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Tutorials and Doc (#191)

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* Tutorial doc update
* update doc tutorial
* doc not compiling

---------

Co-authored-by: Dario Coscia <dcoscia@euclide.maths.sissa.it>
Co-authored-by: Dario Coscia <dariocoscia@Dario-Coscia.local>
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Nicola Demo
2023-10-23 12:48:09 +02:00
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@@ -1,24 +1,10 @@
#!/usr/bin/env python
# coding: utf-8
# # Tutorial 3: resolution of wave equation with hard constraint PINNs.
# ## The problem definition
# In this tutorial we present how to solve the wave equation using hard constraint PINNs. For doing so we will build a costum torch model and pass it to the `PINN` solver.
# # Tutorial: Two dimensional Wave problem with hard constraint
#
# The problem is written in the following form:
# In this tutorial we present how to solve the wave equation using hard constraint PINNs. For doing so we will build a costum `torch` model and pass it to the `PINN` solver.
#
# \begin{equation}
# \begin{cases}
# \Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
# u(x, y, t=0) = \sin(\pi x)\sin(\pi y), \\\\
# u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
# \end{cases}
# \end{equation}
#
# where $D$ is a square domain $[0,1]^2$, and $\Gamma_i$, with $i=1,...,4$, are the boundaries of the square, and the velocity in the standard wave equation is fixed to one.
# First of all, some useful imports.
# In[1]:
@@ -36,6 +22,20 @@ from pina.equation.equation_factory import FixedValue
from pina import Condition, Plotter
# ## The problem definition
# The problem is written in the following form:
#
# \begin{equation}
# \begin{cases}
# \Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
# u(x, y, t=0) = \sin(\pi x)\sin(\pi y), \\\\
# u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
# \end{cases}
# \end{equation}
#
# where $D$ is a square domain $[0,1]^2$, and $\Gamma_i$, with $i=1,...,4$, are the boundaries of the square, and the velocity in the standard wave equation is fixed to one.
# Now, the wave problem is written in PINA code as a class, inheriting from `SpatialProblem` and `TimeDependentProblem` since we deal with spatial, and time dependent variables. The equations are written as `conditions` that should be satisfied in the corresponding domains. `truth_solution` is the exact solution which will be compared with the predicted one.
# In[2]:
@@ -78,7 +78,7 @@ problem = Wave()
# ## Hard Constraint Model
# After the problem, a **torch** model is needed to solve the PINN. Usually, many models are already implemented in `PINA`, but the user has the possibility to build his/her own model in `PyTorch`. The hard constraint we impose is on the boundary of the spatial domain. Specifically, our solution is written as:
# After the problem, a **torch** model is needed to solve the PINN. Usually, many models are already implemented in **PINA**, but the user has the possibility to build his/her own model in `torch`. The hard constraint we impose is on the boundary of the spatial domain. Specifically, our solution is written as:
#
# $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t), $$
#
@@ -92,11 +92,11 @@ class HardMLP(torch.nn.Module):
def __init__(self, input_dim, output_dim):
super().__init__()
self.layers = torch.nn.Sequential(torch.nn.Linear(input_dim, 20),
torch.nn.Tanh(),
torch.nn.Linear(20, 20),
torch.nn.Tanh(),
torch.nn.Linear(20, output_dim))
self.layers = torch.nn.Sequential(torch.nn.Linear(input_dim, 40),
torch.nn.ReLU(),
torch.nn.Linear(40, 40),
torch.nn.ReLU(),
torch.nn.Linear(40, output_dim))
# here in the foward we implement the hard constraints
def forward(self, x):
@@ -106,14 +106,19 @@ class HardMLP(torch.nn.Module):
