Fig. 1. Heterogeneous forecasting scheme using photonic time-delayed reservoir computing combined with an imperfect dynamical model.

Fig. 2. Predicted performance NMSE of isolated TDRC: (a) NMSE as a function of the input gain coefficient γ; (b) NMSE as a function of the self-feedback strength kf; (c) NMSE as a function of the injection strength from the light source σ with frequency detuning Δvs=−4.7  GHz.

Fig. 3. Predictive performance of isolated TDRC (a), (b) compared to knowledge-based photonic reservoir computing with respect to error coefficients η1 for deviated self-feedback strength (c) and dynamical state mismatch η2 (d).

Fig. 4. Predictive performance for chaotic dynamics generated by a VCSEL with (a) isolated TDRC and (b) heterogeneous TDRC; the prediction error between the original data and isolated TDRC (c), heterogeneous predicted signal (d).

Fig. 5. Robustness of the prediction performance of heterogeneous TDRC and isolated TDRC for different original data. NMSE as a function of self-feedback strength γf (a) and current coefficient μ (b) in spin-flip model of a VCSEL as shown in Eqs. (1)–(4); (c),(d) are the zoomed-in views of (a), (b).

Fig. 6. Effects of the complexity of the original signal on predictive performance. Evolutions of the polarization state plotted on the normalized Poincaré sphere for polarizer angles (a) θp=10° and (b) θp=50°; (c) NMSE as a function of polarizer angle for isolated TDRC and heterogeneous TDRC; (d) zoomed-in view of (c).

Fig. 7. Two-dimensional evolution of NMSE in parameter spaces of pf×Δfs and σ×kf for (a), (b) isolated TDRC and (c), (d) heterogeneous TDRC.

Fig. 8. Prediction accuracy NMSE of heterogeneous TDRC and isolated TDRC with different number of virtual nodes NR (a1); virtual node intervals θ (a2); size of training dataset Kv (a3). (b1)–(b3) show the zoomed-in view of NMSE for heterogeneous TDRC.

Fig. 9. Diagram of the heterogeneous TDRC experimental setup (includes the input and reservoir layers shown in Fig. 1). AWG, arbitrary waveform generator; OSC, oscilloscope; EDFA, erbium-doped fiber amplifier.

Fig. 10. Predictive performance NMSE of isolated TDRC and heterogeneous TDRC under different (a) self-feedback strengths with σ=5  mW and (b) optical injection strengths with kf=2  μW. In experiments, predictive performance for chaotic dynamics generated by a VCSEL with (c) isolated TDRC prediction and the prediction error between the original data and isolated TDRC, and (d) heterogeneous TDRC prediction and the prediction error between the original data and heterogeneous TDRC.