Tutorial: PINA and PyTorch Lightning, training tips and visualizations

In this tutorial, we will delve deeper into the functionality of the Trainer class, which serves as the cornerstone for training PINA Solvers.

The Trainer class offers a plethora of features aimed at improving model accuracy, reducing training time and memory usage, facilitating logging visualization, and more thanks to the amazing job done by the PyTorch Lightning team!

Our leading example will revolve around solving the SimpleODE problem, as outlined in the Introduction to PINA for Physics Informed Neural Networks training. If you haven't already explored it, we highly recommend doing so before diving into this tutorial.

Let's start by importing useful modules, define the SimpleODE problem and the PINN solver.

try:
    import google.colab

    IN_COLAB = True
except:
    IN_COLAB = False
if IN_COLAB:
    !pip install "pina-mathlab"

import torch
import warnings

from pina import Condition, Trainer
from pina.solver import PINN
from pina.model import FeedForward
from pina.problem import SpatialProblem
from pina.operator import grad
from pina.domain import CartesianDomain
from pina.equation import Equation, FixedValue

warnings.filterwarnings("ignore")

Define problem and solver.

# 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"]))


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

# 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)

Till now we just followed the extact step of the previous tutorials. The Trainer object can be initialized by simiply passing the PINN solver

trainer = Trainer(solver=pinn)

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

Trainer Accelerator

When creating the trainer, by defualt the Trainer will choose the most performing accelerator for training which is available in your system, ranked as follow:

1. TPU
2. IPU
3. HPU
4. GPU or MPS
5. CPU

For setting manually the accelerator run:

• accelerator = {'gpu', 'cpu', 'hpu', 'mps', 'cpu', 'ipu'} sets the accelerator to a specific one

trainer = Trainer(solver=pinn, accelerator="cpu")

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

as you can see, even if in the used system GPU is available, it is not used since we set accelerator='cpu'.

Trainer Logging

In PINA you can log metrics in different ways. The simplest approach is to use the MetricTraker class from pina.callbacks as seen in the Introduction to PINA for Physics Informed Neural Networks training tutorial.

However, expecially when we need to train multiple times to get an average of the loss across multiple runs, pytorch_lightning.loggers might be useful. Here we will use TensorBoardLogger (more on logging here), but you can choose the one you prefer (or make your own one).

We will now import TensorBoardLogger, do three runs of training and then visualize the results. Notice we set enable_model_summary=False to avoid model summary specifications (e.g. number of parameters), set it to true if needed.

from lightning.pytorch.loggers import TensorBoardLogger

# three run of training, by default it trains for 1000 epochs
# we reinitialize the model each time otherwise the same parameters will be optimized
for _ in range(3):
    model = FeedForward(
        layers=[10, 10],
        func=torch.nn.Tanh,
        output_dimensions=len(problem.output_variables),
        input_dimensions=len(problem.input_variables),
    )
    pinn = PINN(problem, model)
    trainer = Trainer(
        solver=pinn,
        accelerator="cpu",
        logger=TensorBoardLogger(save_dir="training_log"),
        enable_model_summary=False,
        train_size=1.0,
        val_size=0.0,
        test_size=0.0,
    )
    trainer.train()

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

Missing logger folder: training_log/lightning_logs

`Trainer.fit` stopped: `max_epochs=1000` reached.

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.

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 now visualize the logs by simply running tensorboard --logdir=training_log/ on terminal, you should obtain a webpage as the one shown below:

as you can see, by default, PINA logs the losses which are shown in the progress bar, as well as the number of epochs. You can always insert more loggings by either defining a callback (more on callbacks), or inheriting the solver and modify the programs with different hooks (more on hooks).

Trainer Callbacks

Whenever we need to access certain steps of the training for logging, do static modifications (i.e. not changing the Solver) or updating Problem hyperparameters (static variables), we can use Callabacks. Notice that Callbacks allow you to add arbitrary self-contained programs to your training. At specific points during the flow of execution (hooks), the Callback interface allows you to design programs that encapsulate a full set of functionality. It de-couples functionality that does not need to be in PINA Solvers. Lightning has a callback system to execute them when needed. Callbacks should capture NON-ESSENTIAL logic that is NOT required for your lightning module to run.

The following are best practices when using/designing callbacks.

• Callbacks should be isolated in their functionality.
• Your callback should not rely on the behavior of other callbacks in order to work properly.
• Do not manually call methods from the callback.
• Directly calling methods (eg. on_validation_end) is strongly discouraged.
• Whenever possible, your callbacks should not depend on the order in which they are executed.

We will try now to implement a naive version of MetricTraker to show how callbacks work. Notice that this is a very easy application of callbacks, fortunately in PINA we already provide more advanced callbacks in pina.callbacks.

from lightning.pytorch.callbacks import Callback
from lightning.pytorch.callbacks import EarlyStopping
import torch


# define a simple callback
class NaiveMetricTracker(Callback):
    def __init__(self):
        self.saved_metrics = []

    def on_train_epoch_end(
        self, trainer, __
    ):  # function called at the end of each epoch
        self.saved_metrics.append(
            {key: value for key, value in trainer.logged_metrics.items()}
        )

Let's see the results when applyed to the SimpleODE problem. You can define callbacks when initializing the Trainer by the callbacks argument, which expects a list of callbacks.

model = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)
pinn = PINN(problem, model)
trainer = Trainer(
    solver=pinn,
    accelerator="cpu",
    logger=True,
    callbacks=[NaiveMetricTracker()],  # adding a callbacks
    enable_model_summary=False,
    train_size=1.0,
    val_size=0.0,
    test_size=0.0,
)
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/tutorial11/lightning_logs

`Trainer.fit` stopped: `max_epochs=1000` reached.

