Dynamic Mode Decomposition
Fully data-driven ROM approach
presentation is made by Pavel Temirchev
Motivation
Assume we have a dynamical system:
Dynamics is either unknown
or given as a black-box
Dimensionality is huge
We are interested in:
- system identification
(for computer simulations) - decreasing the computational complexity
- analysis of the system behavior
We can obtain data from the process
\frac{\partial x}{\partial t} = f(x)
x \in \mathbb{R}^n
Example of the dynamics
Fluid flow
Linearity assumption
Approximate the continuous dynamics by a discrete-time linear process:
x_{k+1} = Ax_k
We call a snapshot
x_k
Multidimensional data can be unravel into a vector of size
n
Linearity assumption
Let's collect the snapshot matrices as the data from the observed process:
X =
\begin{bmatrix}
| & | & & | \\
x_0 & x_1 & \cdots & x_{m-1} \\
| & | & & |
\end{bmatrix}
X' =
\begin{bmatrix}
| & | & & | \\
x_1 & x_2 & \cdots & x_m \\
| & | & & |
\end{bmatrix}
Time
So, the dynamics became:
X' = AX
Linearity assumption
One can compute as follows:
where is a pseudo-inverse of
A = X'X^\dagger
A
X^\dagger
X
However,
A \in \mathbb{R}^{n \times n}
Which is VERY HUGE!
Reduced Linear Model
Let us make Singular Value Decomposition of
(also known as Proper Orthogonal Decomposition in this snapshot-setting)
X
X = U \Sigma V^T
=
\times
\times
TRUNCATED
\approx \tilde{U} \tilde\Sigma \tilde V^T
Crop up to first biggest singular values
r
Reduced Linear Model
Instead of computing
we can compute its projection on the low-rank POD basis:
A
\tilde A = \tilde U^T A \tilde U
= \tilde U^T X'X^\dag \tilde U
= \tilde U^T X' \tilde V\Sigma^{-1}\tilde U^T \tilde U
= \tilde U^T X' \tilde V\Sigma^{-1}
\tilde A \in \mathbb{R}^{r \times r}
Making Predictions
So, now we are interested in predictions of future states:
\tilde A
x_{k+1} = Ax_k
But we can't do it with the truncated directly!
Luckily, we can find a way to not compute the full
A
Reminder:
Eigenvalue decomposition
A \Phi = \Phi \Lambda
A = \Phi \Lambda \Phi^\dag
x_{k+1} = \Phi \Lambda \Phi^\dag x_k
Both and
can be computed
relatively easily
\Phi
\Lambda
Making Predictions
- Full and truncated share the same eigenvalues
\tilde A
A
\tilde A W = W \Lambda
\Lambda
- Full eigenvectors can be computed as follows:
\Phi
We can make predictions as follows
x_{1} = \Phi \Lambda \Phi^\dag x_0
x_{2} = \Phi \Lambda \Phi^\dag \Phi \Lambda \Phi^\dag x_0
x_{2} = \Phi \Lambda^2 \Phi^\dag x_0
x_{k} = \Phi \Lambda^k \Phi^\dag x_0
Element-wise exponentiation!
\Phi = X' \tilde V \tilde \Sigma^{-1} W
Dynamic Modes and
Stability Analysis
Eigenvectors are called Dynamic Modes
\Phi \in \mathbb{C}^{n \times r}
Dynamic modes (columns of ) represent spatio-temporal patterns of your data
\Phi
You can plot them:
Dynamic Modes and
Stability Analysis
Eigenvalues can be used for the
stability analysis of a corresponding dynamic mode
\Lambda \in \mathbb{C}^{r \times r}
We can plot them too:
|\lambda_i| < 1
|\lambda_i| = 1
|\lambda_i| > 1
Im(\lambda_i) \neq 0
Im(\lambda)
Re(\lambda)
if - mode is decaying
if - mode is stable
if - mode is growing
if - mode oscillates
No way I can represent wells in this model...
But you can
You need DMD with control:
x_{k+1} = Ax_k + Bu_k
Comparison to other ROMs
| property | Neural Network ROM | Dynamic Mode Decomposition | POD-Galerkin projection |
|---|---|---|---|
| Parametric dynamical systems | yes | not really | not really |
| Can work with unknown dynamics | yes | yes | no |
| Easy to implement | no | yes | not really |
| Training speed | slow | moderate | moderate |
| Scalability within spatial dimensionality | yes | no | no |
| Interpretability | not really | yes | not really |
Dynamic Mode Decomposition
By cydoroga
