Optimising Lagrangian Neural Networks

This project is a demonstration of Lagrangian Neural Networks (LNNs) superiority to Standard MLP due the utilisation of Euler-Lagrange Equation which allows the LNN to follow constraints of Energy and Lagrangian Mechanics. To make the analysis and comparison fair, the LNNs and Standard MLPs are optimised for their architecture stack and then compared.

Different kinds of systems which follow Lagrangian Dynamics are considered for comparison this includes

• Simple Pendulums
• Compound Pendulums

Project Structure

Optimising Lagrangian Neural Networks
|
|--- src
|    |--- __init.py__
|    |--- Simulation.py
|    |--- Model.py
|    |--- Training.py
|--- main.py
|--- requirements.txt
|--- README.md
|--- LICENSE

Cloning Instructions

Installation

git clone https://github.com/Sammybro11/Optimising-Lagrangian-Neural-Networks
cd Optimising-Lagrangian-Neural-Networks

Python Virtual Environment

python3 -m venv env
source env/bin/activate

Installing required Packages

pip install -r requirements.txt

---

Theory

Let us consider the case of a simple Pendulum

The Lagrangian $L(\phi, \dot{\phi}, t)$ describes the motion of the pendulum, where:

• $\phi $ = Vertical angle of Pendulum from support
• $\dot{\phi}$ = Angular velocity of Pendulum from support
• $t$ = Time variable

The Euler-Lagrange equation is:

$$\frac{d}{dt} \nabla_{\dot{\phi}} L = \nabla_\phi L$$

where:

• $\nabla_\phi L$ is the gradient w.r.t. $\phi$
• $\nabla_{\dot{\phi}} L$ is the gradient w.r.t. $\dot{\phi}$

We want to solve for $\ddot{\phi}$:

$$\frac{d}{dt} \nabla_{\dot{\phi}} L = \nabla_\phi L$$

By applying Chain Rule

$$\frac{\partial}{\partial t} \nabla_{\dot{\phi}} L + \nabla^2_{\dot{\phi} \dot{\phi}} L \, \ddot{\phi} + \nabla^2_{\phi \dot{\phi}} L \, \dot{\phi} = \nabla_\phi L$$ $$\nabla^2_{\dot{\phi} \dot{\phi}} L \, \ddot{\phi} = \nabla_\phi L - \nabla^2_{\phi \dot{\phi}} L \, \dot{\phi} - \frac{\partial}{\partial t} \nabla_{\dot{\phi}} L$$

Therefore,

$$\ddot{\phi} = [\nabla^2_{\dot{\phi} \dot{\phi}} L]^{-1} [\nabla_\phi L - \nabla^2_{\phi \dot{\phi}} L \, \dot{\phi} - \frac{\partial}{\partial t} \nabla_{\dot{\phi}} L]$$

---

Model Structure

The second time derivative obtained from the LNN + Euler Lagrange Equation ( src/Model.py ) is used to obtain the next time step values of the state variables of the system using RK4 Method ( src/Simulation.py ). This is then used to obtain Mean Squared Error Loss used in backpropagation using Adam Optimizer.

Analysis of Simple Pendulum

Analysis of Pendulum on Moving Support

Analysis of Duffing Oscillator

📜 License

Apache License 2.0. Free to use and modify.