A Physics-Informed Neural Network (PINN) that solves the damped harmonic oscillator ODE without any training data — using only the physics equation itself as the loss function.
| Parameter | Value |
|---|---|
| Initial position x(0) | 0.7 |
| Initial velocity x'(0) | 1.2 |
| Damping range ξ | [0.1, 0.4] |
| Time domain z | [0, 20] |
A standard neural network learns from labeled data. A PINN learns from physics. Instead of minimizing prediction error against a dataset, the loss function directly penalizes violations of the governing differential equation:
Total Loss = Physics Loss + Initial Condition Loss
Physics Loss = mean( x'' + 2ξx' + x )² ← ODE residual at random points
IC Loss = (x(0) - 0.7)² + (x'(0) - 1.2)²
Derivatives x' and x'' are computed exactly using PyTorch autograd — no finite differences, no approximations.
| ξ | L2 Relative Error |
|---|---|
| 0.1 | 26.5% |
| 0.2 | 3.3% |
| 0.3 | 2.6% |
| 0.4 | 4.5% |
The ξ=0.1 case (least damping, most oscillations) is hardest due to spectral bias — neural networks naturally learn low-frequency patterns first and struggle with rapid oscillations over a long time domain.
PINN-Damped-Oscillator/
├── pinn_oscillator.py # model architecture, training, evaluation
├── outputs/
│ ├── pinn_results.png # comparison plots vs analytical solution
│ └── pinn_oscillator.pth # saved model weights
├── requirements.txt
└── README.md
python3 -m venv pinn_env
source pinn_env/bin/activate
pip install -r requirements.txt
python3 pinn_oscillator.pytorch
numpy
matplotlib
- Input: (z, ξ) — time and damping ratio
- Output: x(z, ξ) — predicted position
- Layers: 6 hidden layers, 128 units each, tanh activation
- Why tanh: ReLU has zero second derivative everywhere — autograd can't compute x'' through it. tanh is smooth and infinitely differentiable.
- Training: 30,000 epochs, Adam optimizer (lr=5e-4), 3,000 collocation points per epoch
This evaluation task directly demonstrates the core mechanism behind the GENIE PINNDE project. The reverse-time diffusion equation used in particle shower generation is a high-dimensional PDE — the same PINN approach (physics residual as loss + autograd derivatives) scales to that setting. This oscillator serves as a 1D proof-of-concept for the full pipeline.
Binoy Saha | github.com/binoysaha025 | binoysaha025@gmail.com