Equivariance and partial observations in Koopman operator theory for partial differential equations

Figure 1.  The extended Koopman operator concept for partially observed or unknown states. Instead of directly learning the Koopman operator for the observable $ f: \mathcal{Y} \rightarrow \mathbb{R}^q $, we introduce the core dynamical system $ \varphi^\tau $ as an intermediate model that—given a sufficiently large embedding dimension $ q $—has a one-to-one correspondence to $ \Phi^\tau $ on the attractor. The Koopman operator is then defined in the standard ODE setting using a new observable function $ h\in \mathcal{H} $, $ h: \mathbb{R}^q \rightarrow \mathbb{R}^q $. For simplicity, we choose $ h = \operatorname{id} $ here

Figure 2.  Schematic of the local Koopman approach. We consider a local Koopman matrix $ \hat{K}^\tau \in \mathbb{R}^{q \times q} $. (a) The same approximation $ \hat{K}^\tau $ can be applied anywhere in the domain such that we obtain a global matrix $ \tilde{K}^\tau $ with identical blocks $ \hat{K}^\tau $. (b) The shaded $ B $ terms represent coupling terms to neighboring local models if we pursue a DMDc-like approach

Figure 3.  PDE solution vs. global Koopman-approximation for $ \mu = 15 $. To compute $ K $, we have used the standard DMD algorithm on the full state observable (i.e., $ f = \Psi = \operatorname{id} $)

Figure 4.  Eigenvalues of $ K $ vs. $ \hat{K} $ for varying $ q $ values

Figure 5.  Predictions using $ \tilde{K} $ with varying $ q $ values

Figure 6.  One-step prediction error for varying window widths $ q_w $

Figure 7.  Local DMDc-approximation according to Fig. 2 (b), with $ q = 1 $ and an additional control input from left and right, respectively

Figure 8.  PDE vs. local model with $ q_w = 8 $ and $ q_d = 50 $ versus DMDc model with $ q_w = 1 $ and $ q_d = 50 $

Figure 9.  Numerical PDE solution versus global Koopman-approximation for $ \mu = 18 $

Figure 10.  Eigenvalues of $ K $ vs. $ \hat{K} $ for varying $ q $ values

Figure 11.  Predictions using $ \tilde{K} $ with varying $ q $ values

Figure 12.  One-step prediction error of $ \tilde{K} $-predictions for varying $ q $ values

Figure 13.  Local DMDc-approximation according to Fig. 2 (b), with $ q_d = 50 $, $ q_w = 1 $, and an additional control input from left and right (all $ q_d = 50 $ delays), respectively