Data-driven prediction of large-scale spatiotemporal chaos with distributed low-dimensional models

Spatiotemporal chaos in systems such as turbulent flows often resides on finite-dimensional attractors, enabling the construction of reduced-order models. Unfortunately, as the dimension of these attractors increases it becomes more difficult to train reduced-order models because more data are needed to sample states of the system. For example, the attractor dimension often scales linearly with the domain size in a single direction for large-scale spatially extended systems, thus we need methods for decomposing these systems to overcome the burden of increasing dimensionality. Here, we develop a framework that constructs local reduced-order models by decomposing spatially extended systems into patches. Each patch uses autoencoders for dimension reduction and neural ordinary differential equations for learning the temporal dynamics locally. We apply this framework to the Kuramoto-Sivashinsky equation and two-dimensional Kolmogorov flow. Our approach reduces the dimension by up to two orders of magnitude while accurately capturing both short-term dynamics and long-term statistics. This framework is applicable to any dissipative partial differential equations, thereby offering broad implications for a wide range of physical and engineering systems.