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Reservoir Computing for the Lorenz System

This repository contains an implementation of a Reservoir Computing (RC) system to predict the Lorenz system's dynamics. The Lorenz system is a set of ordinary differential equations that model atmospheric convection and is known for its chaotic solutions.

Overview

Reservoir Computing is a type of recurrent neural network (RNN) where only the output weights are trained, making it computationally efficient. This project uses RC to predict the Lorenz system's behavior given initial conditions.

Files

  • RC.ipynb: Jupyter notebook containing the implementation of Reservoir Computing for the Lorenz system. It includes data generation, model training, prediction, and error analysis.

Key Components

Data Generation

The Lorenz system's equations are defined as:

  • dx/dt = sigma * (y - x)
  • dy/dt = x * (rho - z) - y
  • dz/dt = x * y - beta * z

Where:

  • sigma is the Prandtl number.
  • rho is the Rayleigh number.
  • beta is a geometric factor.

Reservoir Computer

The reservoir computer consists of:

  • Input Layer: Maps the input signal to a high-dimensional space.
  • Reservoir Layer: A fixed, recurrent network of neurons.
  • Output Layer: Trained to produce the desired output.

Implementation Details

  • Data Preparation: Generate training and testing data from the Lorenz system.
  • Training: Train the reservoir computer using the training data.
  • Prediction: Use the trained model to predict the Lorenz system's future states.
  • Error Analysis: Compare the predicted states with the actual states and compute the prediction errors.

Usage

  1. Clone the repository:

    git clone https://github.com/yourusername/lorenz-reservoir-computing.git
cd lorenz-reservoir-computing

  2. Install the necessary dependencies:

    pip install numpy matplotlib torch

  3. Run the Jupyter notebook: Open HW D.ipynb in Jupyter Notebook or Jupyter Lab and execute the cells to generate data, train the model, and visualize the results.

Example Output

The notebook will generate plots showing:

  • The predicted vs. actual trajectories of the Lorenz system.
  • The cumulative prediction error over time.

Dependencies

  • numpy
  • matplotlib
  • torch

Install the dependencies using:

pip install numpy matplotlib torch