Discrete Dataset
Consider a simple nonlinear discrete spectrum system, described as follows:
\dot{x_{1}} = \mu x_{1}
\dot{x_{2}} = \lambda (x_{2} - x_{1}^{2})
Analytic Koopman Embedding
This system has a three-dimensional Koopman invariant space.
y_{1} = x_{1}
y_{2} = x_{2}
y_{3} = x_{1}^{2}
Therefore, the nonlinear system x can be transformed to a new coordinate system where dynamics are linear (in the form of equation (1) and (2)).
\dot{y_{1}} = \dot{x_{1}} = \mu x_{1} = \mu y_{1}
\dot{y_{2}} = \dot{x_{2}} = \lambda (x_{2} - x_{1}^{2}) = \lambda (y_{2} - y_{3})
\dot{y_{3}} = \dot{x_{1}^{2}} = 2 x_{1} \dot{x_{1}} = 2 x_{1} \mu x_{1} = 2 \mu x_{1}^{2}
\begin{equation} \label{eq:1}
\frac{d}{dt}
\begin{bmatrix}
y_{1}\\
y_{2}\\
y_{3}
\end{bmatrix} =
\begin{bmatrix}
\mu & 0 & 0 \\
0 & \lambda & -\lambda\\
0 & 0 & 2\mu
\end{bmatrix}
\begin{bmatrix}
y_{1}\\
y_{2}\\
y_{3}
\end{bmatrix}
\end{equation}
In order to compare the Dynamic Mode Decomposition reconstruction accuracy on the two coordinates x and y, we evaluated
\begin{equation} \label{eq:6}
\begin{aligned}
\left\| X'- AX \right\| _{F}^{2} = \left\| X'- (X' V \Sigma^{-1} U^{*}) X \right\| _{F}^{2}\\
= \left\| X'- (X' V \Sigma^{-1} U^{*}) ( U \Sigma V^{T}) \right\| _{F}^{2} \\ =\left\| X' ( I - V V^{T}) \right\| _{F}^{2}
\end{aligned}
\end{equation}
where in the y coordinate system dynamic mode decomposition reconstruction was able to recover twice as many significant figures in comparison to the x coordinate.
- Related notebooks:
dmd_autoencoder_discrete_train.ipynbandcompare_full_machine_results_discrete_dataset.ipynb
DMD Autoencoder Results
The Dynamic Mode Decomposition Autoencoder attempted to find a nonlinear mapping g which will map the trajectories to a space where the dynamics are approximately linear.