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] 𝒘 𝐮+𝟐 ℒ ℳ 𝑕 𝒈 𝒘 𝒈 𝒧 𝒘 𝑢 𝑕 𝒘 𝒘 𝑢+1 = 𝒈 𝒘 𝑢 𝑕 𝒈 𝒘 = 𝒧𝑕 𝒘 𝒘 ∈ ℳ, 𝒈: ℳ → ℳ 𝑕 ∈ ℒ: ℳ → ℂ, 𝒧: ℒ → ℒ 𝒈 may be nonlinear 𝒧 is a linear operator usually, dim ℳ < ∞ in general, dim ℒ = ∞ 2

Modal Decomposition Based on Koopman Operator [ Mezić 05] For simplicity, suppose 𝒧 has only discrete spectra (eigenvalues) 𝒧𝜒 𝑘 𝒘 = 𝜇 𝑘 𝜒 𝑘 𝒘 ( 𝜇 𝑘 ∈ ℂ , 𝜒: ℳ → ℂ , and 𝑘 ∈ ℕ ) Assume that observable 𝑕 is in the span of the eigenfunctions: 𝑕 𝒘 = ෍ 𝑥 𝑘 𝜒 𝑘 𝒘 𝑘 With these assumptions, because 𝜒 𝑘 𝒈 𝒘 = 𝒧𝜒 𝑘 𝒘 = 𝜇𝜒 𝑘 𝒘 , 𝑢 𝑥 𝑕 𝒈 𝑢 𝒘 = ෍ 𝜇 𝑘 𝑘 𝜒 𝑘 𝒘 𝑘 frequency / mode decay rate 3

Dynamic Mode Decomposition [Rowley+ 09; Schmid 10] Dynamic mode decomposition (DMD) can compute the Koopman-based modal decomposition under some conditions. Let 𝒉: ℳ → ℂ 𝑛 or ℝ 𝑛 (vector-valued observable). Suppose we have time-series data from time 𝑢 0 to 𝑢 𝑜 ( 𝑢 𝑗 = 𝑢 0 + 𝑗Δ𝑢 ). 𝒉 𝒘 𝑢 0 , 𝒉 𝒘 𝑢 1 , … , 𝒉 𝒘 𝑢 𝑗 , … , 𝒉 𝒘 𝑢 𝑜−1 , 𝒉 𝒘 𝑢 𝑜 𝒀 = 𝒉 𝒘 𝑢 0 ⋯ 𝒉 𝒘 𝑢 𝑜−1 and 𝒁 = 𝒉 𝒘 𝑢 1 ⋯ 𝒉 𝒘 𝑢 𝑜 DMD computes eigenvalues 𝜇 & eigenvectors 𝒙 of 𝑩 = 𝒁𝒀 + . 𝑗 𝒙 𝑘 𝒜 𝑘 ∗ 𝒉 𝒘 𝑢 0 𝒉 𝒘 𝑢 𝑗 = ෍ 𝜇 𝑘 Under some conditions, these yield 𝑘 4

Dynamic Mode Decomposition (cont’d) [Rowley+ 09; Schmid 10] frequency / data decay rate coherent mode 𝑗 𝒙 𝑘 𝒜 𝑘 ∗ 𝒉 𝒘 𝑢 0 𝒉 𝒘 𝑢 𝑗 = ෍ 𝜇 𝑘 𝑘 × + = × 5

Limitation of Standard DMDs Within the dataset at hand, system is assumed to be time-invariant, and only 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 × + = × off × + = off off × → Implement this idea via probabilistic formulation. 7

Preliminary: Probabilistic DMD [Takeishi+ 17] Dataset: 𝒀 = 𝒉 𝒘 𝑢 0 ⋯ 𝒉 𝒘 𝑢 𝑜−1 and 𝒁 = 𝒉 𝒘 𝑢 1 ⋯ 𝒉 𝒘 𝑢 𝑜 = 𝒚 1 ⋯ 𝒚 𝑜 = 𝒛 1 ⋯ 𝒛 𝑜 𝝌 𝑗 Likelihood (observation model): 𝒚 𝑗 σ 𝑘 𝒙 𝑘 𝜒 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒚 𝑗 ∣ 𝝌 𝑗 = 𝒟𝒪 𝒛 𝑗 σ 𝑘 𝜇 𝑘 𝒙 𝑘 𝜒 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒛 𝑗 ∣ 𝝌 𝑗 = 𝒟𝒪 𝒚 𝑗 𝒛 𝑗 Prior: 𝒙, 𝜇, 𝜏 2 𝑞 𝜒 𝑗,𝑘 = 𝒟𝒪 𝜒 𝑗,𝑘 0, 1 for 𝑗 = 1, … , 𝑜 → MLE in 𝜏 2 → 0 coincides with TLS-DMD 8

