Fig. 2: Data-driven PDE for the chaotic dynamics in the complex Ginzburg-Landau equation.
From: Learning emergent partial differential equations in a learned emergent space
a The real part of the complex field W(x, t) obtained from simulating Eq. (2) with N = 256 mesh points after initial transients have decayed. b Removing the spatial label yields a collection of N time series plotted here in random sequence. c Using manifold learning (here diffusion maps), one finds that there exists two modes ϕ1 and ϕ2 parametrizing these time series. Each point corresponds to one of the N time series, and is colored by its scrambled spatial location x. d Having obtained the embedding, we can introduce an emergent coordinate \(\tilde{x}\) parametrizing the circle spanned by ϕ1 and ϕ2. e The real parts of the time series parametrized by \(\tilde{x}\). f Real part of simulation predictions for the complex variable W starting from an initial condition in our test set, using the partial differential equation model learned with \(\tilde{x}\) as the spatial variable and a periodic domain.
