Tutorial: Two dimensional Wave problem with hard constraint¶
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.
First of all, some useful imports.
## routine needed to run the notebook on Google Colab
try:
import google.colab
IN_COLAB = True
except:
IN_COLAB = False
if IN_COLAB:
!pip install "pina-mathlab"
import torch
import matplotlib.pyplot as plt
import warnings
from pina import Condition, LabelTensor, Trainer
from pina.problem import SpatialProblem, TimeDependentProblem
from pina.operator import laplacian, grad
from pina.domain import CartesianDomain
from pina.solver import PINN
from pina.equation import Equation, FixedValue
from pina.callback import MetricTracker
warnings.filterwarnings("ignore")
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 squared 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. solution is the exact solution which will be compared with the predicted one.
def wave_equation(input_, output_):
u_t = grad(output_, input_, components=["u"], d=["t"])
u_tt = grad(u_t, input_, components=["dudt"], d=["t"])
nabla_u = laplacian(output_, input_, components=["u"], d=["x", "y"])
return nabla_u - u_tt
def initial_condition(input_, output_):
u_expected = torch.sin(torch.pi * input_.extract(["x"])) * torch.sin(
torch.pi * input_.extract(["y"])
)
return output_.extract(["u"]) - u_expected
class Wave(TimeDependentProblem, SpatialProblem):
output_variables = ["u"]
spatial_domain = CartesianDomain({"x": [0, 1], "y": [0, 1]})
temporal_domain = CartesianDomain({"t": [0, 1]})
domains = {
"g1": CartesianDomain({"x": 1, "y": [0, 1], "t": [0, 1]}),
"g2": CartesianDomain({"x": 0, "y": [0, 1], "t": [0, 1]}),
"g3": CartesianDomain({"x": [0, 1], "y": 0, "t": [0, 1]}),
"g4": CartesianDomain({"x": [0, 1], "y": 1, "t": [0, 1]}),
"initial": CartesianDomain({"x": [0, 1], "y": [0, 1], "t": 0}),
"D": CartesianDomain({"x": [0, 1], "y": [0, 1], "t": [0, 1]}),
}
conditions = {
"g1": Condition(domain="g1", equation=FixedValue(0.0)),
"g2": Condition(domain="g2", equation=FixedValue(0.0)),
"g3": Condition(domain="g3", equation=FixedValue(0.0)),
"g4": Condition(domain="g4", equation=FixedValue(0.0)),
"initial": Condition(
domain="initial", equation=Equation(initial_condition)
),
"D": Condition(domain="D", equation=Equation(wave_equation)),
}
def solution(self, pts):
f = (
torch.sin(torch.pi * pts.extract(["x"]))
* torch.sin(torch.pi * pts.extract(["y"]))
* torch.cos(
torch.sqrt(torch.tensor(2.0)) * torch.pi * pts.extract(["t"])
)
)
return LabelTensor(f, self.output_variables)
# define problem
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 torch. The hard constraint we impose is on the boundary of the spatial domain. Specifically, our solution is written as:
$$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t), $$
where $NN$ is the neural net output. This neural network takes as input the coordinates (in this case $x$, $y$ and $t$) and provides the unknown field $u$. By construction, it is zero on the boundaries. The residuals of the equations are evaluated at several sampling points (which the user can manipulate using the method discretise_domain) and the loss minimized by the neural network is the sum of the residuals.
class HardMLP(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 = (
x.extract(["x"])
* (1 - x.extract(["x"]))
* x.extract(["y"])
* (1 - x.extract(["y"]))
)
return hard * self.layers(x)
Train and Inference¶
In this tutorial, the neural network is trained for 1000 epochs with a learning rate of 0.001 (default in PINN). As always, we will log using Tensorboard.
