Tutorial: Applying Periodic Boundary Conditions in PINNs to solve the Helmotz Problem

This tutorial demonstrates how to solve a one-dimensional Helmholtz equation with periodic boundary conditions (PBC) using Physics-Informed Neural Networks (PINNs).
We will use standard PINN training, augmented with a periodic input expansion as introduced in An Expert’s Guide to Training Physics-Informed Neural Networks.

Let's start with 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[tutorial]"

import torch
import matplotlib.pyplot as plt
import warnings

from pina import Condition, Trainer
from pina.problem import SpatialProblem
from pina.operator import laplacian
from pina.model import FeedForward
from pina.model.block import PeriodicBoundaryEmbedding  # The PBC module
from pina.solver import PINN
from pina.domain import CartesianDomain
from pina.equation import Equation
from pina.callback import MetricTracker

warnings.filterwarnings("ignore")

Problem Definition

The one-dimensional Helmholtz problem is mathematically expressed as:

$$ \begin{cases} \frac{d^2}{dx^2}u(x) - \lambda u(x) - f(x) &= 0 \quad \text{for } x \in (0, 2) \\ u^{(m)}(x = 0) - u^{(m)}(x = 2) &= 0 \quad \text{for } m \in \{0, 1, \dots\} \end{cases} $$

In this case, we seek a solution that is $C^{\infty}$ (infinitely differentiable) and periodic with period 2, over the infinite domain $x \in (-\infty, \infty)$.

A classical PINN approach would require enforcing periodic boundary conditions (PBC) for all derivatives—an infinite set of constraints—which is clearly infeasible.

To address this, we adopt a strategy known as coordinate augmentation. In this approach, we apply a coordinate transformation $v(x)$ such that the transformed inputs naturally satisfy the periodicity condition:

$$ u^{(m)}(x = 0) - u^{(m)}(x = 2) = 0 \quad \text{for } m \in \{0, 1, \dots\} $$

For demonstration purposes, we choose the specific parameters:

• $\lambda = -10\pi^2$
• $f(x) = -6\pi^2 \sin(3\pi x) \cos(\pi x)$

These yield an analytical solution:

$$ u(x) = \sin(\pi x) \cos(3\pi x) $$

def helmholtz_equation(input_, output_):
    x = input_.extract("x")
    u_xx = laplacian(output_, input_, components=["u"], d=["x"])
    f = (
        -6.0
        * torch.pi**2
        * torch.sin(3 * torch.pi * x)
        * torch.cos(torch.pi * x)
    )
    lambda_ = -10.0 * torch.pi**2
    return u_xx - lambda_ * output_ - f


class Helmholtz(SpatialProblem):
    output_variables = ["u"]
    spatial_domain = CartesianDomain({"x": [0, 2]})

    # here we write the problem conditions
    conditions = {
        "phys_cond": Condition(
            domain=spatial_domain, equation=Equation(helmholtz_equation)
        ),
    }

    def solution(self, pts):
        return torch.sin(torch.pi * pts) * torch.cos(3.0 * torch.pi * pts)


problem = Helmholtz()

# let's discretise the domain
problem.discretise_domain(200, "grid", domains=["phys_cond"])

As usual, the Helmholtz problem is implemented in PINA as a class. The governing equations are defined as conditions, which must be satisfied within their respective domains. The solution represents the exact analytical solution, which will be used to evaluate the accuracy of the predicted solution.

For selecting collocation points, we use Latin Hypercube Sampling (LHS), a common strategy for efficient space-filling in high-dimensional domains

Solving the Problem with a Periodic Network

Any $\mathcal{C}^{\infty}$ periodic function $u : \mathbb{R} \rightarrow \mathbb{R}$ with period $L \in \mathbb{N}$
can be constructed by composing an arbitrary smooth function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ with a smooth, periodic mapping$v : \mathbb{R} \rightarrow \mathbb{R}^n$ of the same period $L$. That is,

$$ u(x) = f(v(x)). $$

This formulation is general and can be extended to arbitrary dimensions.
For more details, see A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks.

