Learning Objectives:

• Understand how to incorporate physics constraints into DeepONet training
• Implement soft-constrained physics-informed DeepONet (PI-DeepONet)
• Apply PI-DeepONet to the 1D Poisson equation with Dirichlet boundary conditions
• Compare physics-informed vs data-driven approaches

Problem: Learn the solution operator for the 1D Poisson equation:

with Dirichlet boundary conditions: \(u(0) = 0\), \(u(1) = 0\)

Physics-Informed Approach: Incorporate the differential equation and boundary conditions directly into the loss function.