Optical next generation reservoir computing

TL;DR

An optical implementation of the “next generation reservoir computing” (NGRC) algorithm, leveraging large-scale computations via light scattering through complex media, which can accurately forecast both short-term and long-term dynamics of chaotic time series and infer unmeasured variables from observables.

Abstract

Artificial neural networks with internal dynamics exhibit remarkable capability in processing information. Reservoir computing (RC) is a canonical example that features rich computing expressivity and compatibility with physical implementations for enhanced efficiency. Recently, a new RC paradigm known as next generation reservoir computing (NGRC) further improves expressivity but compromises its physical openness, posing challenges for realizations in physical systems. Here we demonstrate optical NGRC with computations performed by light scattering through disordered media. In contrast to conventional optical RC implementations, we drive our optical reservoir directly with time-delayed inputs. Much like digital NGRC that relies on polynomial features of delayed inputs, our optical reservoir also implicitly generates these polynomial features for desired functionalities. By leveraging the domain knowledge of the reservoir inputs, we show that the optical NGRC not only predicts the short-term dynamics of the low-dimensional Lorenz63 and large-scale Kuramoto-Sivashinsky chaotic time series, but also replicates their long-term ergodic properties. Optical NGRC shows superiority in shorter training length, increased interpretability and fewer hyperparameters compared to conventional optical RC based on scattering media, while achieving better forecasting performance. Our optical NGRC framework may inspire the realization of NGRC in other physical RC systems, new applications beyond time-series processing, and the development of deep and parallel architectures broadly.

Figures (4)

• Figure 1: Optical next generation reservoir computing.(a) Reservoir computing (RC) is a special type of recurrent neural network and is compatible with various physical implementations. In the training phase, RC sequentially maps the current input ($\boldsymbol{u}_t$, blue) and the current reservoir state ($\boldsymbol{r}_{t}$) into the next reservoir state ($\boldsymbol{r}_{t+1}$, orange). After that, only a linear readout layer $\boldsymbol{W}_{out}$ is trained to match $\boldsymbol{\hat{o}}_{t} = \boldsymbol{W}_{out}\boldsymbol{r}_{t}$ with the desired output ($\boldsymbol{o}_{t}$, purple), which is often the input (i.e. $\boldsymbol{o}_{t}=\boldsymbol{u}_{t}$) in the time series prediction tasks. In the prediction phase, by feeding back the predicted output to the input, RC can autonomously evolve as a dynamical system. (b) Different from the conventional RC scheme, next generation reservoir computing (NGRC) directly synthesizes reservoir features by constructing the polynomial functions of the time-delayed inputs (e.g., $\boldsymbol{u}_{t}$ and $\boldsymbol{u}_{t-1}$), without relying on an actual reservoir. (c) Similar to the digital NGRC, the proposed optical NGRC also drives the optical reservoir with time-delayed inputs. The optical process generates the polynomial features of the inputs implicitly. (d) The schematic experimental setup for the optical NGRC. First, input data at the current and the previous time steps ($\boldsymbol{u}_t$ and $\boldsymbol{u}_{t-1}$) as well as a bias $\boldsymbol{b}$, are encoded onto the phase front of a laser beam via a spatial light modulator (SLM). Then, the modulated coherent light illuminates a disordered scattering medium, which provides rich mixing of the input and generates speckle patterns at the output. Finally, the reservoir features are obtained by measuring the intensity of the speckles in a camera. A computer (PC) is used to interface the SLM and the camera, as well as training and implementing readout layer. (e) The mathematical model of optical NGRC. The nonlinear, implicit reservoir speckle features $\boldsymbol{r}_{t+1}$ can be approximated as the multiplication of a system-given matrix $\boldsymbol{M}_s$ by a library of explicit polynomial feature terms $\boldsymbol{\Theta_t}$
• Figure 2: Optical NGRC for Lorenz63 attractor forecasting.(a) Time series of the Lorenz63 attractor (state variables $u_1, u_2, u_3$) that drives the optical NGRC. At each time step of the training phase, the input states from the current ($\boldsymbol{u}_t$) and the previous ($\boldsymbol{u}_{t-1}$) time steps are encoded to the optical system to generate reservoir features ($\boldsymbol{r}_{t+1}$). (b) The temporal evolution of 10 randomly selected optical reservoir nodes (out of 2,000 nodes), which resembles the dynamics of the input data. After training iterations of 4,000 time steps, a linear estimator $\boldsymbol{W}_{out}$ is trained to match the weighted sums of the reservoir features ($\boldsymbol{\hat{o}}_{t} = \boldsymbol{W}_{out}\boldsymbol{r}_{t}$) with the input data at the next time step ($\boldsymbol{u}_{t}$), i.e., $\boldsymbol{\hat{o}}_{t}\approx \boldsymbol{u}_{t}$. (c) Once $\boldsymbol{W}_{out}$ 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 8,000 time steps. (e) The return map of the ground truth (blue) and the experimental prediction (red).
• Figure 3: Optical NGRC for Kuramoto-Sivashinsky time series forecasting.(a) Experimental short-term prediction results of the Kuramoto-Sivashinsky (KS) time series with a domain size of $L=22$ and a spatial sampling of $S=64$. An optical NGRC with 2,500 optical reservoir nodes is used for KS forecasting, which employs the current ($\boldsymbol{u}_t$) and the previous ($\boldsymbol{u}_{t-1}$) time steps in each training iteration for a total training length of 6,000 time steps. The error subfigure (bottom) is the element-wise difference between the ground truth (top) and the experimental prediction (middle). The temporal axis is normalized by its largest Lyapunov time ($\lambda_{max}=0.043$). (b) A part of the long-term prediction results by optical NGRC (between $t_1$ and $t_2$, where the prediction starts at $t_0$). Albeit the complete deviation between the KS ground truth (top) and the optical NGRC predicted output (bottom) at the element-wise level, the optical NGRC replicates the long-term behavior of the KS chaotic system. (c) The power spectra of the long-term prediction in (b) (red), the KS ground truth (blue) and a random noise signal (yellow). The power spectra of the ground truth and optical NGRC predictions are in good agreement, in stark contrast to the power spectrum of the random noise background.
• Figure 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 $u_1$ and $u_2$ (blue) of the Lorenz63 system are provided as observables to infer the third variable $u_3$. 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.