Tutorial: Physics Informed Neural Networks on PINA

In this tutorial, we will demonstrate a typical use case of PINA on a toy problem, following the standard API procedure.

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:

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.Tensors 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:

## 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 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:

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.

# 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.

# 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

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

Input points: {'x0': LabelTensor([[0.]]), 'D': LabelTensor([[0.3416],
             [0.0857],
             [0.5368],
             [0.7287],
             [0.4425],
             [0.6176],
             [0.6806],
             [0.0268],
             [0.3685],
             [0.1342],
             [0.9353],
             [0.2686],
             [0.2114],
             [0.8439],
             [0.7916],
             [0.1877],
             [0.9715],
             [0.4534],
             [0.5888],
             [0.8793]])}
Input points labels: ['x']

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

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()

<matplotlib.legend.Legend at 0x7f747f521fd0>

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.

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()

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

Missing logger folder: tutorial_logs/lightning_logs

`Trainer.fit` stopped: `max_epochs=1500` reached.

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

# inspecting final loss
trainer.logged_metrics

{'bound_cond_loss': tensor(2.5384e-08),
 'phys_cond_loss': tensor(2.1712e-05),
 'train_loss': tensor(2.1738e-05)}

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

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()

<matplotlib.legend.Legend at 0x7f747eda3af0>

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

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

To load TensorBoard run load_ext tensorboard on your terminal
To visualize the loss you can run tensorboard --logdir 'tutorial_logs' on your terminal

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:

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")

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/tutorial1/lightning_logs

`Trainer.fit` stopped: `max_epochs=1500` reached.

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...