WAVEFRONT CONTROL FOCUS
For both configurations, the system remains fixed, only a linear readout is trained to predict the future value of the time series. Importantly, the same input sequence, training procedure and sampling of the measured speckle patterns are used for both reservoir architectures. Since both single- and doublepass speckle patterns can be acquired from complementary DMD paths, this provides a direct comparison of their computational performance under the same experimental conditions.
The forecasting performance is quantified using the normalized mean square error( NMSE) shown in Figure 2b, as a function of the prediction horizon expressed in units of the Lyapunov time. The Lyapunov time characterizes the time scale over which initially close trajectories of a chaotic system diverge, hence providing a natural timescale for assessing predictability. To limit the influence of individual test samples, which can introduce significant variability in the outcome, the NMSE is first averaged over the evaluation samples and then over 14 realizations of the Mackey-Glass time series, each generated from a different initial condition. As shown in Figure 2b, the structurally nonlinear reservoir consistently achieves a lower NMSE than the conventional linear reservoir. Interestingly, beyond approximately one Lyapunov time, the NMSE curves of the two reservoir configurations look parallel. This suggests that structural nonlinearity provides an improvement during the earlier prediction stages, which is subsequently maintained as both reservoirs accumulate error at a comparable scale. Structural nonlinearity also improves the fidelity of longer-term forecasts, for which pointwise comparison such as the NMSE becomes less meaningful due to the inherent divergence of chaotic trajectories. Indeed, the structurally nonlinear reservoir reproduces the statistical properties of the Mackey-Glass dynamics, with lower errors in both the spectral content and the distribution of the predicted trajectories [ 6 ]. This indicates that the benefit of structural nonlinearity extends beyond short term prediction accuracy to more longer term dynamics of the chaotic system.
CONCLUSION Wavefront shaping can introduce structural nonlinearity into optical computing systems using only linear components. By directing light to interact multiple times with an encoded spatial pattern, the relationships between the scattering potential encoded on the spatial light modulator and the output field becomes nonlinear, without relying on high optical powers or nonlinear materials. We experimentally demonstrated improved short term and long term forecasting of a chaotic time series compared with a conventional optical reservoir. Looking forward, increasing the number of interactions between the beam and the spatial light modulator could provide access to higher nonlinear orders. Combined with faster phase modulation, rather than binary amplitude modulation, this could open interesting opportunities for wavefront shaping in optical machine learning and other optical computing applications.
REFERENCES
[ 1 ] P. L. McMahon, Nat. Rev. Phys. 5, 717( 2023) [ 2 ] A. Saade et al., Proc. IEEE ICASSP, 6215( 2016) [ 3 ] J. Dong et al., IEEE J. Sel. Top. Quantum Electron. 26, 1( 2020) [ 4 ] F. Xia et al., Nat. Photonics 18, 1067( 2024) [ 5 ] M. Yildirim et al., Nat. Photonics 18, 1076( 2024)
[ 6 ] L. R. Venâncio, J. Dong, M. Mounaix et J. Bertolotti,“ Optical Reservoir Computing with Structural Nonlinearity for Forecasting Chaotic Time Series”, arXiv: 2609.02733( 2026)
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