# generate the data
problem.discretise_domain(1000, "random", domains="all")
# define model
model = HardMLP(len(problem.input_variables), len(problem.output_variables))
# crete the solver
pinn = PINN(problem=problem, model=model)
# create trainer and train
trainer = Trainer(
solver=pinn,
max_epochs=1000,
accelerator="cpu",
enable_model_summary=False,
train_size=1.0,
val_size=0.0,
test_size=0.0,
callbacks=[MetricTracker(["train_loss", "initial_loss", "D_loss"])],
)
trainer.train()
GPU available: False, used: False
TPU available: False, using: 0 TPU cores
HPU available: False, using: 0 HPUs
Missing logger folder: /home/runner/work/PINA/PINA/tutorials/tutorial3/lightning_logs
`Trainer.fit` stopped: `max_epochs=1000` reached.
Let's now plot the losses inside MetricTracker to see how they vary during training.
trainer_metrics = trainer.callbacks[0].metrics
for metric, loss in trainer_metrics.items():
plt.plot(range(len(loss)), loss, label=metric)
# plotting
plt.xlabel("epoch")
plt.ylabel("loss")
plt.yscale("log")
plt.legend()
<matplotlib.legend.Legend at 0x7f66c8f02610>
Notice that the loss on the boundaries of the spatial domain is exactly zero, as expected! After the training is completed one can now plot some results using the matplotlib. We plot the predicted output on the left side, the true solution at the center and the difference on the right side using the plot_solution function.
@torch.no_grad()
def plot_solution(solver, time):
# get the problem
problem = solver.problem
# get spatial points
spatial_samples = problem.spatial_domain.sample(30, "grid")
# get temporal value
time = LabelTensor(torch.tensor([[time]]), "t")
# cross data
points = spatial_samples.append(time, mode="cross")
# compute pinn solution, true solution and absolute difference
data = {
"PINN solution": solver(points),
"True solution": problem.solution(points),
"Absolute Difference": torch.abs(
solver(points) - problem.solution(points)
),
}
# plot the solution
plt.suptitle(f"Solution for time {time.item()}")
for idx, (title, field) in enumerate(data.items()):
plt.subplot(1, 3, idx + 1)
plt.title(title)
plt.tricontourf( # convert to torch tensor + flatten
points.extract("x").tensor.flatten(),
points.extract("y").tensor.flatten(),
field.tensor.flatten(),
)
plt.colorbar(), plt.tight_layout()
Let's take a look at the results at different times, for example 0.0, 0.5 and 1.0:
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=0)
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=0.5)
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=1)
The results are not so great, and we can clearly see that as time progresses the solution gets 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
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(2.0)) * 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 the previous test
# define model
model = HardMLPtime(len(problem.input_variables), len(problem.output_variables))
# crete the solver
pinn = PINN(problem=problem, model=model)
# create trainer and train
trainer = Trainer(
solver=pinn,
max_epochs=1000,
accelerator="cpu",
enable_model_summary=False,
train_size=1.0,
val_size=0.0,
test_size=0.0,
callbacks=[MetricTracker(["train_loss", "initial_loss", "D_loss"])],
)
trainer.train()
GPU available: False, used: False
TPU available: False, using: 0 TPU cores
HPU available: False, using: 0 HPUs
`Trainer.fit` stopped: `max_epochs=1000` reached.
We can clearly see that the loss is way lower now. Let's plot the results
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=0)
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=0.5)
plt.figure(figsize=(12, 6))
plot_solution(solver=pinn, time=1)
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 time evolved. By imposing the initial condition the network is able to correctly solve the problem.
What's next?¶
Congratulations on completing the two dimensional Wave tutorial of PINA! There are multiple directions you can go now:
Train the network for longer or with different layer sizes and assert the finaly accuracy
Propose new types of hard constraints in time, e.g. $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t)(1-\exp(-t)) + \cos(\sqrt{2}\pi t)sin(\pi x)\sin(\pi y), $$
Exploit extrafeature training for model 1 and 2
Many more...