We can easily access the data by calling trainer.callbacks[0].saved_metrics (notice the zero representing the first callback in the list given at initialization).

trainer.callbacks[0].saved_metrics[:3]  # only the first three epochs

[{'bound_cond_loss': tensor(0.6365),
  'phys_cond_loss': tensor(0.1283),
  'train_loss': tensor(0.7647)},
 {'bound_cond_loss': tensor(0.6307),
  'phys_cond_loss': tensor(0.1309),
  'train_loss': tensor(0.7616)},
 {'bound_cond_loss': tensor(0.6250),
  'phys_cond_loss': tensor(0.1335),
  'train_loss': tensor(0.7585)}]

PyTorch Lightning also has some built in Callbacks which can be used in PINA, here an extensive list.

We can for example try the EarlyStopping routine, which automatically stops the training when a specific metric converged (here the train_loss). In order to let the training keep going forever set max_epochs=-1.

model = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)
pinn = PINN(problem, model)
trainer = Trainer(
    solver=pinn,
    accelerator="cpu",
    max_epochs=-1,
    enable_model_summary=False,
    enable_progress_bar=False,
    val_size=0.2,
    train_size=0.8,
    test_size=0.0,
    callbacks=[EarlyStopping("val_loss")],
)  # adding a callbacks
trainer.train()

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

As we can see the model automatically stop when the logging metric stopped improving!

Trainer Tips to Boost Accuracy, Save Memory and Speed Up Training

Untill now we have seen how to choose the right accelerator, how to log and visualize the results, and how to interface with the program in order to add specific parts of code at specific points by callbacks. Now, we well focus on how boost your training by saving memory and speeding it up, while mantaining the same or even better degree of accuracy!

There are several built in methods developed in PyTorch Lightning which can be applied straight forward in PINA, here we report some:

• Stochastic Weight Averaging to boost accuracy
• Gradient Clippling to reduce computational time (and improve accuracy)
• Gradient Accumulation to save memory consumption
• Mixed Precision Training to save memory consumption

We will just demonstrate how to use the first two, and see the results compared to a standard training. We use the Timer callback from pytorch_lightning.callbacks to take the times. Let's start by training a simple model without any optimization (train for 2000 epochs).

from lightning.pytorch.callbacks import Timer
from lightning.pytorch import seed_everything

# setting the seed for reproducibility
seed_everything(42, workers=True)

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

pinn = PINN(problem, model)
trainer = Trainer(
    solver=pinn,
    accelerator="cpu",
    deterministic=True,  # setting deterministic=True ensure reproducibility when a seed is imposed
    max_epochs=2000,
    enable_model_summary=False,
    callbacks=[Timer()],
)  # adding a callbacks
trainer.train()
print(f'Total training time {trainer.callbacks[0].time_elapsed("train"):.5f} s')

Seed set to 42

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

`Trainer.fit` stopped: `max_epochs=2000` reached.

Total training time 22.90211 s

Now we do the same but with StochasticWeightAveraging

from lightning.pytorch.callbacks import StochasticWeightAveraging

# setting the seed for reproducibility
seed_everything(42, workers=True)

model = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)
pinn = PINN(problem, model)
trainer = Trainer(
    solver=pinn,
    accelerator="cpu",
    deterministic=True,
    max_epochs=2000,
    enable_model_summary=False,
    callbacks=[Timer(), StochasticWeightAveraging(swa_lrs=0.005)],
)  # adding StochasticWeightAveraging callbacks
trainer.train()
print(f'Total training time {trainer.callbacks[0].time_elapsed("train"):.5f} s')

Seed set to 42

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

Swapping scheduler `ConstantLR` for `SWALR`

`Trainer.fit` stopped: `max_epochs=2000` reached.

Total training time 23.68921 s

As you can see, the training time does not change at all! Notice that around epoch 1600 the scheduler is switched from the defalut one ConstantLR to the Stochastic Weight Average Learning Rate (SWALR). This is because by default StochasticWeightAveraging will be activated after int(swa_epoch_start * max_epochs) with swa_epoch_start=0.7 by default. Finally, the final mean_loss is lower when StochasticWeightAveraging is used.

We will now now do the same but clippling the gradient to be relatively small.

# setting the seed for reproducibility
seed_everything(42, workers=True)

model = FeedForward(
    layers=[10, 10],
    func=torch.nn.Tanh,
    output_dimensions=len(problem.output_variables),
    input_dimensions=len(problem.input_variables),
)
pinn = PINN(problem, model)
trainer = Trainer(
    solver=pinn,
    accelerator="cpu",
    max_epochs=2000,
    enable_model_summary=False,
    gradient_clip_val=0.1,  # clipping the gradient
    callbacks=[Timer(), StochasticWeightAveraging(swa_lrs=0.005)],
)
trainer.train()
print(f'Total training time {trainer.callbacks[0].time_elapsed("train"):.5f} s')

Seed set to 42

GPU available: False, used: False

TPU available: False, using: 0 TPU cores

HPU available: False, using: 0 HPUs

Swapping scheduler `ConstantLR` for `SWALR`

`Trainer.fit` stopped: `max_epochs=2000` reached.

Total training time 24.08936 s

As we can see we by applying gradient clipping we were able to even obtain lower error!

What's next?

Now you know how to use efficiently the Trainer class PINA! There are multiple directions you can go now:

1. Explore training times on different devices (e.g.) TPU

2. Try to reduce memory cost by mixed precision training and gradient accumulation (especially useful when training Neural Operators)

3. Benchmark Trainer speed for different precisions.