(554b) Modeling Chaotic Spatiotemporal Dynamics with a Minimal Representation Using Neural ODEs

Many dissipative partial differential equations (PDE) that exhibit chaotic dynamics have solutions that collapse onto a finite-dimensional manifold. When these manifolds exist, the long-time trajectories of infinite dimensional solutions to PDEs can be exactly described using only a finite number of dimensions. As a step toward constructing such a description for complex fluid flow problems, we describe a data-driven reduced order modelling (ROM) method where we use data in space and time to first find the coordinates on this manifold, and then find an ordinary differential equation (ODE) in these new coordinates. This type of ROM is advantageous because it is completely data-driven, may be computationally less expensive to evaluate than the full system (this may be useful for calculating statistics, control problems, or finding exact coherent states), and the intrinsic coordinates of the manifold may be physically meaningful.

In our approach, the change of coordinates is found by reducing the system dimension via an undercomplete autoencoder – a neural network (NN) that reduces then expands dimension. By varying the dimension of the reduced space, we find a minimal representation of the state. We then train a Neural ODE – a NN that approximates the right-hand side of an ODE – in the manifold coordinates. By learning an ODE, instead of the common approach of finding discrete time-maps, trajectories can be evolved arbitrarily far forward in time, and training data need not be evenly spaced in time. We test this method on a proxy for turbulence, the Kuramoto-Sivashinsky equation that has spatiotemporally chaotic dynamics, and find that accurate models can be generated for multiple domain sizes using data separated by 0.7 Lyapunov times. Finally, we apply this method to turbulent plane Couette flow.