(388b) Stabilized Neural ODEs for Data-Driven Reduced-Order Modeling of Plane Couette Flow

For dissipative partial differential equations we expect at long-times that trajectories collapse onto a finite-dimensional manifold. This implies that, once on this manifold, the system can be exactly described by a coordinate system parameterizing the manifold. Describing the system in these coordinates drastically reduces the dimension of the problem, which allows for much faster simulations (this is useful for calculating statistics, control problems, or finding invariant solutions) and the intrinsic coordinates of the manifold may be physically meaningful. We show this description of the dynamics is possible using a data-driven reduced order modeling method in which we use an autoencoder to find a manifold coordinate system and stabilized neural ordinary differential equations to evolve these coordinates forward in time.

Specifically, we apply this method to turbulent plane Couette flow direct numerical simulations that require O(10⁶) points to simulate. First, we use data from the full simulation and show our method gives excellent statistical agreement using only O(10) manifold coordinates. Then, we show the same performance can be achieved when only providing the method with wall shear observations. We highlight the usefulness of these models by using them for rapid discovery of unstable solutions that we find exist in the true underlying system.