Fig. 2: Optical NGRC for Lorenz63 attractor forecasting.

a Time series of the Lorenz63 attractor (state variables u₁, u₂, u₃) that drives the optical NGRC. At each time step of the training phase, the input states from the current (u_t) and the previous (u_t−1) time steps are encoded to the optical system to generate reservoir features (r_t+1). b The temporal evolution of 10 randomly selected optical reservoir nodes (out of 2000 nodes), which resembles the dynamics of the input data. After training iterations of 4000 time steps, a linear estimator W_out is trained to match the weighted sums of the reservoir features (\({\hat{{\boldsymbol{o}}}}_{t}={{\boldsymbol{W}}}_{out}{{\boldsymbol{r}}}_{t}\)) with the input data at the next time step (u_t), i.e., \({\hat{{\boldsymbol{o}}}}_{t}\approx {{\boldsymbol{u}}}_{t}\). c Once W_out is optimized, the optical NGRC is switched to the autonomous mode and experimentally predicts short-term results for 400 time steps. The normalized root mean square error (NRMSE) over the first 5 time units of the prediction phase is 0.0971. d The optical NGRC projects onto an attractor similar to the Lorenz63 attactor, experimentally obtained by the long-term forecasting results of 8000 time steps. e The return map of the ground truth (blue) and the experimental prediction (red)