FastVPINNs - Training Tutorial - Helmholtz Problem

In this notebook, we will explore how to use FastVPINNs to solve the Helmholtz equation using custom loss functions and neural networks.

hp-Variational Physics-Informed Neural Networks (hp-VPINNs)

Variational Physics-Informed Neural Networks (VPINNs) are a specialized class of Physics-Informed Neural Networks (PINNs) that are trained using the variational formulation of governing equations. The hp-VPINNs extend this concept by incorporating hp-FEM principles such as h- and p-refinement to enhance solution accuracy.

Mathematical Formulation

The 2D Helmholtz equation is given by:

\[ -\Delta u + ku = f \]

where: - \(u\) is the solution - \(k\) is the Helmholtz coefficient - \(f\) is the source term - \(\Delta\) is the Laplacian operator

To obtain the weak form, we multiply by a test function \(v\) and integrate over the domain \(\Omega\):

\[ \int_{\Omega} (-\Delta u + ku)v \, dx = \int_{\Omega} f v \, dx \]

Applying integration by parts to the Laplacian term:

\[ \int_{\Omega} \nabla u \cdot \nabla v \, dx + \int_{\Omega} kuv \, dx - \int_{\partial \Omega} \nabla u \cdot \mathbf{n} v \, ds = \int_{\Omega} f v \, dx \]

With \(v = 0\) on \(\partial \Omega\), the boundary integral vanishes, giving us the weak form:

\[ \int_{\Omega} \nabla u \cdot \nabla v \, dx + \int_{\Omega} kuv \, dx = \int_{\Omega} f v \, dx \]

Problem Setup

In this example, we solve the Helmholtz equation with the following exact solution:

\[ u(x,y) = (x + y)\sin(\pi x)\sin(\pi y) \]

Key Parameters

Geometry Parameters:

Mesh type: Quadrilateral
Domain: \([0,1] \times [0,1]\)
Cells: \(2\times2\) grid
Boundary points: 400
Test points: \(100\times100\) grid

Finite Element Parameters:

Order: 6 (Legendre)
Quadrature: Gauss-Jacobi (order 10)
Transformation: Bilinear

Neural Network Parameters:

Architecture: \([2, 30, 30, 30, 1]\)
Activation: \(\tanh\)
Learning rate: 0.002
Boundary loss penalty (\(\beta\)): 10
Training epochs: 20,000

Boundary Conditions

The boundary conditions are implemented through four functions defining values on each boundary:

def left_boundary(x, y):
    return (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)

def right_boundary(x, y):
    return (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)

def top_boundary(x, y):
    return (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)

def bottom_boundary(x, y):
    return (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)

### Source Term

The source term f(x,y) is derived through the method of manufactured solutions:

```python
def rhs(x, y):
    term1 = 2 * np.pi * np.cos(np.pi * y) * np.sin(np.pi * x)
    term2 = 2 * np.pi * np.cos(np.pi * x) * np.sin(np.pi * y)
    term3 = (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)
    term4 = -2 * (np.pi**2) * (x + y) * np.sin(np.pi * x) * np.sin(np.pi * y)
    return term1 + term2 + term3 + term4

Implementation Steps

Mesh Generation:
Create a 2×2 quadrilateral mesh
Generate boundary points
Set up test points for solution evaluation
Finite Element Space:
Define Legendre basis functions
Set up quadrature rules
Configure boundary conditions
Neural Network Model:
Create a dense neural network with 3 hidden layers
Configure the Helmholtz loss function
Set up the training parameters
Training Process:
Train for 20,000 epochs
Track loss and timing metrics
Evaluate solution accuracy
Visualization and Analysis:
Plot training loss
Compare exact and predicted solutions
Generate error plots
Calculate error metrics (L2, L1, L∞ norms)

Key Differences from Poisson Problem

PDE Structure:
Addition of the ku term
Modified variational form
Different source term computation
Loss Function:
Uses pde_loss_helmholtz instead of pde_loss_poisson
Includes Helmholtz coefficient in bilinear parameters
Boundary Conditions:
Non-zero Dirichlet conditions
More complex boundary value functions

Results Analysis

The implementation generates four key visualizations: 1. Training loss convergence 2. Exact solution contour 3. Predicted solution contour 4. Absolute error distribution

Error metrics are computed in various norms: - L2 norm (global accuracy) - L1 norm (average absolute error) - L∞ norm (maximum error) - Relative versions of each norm

These metrics help assess the solution quality and convergence characteristics of the method.