Class Koopman
The class Koopman is a mapping \(K: \mathrm{span}(\Psi) \rightarrow \mathrm{span}(\Psi)\),
which acts as the finite-dimensional approximation of the Koopman operator \(\mathcal{K}\),
i.e.,
\[ \mathcal{K} \Psi \approx K \Psi. \]
Given \(\phi = \mathbf{a}^T \Psi\), where \(\mathbf{a} \in \mathbb{R}^{N_{\psi}}\), then the application of \(\mathcal{K}\) satisfies
\[ \mathcal{K} \phi = \mathcal{K} \mathbf{a}^T \Psi \approx \mathbf{a}^T K \Psi. \]
Attributes¶
_func(tensor -> tensor): The mapping \(K: \mathrm{span}(\Psi) \rightarrow \mathrm{span}(\Psi)\).
Methods¶
__init__(self, K = None, func = None)K(tensor): The matrix representation of the Koopman operator.func(tensor -> tensor): The mapping \(K: \mathrm{span}(\Psi) \rightarrow \mathrm{span}(\Psi)\),
__call__(self, x): Applies the Koopman operator,xshould satisfy \(x \in \mathbb{R}^{N \times N_{\psi}}\).predict(self, x0, dictionary, dim_nontrain, traj_len): Predicts the trajectory of the system.x0(tensor): The initial state of the system.dictionary(Dictionary): The dictionary used to map the state to the feature space.dim_nontrain(int): The dimension of the non-trainable part of the state.traj_len(int): The length of the trajectory to predict.
Info
Understanding the __call__ method of Koopman: Given the data set \(\{x^{(n)}\}_{n = 1}^N\),
it represents a mapping:
\[ \left[
\begin{array}{cccc}
\psi_1(x^{(1)})&\psi_2(x^{(1)})&\cdots&\psi_{N_{\psi}}(x^{(1)})\\
\psi_1(x^{(2)})&\psi_2(x^{(2)})&\cdots&\psi_{N_{\psi}}(x^{(2)})\\
\vdots&\vdots&\ddots&\vdots\\
\psi_1(x^{(N)})&\psi_2(x^{(N)})&\cdots&\psi_{N_{\psi}}(x^{(N)})\\
\end{array}
\right] \rightarrow \left[
\begin{array}{cccc}
\mathcal{K}\psi_1(x^{(1)})&\mathcal{K}\psi_2(x^{(1)})&\cdots&\mathcal{K}\psi_{N_{\psi}}(x^{(1)})\\
\mathcal{K}\psi_1(x^{(2)})&\mathcal{K}\psi_2(x^{(2)})&\cdots&\mathcal{K}\psi_{N_{\psi}}(x^{(2)})\\
\vdots&\vdots&\ddots&\vdots\\
\mathcal{K}\psi_1(x^{(N)})&\mathcal{K}\psi_2(x^{(N)})&\cdots&\mathcal{K}\psi_{N_{\psi}}(x^{(N)})\\
\end{array}
\right]\]