First component - jumping rope.

Second component - see-saw.

Third component - flipping wave.

Superposition yields the original signal.

Least-squares definition of $\mathbf{A}$

$$ \underbrace{[\mathbf{x}_2, ..., \mathbf{x}_N]}_{\mathbf{Y}} = A \underbrace{[\mathbf{x}_1, ..., \mathbf{x}_{N-1}]}_{\mathbf{X}} $$

$$ \underset{\mathbf{A}}{\mathrm{argmin}} ||\mathbf{Y}-\mathbf{AX}||_F = \mathbf{YX}^\dagger $$

$^\dagger$ denotes the pseudo inverse

Useful properties of the eigendecomposition $$ \mathbf{A} = \mathbf{\Phi\Lambda\Phi}^{-1} $$

$$ \mathbf{x}_2 = \mathbf{Ax}_1 = \mathbf{\Phi\Lambda\Phi}^{-1} \mathbf{x}_1 $$

$$ \mathbf{x}_3 = \mathbf{AAx}_1 = \mathbf{\Phi\Lambda\Phi}^{-1}\mathbf{\Phi\Lambda\Phi}^{-1} \mathbf{x}_1 = \mathbf{\Phi\Lambda}^{2}\mathbf{\Phi}^{-1} \mathbf{x}_1 $$

$$ \mathbf{x}_n = \mathbf{\Phi\Lambda}^{n-1}\mathbf{\Phi}^{-1} \mathbf{x}_1 $$ Only the $\mathbf{\Lambda}$ term encodes time dependency!

Dealing with large snapshots - DMD:

1. $\mathbf{X} = \mathbf{U\Sigma V}^T$
2. $\tilde{\mathbf{A}} = \mathbf{U}_r^T \mathbf{AU}_r = \mathbf{U}_r^T \mathbf{YX}^\dagger\mathbf{U}_r$, $\quad \tilde{\mathbf{A}} \in \mathbb{R}^{r\times r}$
3. $\tilde{\mathbf{A}} = \mathbf{U}_r^T\mathbf{Y}\mathbf{V}_r\mathbf{\Sigma}_r^{-1}\mathbf{U}_r^T\mathbf{U}_r = \mathbf{U}_r^T\mathbf{Y}\mathbf{V}_r\mathbf{\Sigma}_r^{-1}$
4. $\tilde{\mathbf{A}} =\mathbf{W\Lambda}_r\mathbf{W}^{-1}$, $\quad \mathbf{W}, \mathbf{\Lambda}_r \in \mathbb{C}^{r\times r}$
5. $\mathbf{\Phi}_r = \mathbf{U}_r\mathbf{W}$

Issues with the standard DMD:

• poor reconstruction accuracy
• data overfitting
• sensitive w.r.t. noise/nonlinearity
• sensitive w.r.t. truncation ($r$)

Reconstruction of a single snapshot: $$ \mathbf{x}_n \approx \mathbf{\Phi}_r\mathbf{\Lambda}_r^{n-1}\mathbf{b}_r $$ with $\mathbf{b}_r = \mathbf{\Phi}^{-1} \mathbf{x}_1$

reconstruction of the full dataset

with $\mathbf{b}_r = \mathbf{\Phi}_r^{-1}\mathbf{x}_1$, $M$ - length of $\mathbf{x}$, $N$ - number of snapshots, $r$ - truncation rank

Operator definition in the optimized DMD:

$$ \underset{\mathbf{\lambda}_r,\mathbf{\Phi}_r,\mathbf{b}_r}{\mathrm{argmin}}\left|\left| \mathbf{M}-\mathbf{\Phi}_r\mathbf{D_b}\mathbf{V}_{\mathbf{\lambda}_r} \right|\right|_F $$

$\rightarrow$ "optDMD" problem is nonlinear and non-convex

idea: borrow techniques from ML/DL

• stochastic gradient descent
• automatic differentiation
• train-validation-split
• early stopping

$\rightarrow$ refer to 10.48550/arXiv.2312.12928 for details

Surface-mounted cube at $Re_h = 40000$; image source

reconstruction error improves from $25\%$ to $5\%$