Fig. 6: Data-driven discovery of 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 = 128 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 a one-dimensional parametrization ϕ1 of these time series. Each point corresponds to one of the N time series, and is colored by its actual spatial location x. d The real parts of the time series parametrized by ϕ1. e 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 ϕ1 as the spatial variable. Since no analytical boundary conditions are available, we provide the true values near the boundaries during integration, within a corridor indicated by white vertical lines. f Smallest Euclidean distance d in \({{\mathbb{C}}}^{N}\) between the transients and the true attractor at each time step: true PDE (blue), learned PDE (orange).
