Applied Koopman Theory for Partial Differential Equations and Data-Driven Modeling of Spatio-Temporal Systems

Figure 4

Reconstruction of the NLS dynamics using (a) a standard DMD approximation , (b) the NLS motivated , and (c) a quadratic observable . The Koopman spectra for each observable are demonstrated in (d), (e), and (f) which accompany the observables of (a), (b), and (c), respectively. Note that the observable produces spectra which are approximately purely imaginary which is expected of the 2-soliton evolution. The error between the three observables and the full simulation is shown (g). Note that the observable gives an error reduction of four-orders of magnitude over DMD, while is an order of magnitude worse. This highlights the importance of selecting good observables. The simulation was performed over with 41 equally spaced snapshots taken. The domain was discretized with points on a domain .

(a)

(b)

(c)

(d)

(e)

(f)

(g)