Proposed Model: Factorially-Switching DMD Likelihood (observation model): 𝒚 𝑗 σ 𝑘 𝒙 𝑘 𝜓 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒚 𝑗 ∣ 𝝍 𝑗 = 𝒟𝒪 𝒛 𝑗 σ 𝑘 𝜇 𝑘 𝒙 𝑘 𝜔 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒛 𝑗 ∣ 𝝎 𝑗 = 𝒟𝒪 𝒜 𝝍,𝑗 𝒜 𝝎,𝑗 𝝌 𝑗 Priors: 𝝍 𝑗 𝝎 𝑗 1−𝑨 𝜓,𝑗,𝑘 𝜀 𝜒 𝑗,𝑘 − 𝜓 𝑗,𝑘 𝑨 𝜓,𝑗,𝑘 𝑞 𝜓 𝑗,𝑘 = 𝜀 𝜓 𝑗,𝑘 𝒚 𝑗 𝒛 𝑗 1−𝑨 𝜔,𝑗,𝑘 𝜀 𝜒 𝑗,𝑘 − 𝜔 𝑗,𝑘 𝑨 𝜔,𝑗,𝑘 𝑞 𝜔 𝑗,𝑘 = 𝜀 𝜔 𝑗,𝑘 𝒙, 𝜇, 𝜏 2 𝑞 𝜒 𝑗,𝑘 = 𝒟𝒪 𝜒 𝑗,𝑘 0, 1 → 𝑨 𝑗,𝑘 controls on-off of 𝑘 -th mode at time 𝑗 : 𝑨 𝑗,𝑘 = 1 (on) / 𝑨 𝑗,𝑘 = 0 (off) 9

Proposed Model: Factorially-Switching DMD (cont’d) Likelihood (observation model): GP 𝜈, Σ t 𝜹 𝝍,𝑗 𝜹 𝝎,𝑗 𝒚 𝑗 σ 𝑘 𝒙 𝑘 𝜓 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒚 𝑗 ∣ 𝝍 𝑗 = 𝒟𝒪 𝒛 𝑗 σ 𝑘 𝜇 𝑘 𝒙 𝑘 𝜔 𝑗,𝑘 , 𝜏 2 𝑱 𝑞 𝒛 𝑗 ∣ 𝝎 𝑗 = 𝒟𝒪 𝒜 𝝍,𝑗 𝒜 𝝎,𝑗 𝝌 𝑗 Priors: 𝝍 𝑗 𝝎 𝑗 1−𝑨 𝜓,𝑗,𝑘 𝜀 𝜒 𝑗,𝑘 − 𝜓 𝑗,𝑘 𝑨 𝜓,𝑗,𝑘 𝑞 𝜓 𝑗,𝑘 = 𝜀 𝜓 𝑗,𝑘 𝒚 𝑗 𝒛 𝑗 1−𝑨 𝜔,𝑗,𝑘 𝜀 𝜒 𝑗,𝑘 − 𝜔 𝑗,𝑘 𝑨 𝜔,𝑗,𝑘 𝑞 𝜔 𝑗,𝑘 = 𝜀 𝜔 𝑗,𝑘 𝒙, 𝜇, 𝜏 2 𝑞 𝜒 𝑗,𝑘 = 𝒟𝒪 𝜒 𝑗,𝑘 0, 1 𝑞 𝑨 Τ 𝜓 𝜔,𝑗,𝑘 ∣ 𝛿 Τ 𝜓 𝜔,𝑗,𝑘 = Bernoulli 𝛸 𝛿 Τ ( 𝛸 : cdfof normal) 𝜓 𝜔,𝑗,𝑘 → GPmakes 𝑨 smooth along time 𝑞 𝜹 Τ 𝜓 𝜔,∶,𝑘 = GaussianProcess 𝜈 Τ 𝜓 𝜔,𝑘 𝟐, 𝚻 𝑢 10

Parameter Estimation by Approx. EM Input Number of modes, kernel function, GP 𝜈, Σ t & data matrices 𝒀, 𝒁 𝜹 𝝍,𝑗 𝜹 𝝎,𝑗 1. Initialize quantities using DMD. 𝒜 𝝍,𝑗 𝒜 𝝎,𝑗 𝝌 𝑗 2. E-step Approximate posterior of 𝜒, 𝜓, 𝜔, z, 𝛿 𝝍 𝑗 𝝎 𝑗 using expectation propagation [Minka 01] . 3. M-step 𝒚 𝑗 𝒛 𝑗 Maximize 𝔽 ℒ 𝒙, 𝜇, 𝜏 2 , 𝜈 ( 𝔽 is wrt. distribution from E-step). 𝒙, 𝜇, 𝜏 2 4. Repeat 2. and 3. until convergence. Posterior statistics of 𝜒, z & estimated values of 𝒙, 𝜇, 𝜏 2 , 𝜈 Output 11

Toy Example (Local Traveling Wave) DATA RESULTS DMD FSDMD (proposed) = + bumpy because of sudden changes 12

Transient Fluid Flow DATA RESULTS time on/off states of each dynamic mode 13

Summary Objective ◦ Compute dynamic mode decomposition on time-varying systems & transient phenomena Method ◦ Idea: Introducing on-off switching GP 𝜈, Σ 𝑢 of each dynamic mode at each timestep 𝜹 𝝍,𝑗 𝜹 𝝎,𝑗 ◦ Implemented it via probabilistic modeling/inference 𝒜 𝝍,𝑗 𝒜 𝝎,𝑗 𝝌 𝑗 𝝍 𝑗 𝝎 𝑗 Future Work ◦ Developing faster & more stable inference 𝒚 𝑗 𝒛 𝑗 ◦ Considering interaction between dynamic modes 𝒙, 𝜇, 𝜏 2 14