Helmholtz Mixed BC FEM VarPINN
This example trains a variational PINN for a 2D Helmholtz problem and compares it against a fine FEM reference solution.
Problem Setup
The PDE has the Helmholtz form with mixed Dirichlet and Neumann data and a manufactured exact solution.
Step 1: Write a Weak Residual
The residual is assembled in variational form rather than as a pointwise strong-form loss.
Step 2: Combine Hard Boundary Handling With VPINN Training
The neural ansatz handles some constraints structurally while the weak form captures the PDE and remaining conditions.
Step 3: Compare Against FEM
The script solves a classical FEM reference problem on a finer mesh and visualizes the difference.
What To Notice
- This is a representative VPINN workflow in the tutorial suite.
- Weak forms can improve how derivative-heavy PDEs are handled.
- Direct FEM comparison makes the example especially useful for benchmarking.