We discuss a method for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible fluid flows. The approach is based on a representation of the Koopman operator governing the evolution of observables in a smooth orthonormal basis learned from velocity field data through the diffusion maps algorithm. This representation enables the detection of coherent flow patterns through Koopman eigenfunctions and simulation of the evolution of observables and probability densities under the flow map. We present applications to Gaussian vortex flows and chaotic flows generated by Lorenz 96 systems.
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