# ## Train and Inference
# In this tutorial, the neural network is trained for 3000 epochs with a learning rate of 0.001 (default in `PINN`). Training takes approximately 1 minute.
# In this tutorial, the neural network is trained for 1000 epochs with a learning rate of 0.001 (default in `PINN`). Training takes approximately 3 minutes.
# In[4]:
# generate the data
problem.discretise_domain(1000, 'random', locations=['D', 't0', 'gamma1', 'gamma2', 'gamma3', 'gamma4'])
# crete the solver
pinn = PINN(problem, HardMLP(len(problem.input_variables), len(problem.output_variables)))
problem.discretise_domain(1000, 'random', locations=['D','t0', 'gamma1', 'gamma2', 'gamma3', 'gamma4'])
trainer = Trainer(pinn, max_epochs=3000)
# create trainer and train
trainer = Trainer(pinn, max_epochs=1000, accelerator='cpu', enable_model_summary=False) # we train on CPU and avoid model summary at beginning of training (optional)
trainer.train()
@@ -125,11 +130,93 @@ trainer.train()
plotter = Plotter()
# plotting at fixed time t = 0.0
print('Plotting at t=0')
plotter.plot(trainer, fixed_variables={'t': 0.0})
# plotting at fixed time t = 0.5
print('Plotting at t=0.5')
plotter.plot(trainer, fixed_variables={'t': 0.5})
# plotting at fixed time t = 1.
print('Plotting at t=1')
plotter.plot(trainer, fixed_variables={'t': 1.0})
# The results are not so great, and we can clearly see that as time progress the solution get worse.... Can we do better?
#
# A valid option is to impose the initial condition as hard constraint as well. Specifically, our solution is written as:
#
# $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t)\cdot t + \cos(\sqrt{2}\pi t)sin(\pi x)\sin(\pi y), $$
#
# Let us build the network first
# In[6]:
class HardMLPtime(torch.nn.Module):
def __init__(self, input_dim, output_dim):
super().__init__()
self.layers = torch.nn.Sequential(torch.nn.Linear(input_dim, 40),
torch.nn.ReLU(),
torch.nn.Linear(40, 40),
torch.nn.ReLU(),
torch.nn.Linear(40, output_dim))
# here in the foward we implement the hard constraints
def forward(self, x):
hard_space = x.extract(['x'])*(1-x.extract(['x']))*x.extract(['y'])*(1-x.extract(['y']))
hard_t = torch.sin(torch.pi*x.extract(['x'])) * torch.sin(torch.pi*x.extract(['y'])) * torch.cos(torch.sqrt(torch.tensor(2.))*torch.pi*x.extract(['t']))
return hard_space * self.layers(x) * x.extract(['t']) + hard_t
# Now let's train with the same configuration as thre previous test
# In[7]:
# generate the data
problem.discretise_domain(1000, 'random', locations=['D', 't0', 'gamma1', 'gamma2', 'gamma3', 'gamma4'])
# crete the solver
pinn = PINN(problem, HardMLPtime(len(problem.input_variables), len(problem.output_variables)))
# create trainer and train
trainer = Trainer(pinn, max_epochs=1000, accelerator='cpu', enable_model_summary=False) # we train on CPU and avoid model summary at beginning of training (optional)
trainer.train()
# We can clearly see that the loss is way lower now. Let's plot the results
# In[8]:
plotter = Plotter()
# plotting at fixed time t = 0.0
print('Plotting at t=0')
plotter.plot(trainer, fixed_variables={'t': 0.0})
# plotting at fixed time t = 0.5
print('Plotting at t=0.5')
plotter.plot(trainer, fixed_variables={'t': 0.5})
# plotting at fixed time t = 1.
print('Plotting at t=1')
plotter.plot(trainer, fixed_variables={'t': 1.0})
# We can see now that the results are way better! This is due to the fact that previously the network was not learning correctly the initial conditon, leading to a poor solution when the time evolved. By imposing the initial condition the network is able to correctly solve the problem.
# ## What's next?
#
# Nice you have completed the two dimensional Wave tutorial of **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. Propose new types of hard constraints in time, e.g. $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t)(1-\exp(-t)) + \cos(\sqrt{2}\pi t)sin(\pi x)\sin(\pi y), $$
#
# 3. Exploit extrafeature training for model 1 and 2
#
# 4. Many more...