This project demonstrates how to solve a nonlinear second-order ODE—the classic damped pendulum—using a Physics-Informed Neural Network (PINN) in PyTorch. The network learns to approximate the solution by directly embedding the physics into its training loop—no labeled data required.

To solve:

$$ \theta''(t) + b\theta'(t) + c \sin(\theta(t)) = 0 $$

Where:

• ( \theta(t) ): Angular displacement
• ( b ): Damping coefficient
• ( c ): Gravitational constant

This equation models the motion of a pendulum under damping (like swinging through honey).

1. Define a neural network ( \theta_{\text{NN}}(t) ) to approximate the solution.

2. Use automatic differentiation to compute ( \theta' ) and ( \theta'' ).

3. Construct a physics-informed loss function from the ODE residual:

   $$ \mathcal{L}_{\text{ODE}} = \left| \theta''(t) + b\theta'(t) + c \sin(\theta(t)) \right|^2 $$

4. Train the model by minimizing this loss using a gradient-based optimizer.

• PINN.ipynb: Jupyter Notebook with full implementation, training, and plots.
• README.md: This file—describing the approach, methods, and requirements.

• Python 3.x
• PyTorch
• NumPy
• Matplotlib

• Parameter inference: Estimate b, c from data.
• Noisy or partial data scenarios.
• Extension to chaotic or coupled mechanical systems.

Feel free to fork or contribute to expand the library with more physical systems and optimization techniques.