  Reservoir computing is a powerful tool for forecasting turbulence because its
simple architecture has the computational efficiency to handle large systems.
Its implementation, however, often requires full state-vector measurements and
knowledge of the system nonlinearities. We use nonlinear projector functions to
expand the system measurements to a high dimensional space and then feed them
to a reservoir to obtain forecasts. We demonstrate the application of such
reservoir computing networks on spatiotemporally chaotic systems, which model
several features of turbulence. We show that using radial basis functions as
nonlinear projectors enables complex system nonlinearities to be captured
robustly even with only partial observations and without knowing the governing
equations. Finally, we show that when measurements are sparse or incomplete and
noisy, such that even the governing equations become inaccurate, our networks
can still produce reasonably accurate forecasts, thus paving the way towards
model-free forecasting of practical turbulent systems.
