Dynamic mode decomposition

Core idea

Given snapshot pairs $x_j,y_j\in\mathbb C^d$ , assemble $X=[x_1\ \cdots\ x_M]$ , $Y=[y_1\ \cdots\ y_M]$ . Dynamic mode decomposition (DMD) fits a matrix $A$ minimizing

\|Y-AX\|_F^2,

using the solution of minimum Frobenius norm when the minimizer is not unique, and studies its eigenvalues and associated modes. The norm $\|B\|_F^2=\sum_{i,j}|B_{ij}|^2$ is the sum of squared entry magnitudes.

Koopman connection

If $y_j=T(x_j)$ , this fits future coordinates as linear combinations of present coordinates. These coordinates are a finite dictionary of observables. Their span need not be invariant under the actual Koopman operator; the fitted finite matrix is therefore an approximation, even when the original dynamics is deterministic.

The broader extended DMD construction uses nonlinear observables and makes the projected operator interpretation explicit. Finite-data eigenvalues need not be true Koopman eigenvalues.

References

J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz, “On Dynamic Mode Decomposition: Theory and Applications”, §2.