Decomposition for Koopman Analysis of Time-Variant Systems Naoya - - PowerPoint PPT Presentation

Factorially Switching Dynamic Mode Decomposition for Koopman Analysis of Time-Variant Systems

Naoya Takeishi (RIKEN) Takehisa Yairi (The University of Tokyo) Yoshinobu Kawahara (Osaka University / RIKEN)

1

Koopman Operator

[Koopman 31; Mezić 05]

2

𝒘𝑢+1 = 𝒈 𝒘𝑢 𝒘 ∈ ℳ, 𝒈: ℳ → ℳ 𝒈 may be nonlinear usually, dim ℳ < ∞

𝒘𝑢 𝒘𝐮+𝟐 𝒈 ℳ

𝑕 𝒈 𝒘 = 𝒧𝑕 𝒘 𝑕 ∈ ℒ: ℳ → ℂ, 𝒧: ℒ → ℒ 𝒧 is a linear operator in general, dim ℒ = ∞

𝒧 𝑕 𝒘 𝑕 𝒈 𝒘 ℒ

Modal Decomposition Based on Koopman Operator

[Mezić 05]

For simplicity, suppose 𝒧 has only discrete spectra (eigenvalues) 𝒧𝜒𝑘 𝒘 = 𝜇𝑘𝜒𝑘 𝒘 (𝜇𝑘 ∈ ℂ, 𝜒: ℳ → ℂ, and 𝑘 ∈ ℕ)

3

Assume that observable 𝑕 is in the span of the eigenfunctions: 𝑕 𝒘 = ෍

𝑘

𝑥

𝑘𝜒𝑘 𝒘

frequency / decay rate mode

With these assumptions, because 𝜒𝑘 𝒈 𝒘 = 𝒧𝜒𝑘 𝒘 = 𝜇𝜒𝑘 𝒘 , 𝑕 𝒈𝑢 𝒘 = ෍

𝑘

𝜇𝑘

𝑢𝑥 𝑘𝜒𝑘 𝒘

Dynamic Mode Decomposition

[Rowley+ 09; Schmid 10]

Dynamic mode decomposition (DMD) can compute the Koopman-based modal decomposition under some conditions.

4

𝒉 𝒘𝑢𝑗 = ෍

𝑘

𝜇𝑘

𝑗𝒙𝑘 𝒜𝑘 ∗𝒉 𝒘𝑢0

DMD computes eigenvalues 𝜇 & eigenvectors 𝒙 of 𝑩 = 𝒁𝒀+. Under some conditions, these yield Let 𝒉: ℳ → ℂ𝑛 or ℝ𝑛 (vector-valued observable). Suppose we have time-series data from time 𝑢0 to 𝑢𝑜 (𝑢𝑗 = 𝑢0 + 𝑗Δ𝑢). 𝒉 𝒘𝑢0 , 𝒉 𝒘𝑢1 , … , 𝒉 𝒘𝑢𝑗 , … , 𝒉 𝒘𝑢𝑜−1 , 𝒉 𝒘𝑢𝑜 𝒀 = 𝒉 𝒘𝑢0 ⋯ 𝒉 𝒘𝑢𝑜−1 and 𝒁 = 𝒉 𝒘𝑢1 ⋯ 𝒉 𝒘𝑢𝑜

Dynamic Mode Decomposition (cont’d)

[Rowley+ 09; Schmid 10]

5

𝒉 𝒘𝑢𝑗 = ෍

𝑘

𝜇𝑘

𝑗𝒙𝑘 𝒜𝑘 ∗𝒉 𝒘𝑢0

frequency / decay rate coherent mode data

× × + =

Limitation of Standard DMDs

Within the dataset at hand, system is assumed to be time-invariant, and

• nly a single set of dynamic modes is computed for the dataset.

In practice, however,

• This assumption may not hold. (e.g., switching systems)
• Even if 𝑔 is time-invariant, within finite-data regime,

dynamic modes adequate for different periods of data may vary with time. (e.g., transient phenomena) Existing approaches:

• Manual separation as preprocessing
• Multi-resolution DMD [Kutz+ 16]

6

Core Idea: Introducing “On-off Switching” to Dynamic Modes

→ Implement this idea via probabilistic formulation.

7

× × + = × × + =

• ff
• ff
• ff

Preliminary: Probabilistic DMD

[Takeishi+ 17]

