({"x": [0, 1]})

    domains = {
        "x0": CartesianDomain({"x": 0.0}),
        "D": CartesianDomain({"x": [0, 1]}),
    }

    # conditions to hold
    conditions = {
        "bound_cond": Condition(domain="x0", equation=FixedValue(1.0)),
        "phys_cond": Condition(domain="D", equation=Equation(ode_equation)),
    }

    # defining the true solution
    def solution(self, pts):
        return torch.exp(pts.extract(["x"]))


problem = SimpleODE()


# After we define the `Problem` class, we need to write different class methods, where each method is a function returning a residual. These functions are the ones minimized during PINN optimization, given the initial conditions. For example, in the domain $[0,1]$, the ODE equation (`ode_equation`) must be satisfied. We represent this by returning the difference between subtracting the variable `u` from its gradient (the residual), which we hope to minimize to 0. This is done for all conditions. Notice that we do not pass directly a `python` function, but an `Equation` object, which is initialized with the `python` function. This is done so that all the computations and internal checks are done inside **PINA**.
# 
# Once we have defined the function, we need to tell the neural network where these methods are to be applied. To do so, we use the `Condition` class. In the `Condition` class, we pass the location points and the equation we want minimized on those points (other possibilities are allowed, see the documentation for reference).
# 
# Finally, it's possible to define a `solution` function, which can be useful if we want to plot the results and see how the real solution compares to the expected (true) solution. Notice that the `solution` function is a method of the `PINN` class, but it is not mandatory for problem definition.
# 

# ## Generate data 
# 
# Data for training can come in form of direct numerical simulation results, or points in the domains. In case we perform unsupervised learning, we just need the collocation points for training, i.e. points where we want to evaluate the neural network. Sampling point in **PINA** is very easy, here we show three examples using the `.discretise_domain` method of the `AbstractProblem` class.

# In[ ]:


# sampling 20 points in [0, 1] through discretization in all locations
problem.discretise_domain(n=20, mode="grid", domains="all")

# sampling 20 points in (0, 1) through latin hypercube sampling in D, and 1 point in x0
problem.discretise_domain(n=20, mode="latin", domains=["D"])
problem.discretise_domain(n=1, mode="random", domains=["x0"])

# sampling 20 points in (0, 1) randomly
problem.discretise_domain(n=20, mode="random")


# We are going to use latin hypercube points for sampling. We need to sample in all the conditions domains. In our case we sample in `D` and `x0`.

# In[ ]:


# sampling for training
problem.discretise_domain(1, "random", domains=["x0"])
problem.discretise_domain(20, "lh", domains=["D"])


# The points are saved in a python `dict`, and can be accessed by calling the attribute `input_pts` of the problem 

# In[ ]:


print("Input points:", problem.discretised_domains)
print("Input points labels:", problem.discretised_domains["D"].labels)


# To visualize the sampled points we can use `matplotlib.pyplot`:

# In[ ]:


for location in problem.input_pts:
    coords = (
        problem.input_pts[location].extract(problem.spatial_variables).flatten()
    )
    plt.scatter(coords, torch.zeros_like(coords), s=10, label=location)
plt.legend()


# ## Perform a small training

# Once we have defined the problem and generated the data we can start the modelling. Here we will choose a `FeedForward` neural network available in `pina.model`, and we will train using the `PINN` solver from `pina.solver`. We highlight that this training is fairly simple, for more advanced stuff consider the tutorials in the ***Physics Informed Neural Networks*** section of ***Tutorials***. For training we use the `Trainer` class from `pina.trainer`. Here we show a very short training and some method for plotting the results. Notice that by default all relevant metrics (e.g. MSE error during training) are going to be tracked using a `lightning` logger, by default `CSVLogger`. If you want to track the metric by yourself without a logger, use `pina.callback.MetricTracker`.

# In[ ]:


from pina import Trainer
from pina.solver import PINN
from pina.model import FeedForward
from lightning.pytorch.loggers import TensorBoardLogger
from pina.optim import TorchOptimizer


# build the model
model = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)

# create the PINN object
pinn = PINN(problem, model, TorchOptimizer(torch.optim.Adam, lr=0.005))

# create the trainer
trainer = Trainer(
    solver=pinn,
    max_epochs=1500,
    logger=TensorBoardLogger("tutorial_logs"),
    accelerator="cpu",
    train_size=1.0,
    test_size=0.0,
    val_size=0.0,
    enable_model_summary=False,
)  # we train on CPU and avoid model summary at beginning of training (optional)

# train
trainer.train()


# After the training we can inspect trainer logged metrics (by default **PINA** logs mean square error residual loss). The logged metrics can be accessed online using one of the `Lightning` loggers. The final loss can be accessed by `trainer.logged_metrics`

# In[27]:


# inspecting final loss
trainer.logged_metrics


# By using `matplotlib` we can also do some qualitative plots of the solution. 

# In[ ]:


pts = pinn.problem.spatial_domain.sample(256, "grid", variables="x")
predicted_output = pinn.forward(pts).extract("u").tensor.detach()
true_output = pinn.problem.solution(pts).detach()
fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(8, 8))
ax.plot(pts.extract(["x"]), predicted_output, label="Neural Network solution")
ax.plot(pts.extract(["x"]), true_output, label="True solution")
plt.legend()


# The solution is overlapped with the actual one, and they are barely indistinguishable. We can also take a look at the loss using `TensorBoard`:

# In[ ]:


print("\nTo load TensorBoard run load_ext tensorboard on your terminal")
print(
    "To visualize the loss you can run tensorboard --logdir 'tutorial_logs' on your terminal\n"
)
# # uncomment for running tensorboard
# %load_ext tensorboard
# %tensorboard --logdir=tutorial_logs


# As we can see the loss has not reached a minimum, suggesting that we could train for longer! Alternatively, we can also take look at the loss using callbacks. Here we use `MetricTracker` from `pina.callback`:

# In[ ]:


from pina.callback import MetricTracker

# create the model
newmodel = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)

# create the PINN object
newpinn = PINN(
    problem, newmodel, optimizer=TorchOptimizer(torch.optim.Adam, lr=0.005)
)

# create the trainer
newtrainer = Trainer(
    solver=newpinn,
    max_epochs=1500,
    logger=True,  # enable parameter logging
    callbacks=[MetricTracker()],
    accelerator="cpu",
    train_size=1.0,
    test_size=0.0,
    val_size=0.0,
    enable_model_summary=False,
)  # we train on CPU and avoid model summary at beginning of training (optional)

# train
newtrainer.train()

# plot loss
trainer_metrics = newtrainer.callbacks[0].metrics
loss = trainer_metrics["train_loss"]
epochs = range(len(loss))
plt.plot(epochs, loss.cpu())
# plotting
plt.xlabel("epoch")
plt.ylabel("loss")
plt.yscale("log")


# ## What's next?
# 
# Congratulations on completing the introductory tutorial of **PINA**! There are several directions you can go now:
# 
# 1. Train the network for longer or with different layer sizes and assert the finaly accuracy
# 
# 2. Train the network using other types of models (see `pina.model`)
# 
# 3. GPU training and speed benchmarking
# 
# 4. Many more...