Data-driven reduced-order modeling of spatiotemporal chaos with neural ordinary differential equations

Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced-order modeling method that capitalizes on this fact by finding a coordinate representation for this manifold and then a system of ordinary differential equations (ODEs) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder-a neural network (NN) that reduces and then expands dimension. Then, the ODE, in these coordinates, is determined by a NN using the neural ODE framework. Both of these steps only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. Time-derivative information is not needed. We apply this framework to the Kuramoto-Sivashinsky equation for domain sizes that exhibit chaotic dynamics with again estimated manifold dimensions ranging from 8 to 28. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short- and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ~0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction and find a sweet spot in terms of performance vs dimension.

Automated summary

In this study, we propose a data-driven reduced-order modeling method for chaotic dynamics, which involves finding the coordinate representation of the manifold and describing the dynamics using a system of ordinary differential equations (ODEs) in this coordinate system. We apply this method to a specific system and find that dimension reduction improves performance compared to predictions in the ambient space. Furthermore, we demonstrate that the low-dimensional model is capable of accurately recreating the true dynamics using widely spaced data.