In our specific case, we define the periodic embedding as:

$$ v(x) = \left[1, \cos\left(\frac{2\pi}{L} x\right), \sin\left(\frac{2\pi}{L} x\right)\right], $$

which constitutes the coordinate augmentation. The function $f(\cdot)$ is approximated by a neural network $NN_{\theta}(\cdot)$, resulting in the approximate PINN solution:

$$ u(x) \approx u_{\theta}(x) = NN_{\theta}(v(x)). $$

In PINA, this is implemented using the PeriodicBoundaryEmbedding layer for $v(x)$,
paired with any pina.model to define the neural network $NN_{\theta}$.

Let’s see how this is put into practice!

# we encapsulate all modules in a torch.nn.Sequential container
model = torch.nn.Sequential(
    PeriodicBoundaryEmbedding(input_dimension=1, periods=2),
    FeedForward(
        input_dimensions=3,  # output of PeriodicBoundaryEmbedding = 3 * input_dimension
        output_dimensions=1,
        layers=[64, 64],
    ),
)

As simple as that!

In higher dimensions, you can specify different periods for each coordinate using a dictionary.
For example, periods = {'x': 2, 'y': 3, ...} indicates a periodicity of 2 in the $x$ direction,
3 in the $y$ direction, and so on.

We will now solve the problem using the usual PINN and Trainer classes. After training, we'll examine the losses using the MetricTracker callback from pina.callback.

solver = PINN(problem=problem, model=model)
trainer = Trainer(
    solver,
    max_epochs=2000,
    accelerator="cpu",
    enable_model_summary=False,
    callbacks=[MetricTracker()],
)
trainer.train()

You are using the plain ModelCheckpoint callback. Consider using LitModelCheckpoint which with seamless uploading to Model registry.

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.

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

We are going to plot the solution now!

pts = solver.problem.spatial_domain.sample(256, "grid", variables="x")
predicted_output = solver(pts).extract("u").tensor.detach()
true_output = solver.problem.solution(pts)
plt.plot(pts.extract(["x"]), predicted_output, label="Neural Network solution")
plt.plot(pts.extract(["x"]), true_output, label="True solution")
plt.legend()

<matplotlib.legend.Legend at 0x7fe318a64a00>

Great, they overlap perfectly! This seems a good result, considering the simple neural network used to some this (complex) problem. We will now test the neural network on the domain $[-4, 4]$ without retraining. In principle the periodicity should be present since the $v$ function ensures the periodicity in $(-\infty, \infty)$.

# plotting solution
with torch.no_grad():
    # Notice here we put [-4, 4]!!!
    new_domain = CartesianDomain({"x": [0, 4]})
    x = new_domain.sample(1000, mode="grid")
    fig, axes = plt.subplots(1, 3, figsize=(15, 5))
    # Plot 1
    axes[0].plot(x, problem.solution(x), label=r"$u(x)$", color="blue")
    axes[0].set_title(r"True solution $u(x)$")
    axes[0].legend(loc="upper right")
    # Plot 2
    axes[1].plot(x, solver(x), label=r"$u_{\theta}(x)$", color="green")
    axes[1].set_title(r"PINN solution $u_{\theta}(x)$")
    axes[1].legend(loc="upper right")
    # Plot 3
    diff = torch.abs(problem.solution(x) - solver(x))
    axes[2].plot(x, diff, label=r"$|u(x) - u_{\theta}(x)|$", color="red")
    axes[2].set_title(r"Absolute difference $|u(x) - u_{\theta}(x)|$")
    axes[2].legend(loc="upper right")
    # Adjust layout
    plt.tight_layout()
    # Show the plots
    plt.show()

It's clear that the network successfully captures the periodicity of the solution, with the error also exhibiting a periodic pattern. Naturally, training for a longer duration or using a more expressive neural network could further improve the results.

What's next?

Congratulations on completing the one-dimensional Helmholtz tutorial with PINA! Here are a few directions you can explore next:

1. Train longer or with different architectures: Experiment with extended training or modify the network's depth and width to evaluate improvements in accuracy.

2. Apply PeriodicBoundaryEmbedding to time-dependent problems: Explore more complex scenarios such as spatiotemporal PDEs (see the official documentation for examples).

3. Try extra feature training: Integrate additional physical or domain-specific features to guide the learning process more effectively.

4. ...and many more!: Extend to higher dimensions, test on other PDEs, or even develop custom embeddings tailored to your problem.

For more resources and tutorials, check out the PINA Documentation.