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
∂t∂x=f(x)
\frac{\partial x}{\partial t} = f(x)
x∈Rn
x \in \mathbb{R}^n
Example of the dynamics
Fluid flow
Linearity assumption
Approximate the continuous dynamics by a discrete-time linear process:
xk+1=Axk
x_{k+1} = Ax_k
We call a snapshot
xk
x_k
Multidimensional data can be unravel into a vector of size
n
n
Linearity assumption
Let's collect the snapshot matrices as the data from the observed process:
X=∣x0∣∣x1∣⋯∣xm−1∣
X =
\begin{bmatrix}
| & | & & | \\
x_0 & x_1 & \cdots & x_{m-1} \\
| & | & & |
\end{bmatrix}
X′=∣x1∣∣x2∣⋯∣xm∣
X' =
\begin{bmatrix}
| & | & & | \\
x_1 & x_2 & \cdots & x_m \\
| & | & & |
\end{bmatrix}
Time
So, the dynamics became:
X′=AX
X' = AX
Linearity assumption
One can compute as follows:
where is a pseudo-inverse of
A=X′X†
A = X'X^\dagger
A
A
X†
X^\dagger
X
X
However,
A∈Rn×n
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
X=UΣVT
X = U \Sigma V^T
=
=
×
\times
×
\times
TRUNCATED
≈U~Σ~V~T
\approx \tilde{U} \tilde\Sigma \tilde V^T
Crop up to first biggest singular values
r
r
Reduced Linear Model
Instead of computing
we can compute its projection on the low-rank POD basis:
A
A
A~=U~TAU~
\tilde A = \tilde U^T A \tilde U
=U~TX′X†U~
= \tilde U^T X'X^\dag \tilde U
=U~TX′V~Σ−1U~TU~
= \tilde U^T X' \tilde V\Sigma^{-1}\tilde U^T \tilde U
=U~TX′V~Σ−1
= \tilde U^T X' \tilde V\Sigma^{-1}
A~∈Rr×r
\tilde A \in \mathbb{R}^{r \times r}
Making Predictions
So, now we are interested in predictions of future states:
A~
\tilde A
xk+1=Axk
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
A
Reminder:
Eigenvalue decomposition
AΦ=ΦΛ
A \Phi = \Phi \Lambda
A=ΦΛΦ†
A = \Phi \Lambda \Phi^\dag
xk+1=ΦΛΦ†xk
x_{k+1} = \Phi \Lambda \Phi^\dag x_k
Both and
can be computed
relatively easily
Φ
\Phi
Λ
\Lambda
Making Predictions
- Full and truncated share the same eigenvalues
A~
\tilde A
A
A
A~W=WΛ
\tilde A W = W \Lambda
Λ
\Lambda
- Full eigenvectors can be computed as follows:
Φ
\Phi
We can make predictions as follows
x1=ΦΛΦ†x0
x_{1} = \Phi \Lambda \Phi^\dag x_0
x2=ΦΛΦ†ΦΛΦ†x0
x_{2} = \Phi \Lambda \Phi^\dag \Phi \Lambda \Phi^\dag x_0
x2=ΦΛ2Φ†x0
x_{2} = \Phi \Lambda^2 \Phi^\dag x_0
xk=ΦΛkΦ†x0
x_{k} = \Phi \Lambda^k \Phi^\dag x_0
Element-wise exponentiation!
Φ=X′V~Σ~−1W
\Phi = X' \tilde V \tilde \Sigma^{-1} W
Dynamic Modes and
Stability Analysis
Eigenvectors are called Dynamic Modes
Φ∈Cn×r
\Phi \in \mathbb{C}^{n \times r}
Dynamic modes (columns of ) represent spatio-temporal patterns of your data
Φ
\Phi
You can plot them:
Dynamic Modes and
Stability Analysis
Eigenvalues can be used for the
stability analysis of a corresponding dynamic mode
Λ∈Cr×r
\Lambda \in \mathbb{C}^{r \times r}
We can plot them too:
∣λi∣<1
|\lambda_i| < 1
∣λi∣=1
|\lambda_i| = 1
∣λi∣>1
|\lambda_i| > 1
Im(λi)=0
Im(\lambda_i) \neq 0
Im(λ)
Im(\lambda)
Re(λ)
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:
xk+1=Axk+Buk
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 Fully data-driven ROM approach presentation is made by Pavel Temirchev
Dynamic Mode Decomposition
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