Data-driven Flow Models from Nonlinear Spectral Reduction

Most fluid flows of practical importance admit coexisting steady, periodic of quasiperiodic stationary states, as well as transitions amomg them. No single linearized model can capture such characteristically nonlinear behavior, which explains why reduced-orders for even classic flows, such as Couette and Rayleigh-Bernard flows. have been unavailable. For the same reason, no data-driven modeling approach among the available linear ones has been able to accurately describe characteristically nonlinear fluid-structure interactions problems, such as vortex shedding behind a cylinder or fluid sloshing in a tank.

In this talk, webriefly  review the recent theory of spectral submanifolds (SSMs), which offers an efficient nonlinear alternative to linearized reduced-order modeling. An SSM is  an invariant manifold in the phase space of a flow that acts as the nonlinear continuation of a spectral subspace of the linenarized flow near a stationary state. More specifically, SSMs are low-dimensional attactors that attract all nearby trajectories, and hence their reduced dynamics offers the perfect reduced-order model with which different flow realizations synchronize exponentially fast.

We show on examples how an open-source package, SSMLearn, can identify SSMs and their reduced-order dynamics directly from data. These examples include numerical data sets from Couette and Rayleigh-Bernard flows and experimental data from forced fluid sloshing. In these problems, one- or two-dimensional SSM-based reduced models already capture the full dynamics and  make reliable predictions for non-lineraizable  behavior even outside their training range.