Subspace dynamic mode decomposition for stochastic Koopman analysis

The analysis of nonlinear dynamical systems based on the Koopman operator is attracting attention in various applications. Dynamic mode decomposition (DMD) is a data-driven algorithm for Koopman spectral analysis, and several variants with a wide range of applications have been proposed. However, popular implementations of DMD suffer from observation noise on random dynamical systems and generate inaccurate estimation of the spectra of the stochastic Koopman operator. In this paper, we propose subspace DMD as an algorithm for the Koopman analysis of random dynamical systems with observation noise. Subspace DMD first computes the orthogonal projection of future snapshots to the space of past snapshots and then estimates the spectra of a linear model, and its output converges to the spectra of the stochastic Koopman operator under standard assumptions. We investigate the empirical performance of subspace DMD with several dynamical systems and show its utility for the Koopman analysis of random dynamical systems.

Figure 1

(a) Data generated by the noise-free/noisy Stuart-Landau equation and trigonometric observables, which show the phase diffusion when ${\sigma }_p>0$ (the solid red line). (b, c) The estimated continuous-time eigenvalues, with (b) ${\sigma }_p=0$ and (c) ${\sigma }_p=0.5$. Subspace DMD eliminates the effects of the observation noise, keeping the effects of the process noise.

Figure 2

(a) Data generated by the stochastic Burger's equation with observation noise and (b) the estimated continuous-time eigenvalues. The eigenvalues estimated by subspace DMD agree well with the ones computed with the clean data.

Figure 3

The 95% confidence intervals and averages of the eigenvalues estimated by subspace DMD, standard DMD, tls-DMD, and opt-DMD on the linear systems, (a) $r=1.0$ and (b) $r=0.9$, with process and observation noises for 1000 random trials. When $r=0.9$, only subspace DMD shows consistent results.

Figure 4

Relative errors of the eigenvalues estimated by subspace DMD, standard DMD, tls-DMD, and opt-DMD on the linear systems, (a) $r=1.0$ and (b) $r=0.9$, with process and observation noises against different magnitudes of observation noise ${\sigma }_o$. When $r=0.9$, subspace DMD produces much smaller errors compared to the other two methods.

Figure 5

Relative errors of the eigenvalues estimated by subspace DMD, standard DMD, tls-DMD, and opt-DMD on the linear systems, (a) $r=1.0$ and (b) $r=0.9$, with process and observation noises against different numbers of snapshots $m$ fed into the algorithms. When $r=0.9$, only the output of subspace DMD converges to the true value.

Figure 6

The 95% confidence intervals and averages of the eigenvalues estimated by subspace DMD, standard DMD, tls-DMD, lr-DMD, and OMD on the high-dimensional ($n=500$) low-rank ($r=2$) system for 1000 random trials. The sample sizes are (a) $m=50$ and (b) $m=200$. In both cases, the outputs of subspace DMD distribute around the true values denoted by the black filled circle.

Figure 7

(a) Example snapshot of the vorticity field in the limit-cycle characterized by the Kármán vortex street. (b) Time variation of the vorticity at locations A and B.

Figure 8

(a) Continuous-time eigenvalues estimated on the noise-free and noisy datasets, which are numbered from one to seven according to their frequency (i.e., imaginary part). (b) Dynamic modes computed on the noise-free dataset by standard DMD. (c–e) Dynamic modes computed on the noisy dataset by (c) subspace DMD, (d) standard DMD, and (e) tls-DMD. The upper row corresponds to eigenvalue 1 ($\sim 10i$) and the lower row corresponds to eigenvalue 4 ($\sim 40i$). No adversarial effects are present in the results of subspace DMD, even without the process noise. In (b–e), the magnitude of the dynamic modes are normalized to a common color scheme. Best viewed in color.