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Tutorials: Predictions in Chaotic Dynamical Systems

Tutorials: Predictions in Chaotic Dynamical Systems

This is a tutorial to employ echo state networks (ESNs) and long short-term memory networks (LSTMs) for the prediction and analysis of chaotic dynamics. This library contains both Tensorflow and PyTorch implementations for the LSTM and employs the Magrilab/EchoStateNetwork. Please note that encountered issues may be addressed there.

The example system found here is the Lorenz 63 system, which is found in dynamicalsystems.equations

$$ \begin{aligned} &\dfrac{\mathrm{d}x}{\mathrm{d}t} = \sigma (y-x) \\ &\dfrac{\mathrm{d}y}{\mathrm{d}t} = x (\rho-z) - y \\ &\dfrac{\mathrm{d}z}{\mathrm{d}t} = xy - \beta z. \end{aligned} $$

Tutorials: LSTM and ESN to learn Lorenz-63

The tutorial for the LSTM can be found in LSTM_Tutorial_Lorenz63.ipynb and the ESN can be found in ESB_Tutorial_Lorenz63.ipynb.

Example: Attractor reconstruction by reference (black), LSTM (blue) and ESN (red):

Requirements:

You can find a list of requirements in requirements.txt. We recommend installing the requirements in a conda environment.

For numpy version > 1.15, there may be a np.int error occurring; this is due to a missing bugfix from skopt. Follow the instructions of this issue:Resolve Deprecated Numpy Attribute Error np.int