Loss

The simple DMD Autoencoder loss function is a combination of three evaluations:

Autoencoder reconstruction loss - ensures that the original trajectories can be recovered. \begin{equation} \label{eq:8} L_{1} = MSE \left| X - g^{-1}(\tilde{Y}) \right| \end{equation}

DMD loss - evaluate the linearity of the latent space dynamics. The following is derived in equation (6).

$$\begin{equation} \label{eq:9} L_{2} = \left\| Y' ( I - V V^{T}) \right\| _{F}^{2} \end{equation}$$

DMD reconstruction loss - evaluate the DMD least squares fit of the $A$ matrix.

$$\begin{equation} \label{eq:10} L_{3} = MSE\left\| Y - \tilde{Y} \right\| \end{equation}$$

Linearity loss - evaluate the linearity of the latent space to enhance long-term predictions.

$$\begin{equation} \label{eq:11} L_{4} = MSE\left\| X - g^{-1}(\tilde{Y}) \right\| \end{equation}$$

The final loss function is $L = \alpha_{1}L_{1} + \alpha_{2}L_{2} + \alpha_{3}L_{3} + \alpha_{4}L_{4}$.