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

Abstract

The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems. The main reason is the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. This equally applies to ordinary, stochastic, and partial differential equations (PDEs). Until now, with a few exceptions only, the PDE case is mostly treated rather superficially, and the specific structure of the underlying dynamics is largely ignored. In this paper, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to significantly increase the model efficacy. Moreover, the situation where we only have access to partial observations (i.e., measurements, as is very common for experimental data) has not been treated to its full extent, either. Moreover, we address the highly-relevant case where we cannot measure the full state, where alternative approaches (e.g., delay coordinates) have to be considered. We derive rigorous statements on the required number of observables in this situation, based on embedding theory. We present numerical evidence using various numerical examples including the wave equation and the Kuramoto-Sivashinsky equation.

Figures (13)

• 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.

Theorems & Definitions (18)

• Example 2.1
• Definition 1: Koopman semigroup and generator
• Remark 1
• Remark 2
• Example 3.1
• Remark 3
• Theorem 3.1
• proof
• Example 3.2: Fourier observable
• Definition 2: ZDG19, Definition 2.1