Model-free forecasting of large partially observable spatiotemporally chaotic systems

We implement a reservoir-computing-based recurrent neural network (RC-RNN) in which we first expand the system observables into a high dimensional space using radial basis functions (RBFs), then connect the expanded input to a reservoir from which the output (system forecast) is obtained via a linear readout. The RBFs enable the network to capture the system nonlinearities robustly without any knowledge of the governing equations. With this approach, we achieve the first successful application of an RC-RNN to forecast a large partially observable spatiotemporally chaotic system, namely the Gray-Scott reaction-diffusion system for which only one species concentration is observable. Our method does not require any model reduction and can deal with noisy and sparse measurements. The required reservoir dimension is only two orders of magnitude larger than the system dimension itself, indicating our method's ability to handle large turbulent systems, such as for weather forecasting. We also find that the use of RBFs as nonlinear projectors is interpretable in terms of their nonlinear approximation properties, thus suggesting their generalization to other RNNs.