08 FEM and Variational PINNs

This chapter covers weak-form workflows, including pure finite-element solves and variational PINN comparisons against FEM references.

In This Chapter

• Poisson 2D FEM

  Assemble and solve a classical weak-form Poisson problem.

  Walk through the example

• Stationary Allen-Cahn FEM

  Solve a nonlinear stationary weak-form problem with a classical FEM loop.

  Walk through the example

• Diffusion-Reaction Robin BC

  Compare VPINN and FEM solutions under mixed Dirichlet/Robin conditions.

  Walk through the example

• Helmholtz Mixed BC FEM VarPINN

  Train a variational PINN and compare it against a fine-mesh FEM reference.

  Walk through the example

• Helmholtz 3D

  Solve a 3D weak-form problem on a complex extruded geometry.

  Walk through the example

Download Source Files

• poisson_2d_fem.py
• stationary_allen_cahn_fem.py
• diffusion_reaction_robinBC.py
• helmholtz_mixedBC_fem_varpinn.py
• helmholtz_3D.py