Dataset: 𝒀 = 𝒉 𝒘𝑢0 ⋯ 𝒉 𝒘𝑢𝑜−1 and 𝒁 = 𝒉 𝒘𝑢1 ⋯ 𝒉 𝒘𝑢𝑜 = 𝒚1 ⋯ 𝒚𝑜 = 𝒛1 ⋯ 𝒛𝑜

8 𝒚𝑗 𝒛𝑗 𝝌𝑗 𝒙, 𝜇, 𝜏2

for 𝑗 = 1, … , 𝑜

Likelihood (observation model): 𝑞 𝒚𝑗 ∣ 𝝌𝑗 = 𝒟𝒪

𝒚𝑗 σ𝑘 𝒙𝑘𝜒𝑗,𝑘 , 𝜏2𝑱

𝑞 𝒛𝑗 ∣ 𝝌𝑗 = 𝒟𝒪

𝒛𝑗 σ𝑘 𝜇𝑘𝒙𝑘𝜒𝑗,𝑘 , 𝜏2𝑱

Prior: 𝑞 𝜒𝑗,𝑘 = 𝒟𝒪

𝜒𝑗,𝑘 0, 1

→ MLE in 𝜏2 → 0 coincides with TLS-DMD

Proposed Model: Factorially-Switching DMD

Likelihood (observation model): 𝑞 𝒚𝑗 ∣ 𝝍𝑗 = 𝒟𝒪

𝒚𝑗 σ𝑘 𝒙𝑘𝜓𝑗,𝑘 , 𝜏2𝑱

𝑞 𝒛𝑗 ∣ 𝝎𝑗 = 𝒟𝒪

𝒛𝑗 σ𝑘 𝜇𝑘𝒙𝑘𝜔𝑗,𝑘 , 𝜏2𝑱

Priors: 𝑞 𝜓𝑗,𝑘 = 𝜀 𝜓𝑗,𝑘

1−𝑨𝜓,𝑗,𝑘 𝜀 𝜒𝑗,𝑘 − 𝜓𝑗,𝑘 𝑨𝜓,𝑗,𝑘

𝑞 𝜔𝑗,𝑘 = 𝜀 𝜔𝑗,𝑘

1−𝑨𝜔,𝑗,𝑘 𝜀 𝜒𝑗,𝑘 − 𝜔𝑗,𝑘 𝑨𝜔,𝑗,𝑘

𝑞 𝜒𝑗,𝑘 = 𝒟𝒪

𝜒𝑗,𝑘 0, 1

→ 𝑨𝑗,𝑘 controls on-off of 𝑘-th mode at time 𝑗: 𝑨𝑗,𝑘 = 1 (on) / 𝑨𝑗,𝑘 = 0 (off)

9 𝒚𝑗 𝒛𝑗 𝝍𝑗 𝝎𝑗 𝝌𝑗 𝒜𝝍,𝑗 𝒜𝝎,𝑗 𝒙, 𝜇, 𝜏2

Proposed Model: Factorially-Switching DMD (cont’d)

Likelihood (observation model): 𝑞 𝒚𝑗 ∣ 𝝍𝑗 = 𝒟𝒪

𝒚𝑗 σ𝑘 𝒙𝑘𝜓𝑗,𝑘 , 𝜏2𝑱

𝑞 𝒛𝑗 ∣ 𝝎𝑗 = 𝒟𝒪

𝒛𝑗 σ𝑘 𝜇𝑘𝒙𝑘𝜔𝑗,𝑘 , 𝜏2𝑱

Priors: 𝑞 𝜓𝑗,𝑘 = 𝜀 𝜓𝑗,𝑘

1−𝑨𝜓,𝑗,𝑘 𝜀 𝜒𝑗,𝑘 − 𝜓𝑗,𝑘 𝑨𝜓,𝑗,𝑘

𝑞 𝜔𝑗,𝑘 = 𝜀 𝜔𝑗,𝑘

1−𝑨𝜔,𝑗,𝑘 𝜀 𝜒𝑗,𝑘 − 𝜔𝑗,𝑘 𝑨𝜔,𝑗,𝑘

𝑞 𝜒𝑗,𝑘 = 𝒟𝒪

𝜒𝑗,𝑘 0, 1

𝑞 𝑨 Τ

𝜓 𝜔,𝑗,𝑘 ∣ 𝛿 Τ 𝜓 𝜔,𝑗,𝑘 = Bernoulli 𝛸 𝛿 Τ 𝜓 𝜔,𝑗,𝑘

(𝛸: cdfof normal) 𝑞 𝜹 Τ

𝜓 𝜔,∶,𝑘 = GaussianProcess 𝜈 Τ 𝜓 𝜔,𝑘𝟐, 𝚻 𝑢

→GPmakes 𝑨smooth along time

10 𝒚𝑗 𝒛𝑗 𝝍𝑗 𝝎𝑗 𝝌𝑗 𝒜𝝍,𝑗 𝒜𝝎,𝑗 𝜹𝝍,𝑗 𝜹𝝎,𝑗

GP 𝜈, Σ t

𝒙, 𝜇, 𝜏2

Parameter Estimation by Approx. EM

Input Number of modes, kernel function, & data matrices 𝒀, 𝒁 1. Initialize quantities using DMD. 2. E-step Approximate posterior of 𝜒, 𝜓, 𝜔, z, 𝛿 using expectation propagation [Minka 01]. 3. M-step Maximize 𝔽 ℒ 𝒙, 𝜇, 𝜏2, 𝜈 (𝔽 is wrt. distribution from E-step). 4. Repeat 2. and 3. until convergence. Output Posterior statistics of 𝜒, z & estimated values of 𝒙, 𝜇, 𝜏2, 𝜈

11 𝒚𝑗 𝒛𝑗 𝝍𝑗 𝝎𝑗 𝝌𝑗 𝒜𝝍,𝑗 𝒜𝝎,𝑗 𝜹𝝍,𝑗 𝜹𝝎,𝑗

GP 𝜈, Σ t

𝒙, 𝜇, 𝜏2

Toy Example (Local Traveling Wave)

12

+

DATA

=

FSDMD (proposed) DMD

bumpy because of sudden changes

RESULTS

Transient Fluid Flow

13 DATA RESULTS

• n/off states
• f each dynamic mode

time

Summary

Objective

• Compute dynamic mode decomposition
• n time-varying systems & transient phenomena

Method

• Idea: Introducing on-off switching
• f each dynamic mode at each timestep
• Implemented it via probabilistic modeling/inference

Future Work

• Developing faster & more stable inference
• Considering interaction between dynamic modes

14

𝒚𝑗 𝒛𝑗 𝝍𝑗 𝝎𝑗 𝝌𝑗 𝒜𝝍,𝑗 𝒜𝝎,𝑗 𝜹𝝍,𝑗 𝜹𝝎,𝑗

GP 𝜈, Σ 𝑢

𝒙, 𝜇, 𝜏2