Now the real work begins! Let's implement a Physics-Informed Neural Network step by step.

Step 1: Physics Loss Function

The heart of PINN is computing the physics residual using automatic differentiation:

Step 2: Complete PINN Training Loop

Key Components:

1. Data loss: Fit the sparse measurements
2. Physics loss: Satisfy the differential equation
3. Collocation points: Where we enforce physics (not necessarily data points)
4. Balance parameter \( \lambda \): Controls the trade-off

Step 3: Train the PINN

The moment of truth! Let's train the PINN and see if it can learn both the data and the physics: