Fig. 4: Optical NGRC observer.

a For a dynamical system, often partial information of the full state of the system is measurable, e.g., state variables \({[{u}_{1},...,{u}_{k}]}^{T}\) are observables while \({[{u}_{k+1},...,{u}_{M}]}^{T}\) are unmeasured. The optical NGRC extracts information from measured observables (blue) and predicts unmeasured variables (purple) based on the state of the reservoir (orange). b Two variables u₁ and u₂ (blue) of the Lorenz63 system are provided as observables to infer the third variable u₃. The predicted output by optical NGRC observer (red) matches the ground truth (blue) with high accuracy (NRMSE = 0.0169). c The optical NGRC observer results of the KS time series. 7 out of 64 spatial grids (evenly spaced in the spatial dimension) are input of the optical NGRC to infer the remaining 57 unmeasured variables. Top: ground truth; Middle: reservoir prediction (also including the observables for clarity); Bottom: error. d Performance comparison of the optical NGRC observer and the spline interpolation on the KS time series. The Pearson correlation between the optical NGRC observer prediction and the ground truth is consistently higher than that between the spline interpolation and the ground truth