Physics-Informed Neural Network for Damped Oscillation

This project implements a Physics-Informed Neural Network (PINN) using PyTorch to solve the damped harmonic oscillator by embedding physical laws directly into the learning process. It combines the flexibility of neural networks with the constraints of differential equations to model physical phenomena accurately.

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Problem Statement

We aim to solve the second-order ordinary differential equation:

    d²u/dt² + 2γ du/dt + ω² u = 0

With initial conditions:

    u(0) = 1
    u'(0) = 0

The exact solution is:

    u(t) = exp(-γt) · cos(ωt)

This type of equation models many real-world systems such as:

• Mass-spring-damper mechanical systems
• Electrical RLC circuits
• Vibrating structures with damping

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Loss Function

The total loss function used in training is a combination of physics-informed constraints and data (if available):

    L_total = ω_eq · L_eq + ω_bc · L_bc + ω_dt · L_data

• L_eq: Residual of the differential equation.
• L_bc: Loss enforcing initial conditions.
• L_data (optional): Supervised data loss (if available).

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Model Architecture

The PINN model is a fully-connected feedforward neural network (MLP) consisting of:

• Input layer: maps scalar time input t to hidden dimension
• Hidden layers: configurable number and width, using chosen activation
• Output layer: outputs predicted scalar value u(t)
• Dropout: applied after each layer (configurable)

Supported activation functions include:

• tanh, sigmoid, logsigmoid, softplus
• swish (custom), sine
• elu, sel, elist

Example structure:

Input (1) → [Hidden x N] → Output (1)

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Usage

Clone the repository and install dependencies:

git clone https://github.com/hungtrannam/Physics-Informed-Neural-for-damped-oscillation.git
cd Physics-Informed-Neural-for-damped-oscillation
pip install -r requirements.txt

Run training:

python main/main_PINN2.py

Or use:

./script/main_PINN2.sh

Trained models, plots, and logs are saved in the runs/ directory.

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Configurable Parameters

Argument	Description
--num_epochs	Number of training iterations
--learning_rate	Learning rate for Adam
--num_neurons	Hidden layer size
--num_hidden_layers	Number of hidden layers
--dropout_rate	Dropout rate
--activation_name	Activation function
--omega_eq	Weight for equation loss
--omega_bc	Weight for boundary loss
--omega_dt	Weight for data loss
--seed	Random seed

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Outputs

• Training loss history (equation, boundary, data)
• Model predictions vs. exact solution
• Optional .gif animation of prediction over time
• Trained model weights saved as .pth
• Visualization and evaluation functions in utils/

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Author

Hung Tran Nam
Physics-Informed Machine Learning | PINNs | Neural ODEs

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