Modeling chaotic spatiotemporal dynamics with a minimal representation using Neural ODEs

Solutions to dissipative partial differential equations that exhibit chaotic dynamics often evolve to attractors that exist on finite-dimensional manifolds. We describe a data-driven reduced order modelling (ROM) method to find the coordinates on this manifold and find an ordinary differential equation (ODE) in these coordinates. This ROM is useful because it is data-driven, it is computationally less expensive than the full system, and it provides coordinates which may be physically meaningful. We find the manifold coordinates by reducing the system dimension via an undercomplete autoencoder – a neural network (NN) that reduces then expands dimension. By varying the dimension, we get a minimal representation of the state. Then, in the manifold coordinate system, we train a Neural ODE – a NN that approximates an ODE. Learning an ODE, instead of a discrete time-map, allows us to evolve trajectories arbitrarily far forward, and allows for training on unevenly and/or widely spaced data in time. We test on the Kuramoto-Sivashinsky equation for domain sizes that exhibit spatiotemporally chaos. These ROMs generate accurate short- and long-time statistics with data separated up to 0.7 Lyapunov times. We also study the effect of reducing dimension below the expected manifold dimension.