Double Pendulum PINN (Physics-Informed Neural Network)

A deep learning approach to modeling chaotic systems by enforcing Lagrangian mechanics within a Gated Recurrent Unit (GRU) architecture.

This project implements a Physics-Informed Neural Network (PINN) to predict the trajectory of a double pendulum based on video tracking data. By integrating the Euler-Lagrange equations directly into the loss function, the model learns to respect physical constraints such as energy conservation and rigid-body dynamics, significantly reducing the "drift" common in standard sequence models.

Tech Stack

• Core: Python, PyTorch (Modeling & Autograd)
• Numerical: NumPy (Data Processing), SciPy (Signal Analysis)
• Visualization: Matplotlib (Real-time Animation & Phase Space plots)
• Architecture: Custom Gated Recurrent Unit (GRU)
• Physics Engine: Lagrangian Mechanics (Second-order ODE residuals)

Core Features

• Physics-Informed Loss: Combines standard Mean Squared Error (MSE) with a physical penalty term based on the system's Lagrangian: $Loss = MSE + \lambda \cdot L_{physics}$.
• Custom GRU Architecture: Hand-implemented GRU cells for granular control over hidden state dynamics and gradient flow.
• Modular MSE: Implemented circular topology logic to correctly interpret the $0 = 2\pi$ angular relationship, preventing prediction "snapping" and spinning artifacts.
• Chaos Management: Optimized for high-sensitivity dynamical systems, maintaining trajectory integrity through chaotic transitions.
• Efficient Data Pipeline: Implemented data striding and downsampling to balance computational speed with high-fidelity physical modeling.

Technical Challenges Solved

1. Exploding Gradients: Managed the numerical instability caused by dividing by $dt^2$ in the acceleration terms through adaptive loss annealing and gradient norm clipping.
2. Coordinate Misalignment: Synchronized the mathematical Lagrangian reference frame with the real-world video tracking coordinate system.
3. Local Minima: Utilized a multi-stage training strategy (Warm-up -> Physics Refinement) to ensure the model captures the motion profile before enforcing strict physical laws.

How to Run Locally

1. Clone the Repository

git clone https://github.com/Emwook/double_pendulum.git
cd double-pendulum

2. Install Dependencies

pip install -r requirements.txt

3. Run the Inference Demo

Load the pre-trained 18KB model and visualize the side-by-side comparison between actual tracking and AI prediction.

python scripts/demo.py

4. Run Training

To retrain the model or experiment with different physics weights:

python scripts/training_loop.py

Data Source

The training data for this project consists of high-frequency video tracking coordinates from Myers, Audun; Khasawneh, Firas; Tempelman, Josh; Petrushenko, David (2020), “Low-cost double pendulum for high-quality data collection with open-source video tracking and analysis”. The raw trajectories were processed and converted from pixel-space to angular coordinates (radians) to fit the Lagrangian reference frame. Link to the source:

https://data.mendeley.com/datasets/7yd2ntbh3w/1

Future Work

Energy Conservation: Implementing a Hamiltonian-based loss to further ensure energy symmetry.

Rocket Pathing: This project serves as a prototype for a larger-scale trajectory prediction system for aerospace applications.