Reservoir Computing Generalized

TL;DR

This work introduces generalized reservoir computing (GRC), which shifts the traditional echo state property (ESP) requirement from the reservoir to the final output by leveraging nonlinear readouts with memory to obtain time-invariant outputs from time-variant dynamics. By formalizing Temporal Information Processing Capacity (TIPC), the authors quantify how TI and TV terms are processed and show how TI transformations can recover past inputs even from systems lacking ESP. They validate GRC analytically and numerically using oscillatory and chaotic dynamics, including Lorenz/Rössler models and a spin-torque oscillator, and demonstrate attractor embedding with a Lorenz-96 reservoir for Rössler, Lissajous, and Kuramoto–Sivashinsky targets. The results reveal that spatiotemporal chaos and other non-ESP dynamics can be harnessed for computation via nonlinear readouts with memory, vastly broadening the range of physical substrates usable for neuromorphic and reservoir-based computing.

Abstract

A physical neural network (PNN) has both the strong potential to solve machine learning tasks and intrinsic physical properties, such as high-speed computation and energy efficiency. Reservoir computing (RC) is an excellent framework for implementing an information processing system with a dynamical system by attaching a trained readout, thus accelerating the wide use of unconventional materials for a PNN. However, RC requires the dynamics to reproducibly respond to input sequence, which limits the type of substance available for building information processors. Here we propose a novel framework called generalized reservoir computing (GRC) by turning this requirement on its head, making conventional RC a special case. Using substances that do not respond the same to identical inputs (e.g., a real spin-torque oscillator), we propose mechanisms aimed at obtaining a reliable output and show that processed inputs in the unconventional substance are retrievable. Finally, we demonstrate that, based on our framework, spatiotemporal chaos, which is thought to be unusable as a computational resource, can be used to emulate complex nonlinear dynamics, including large scale spatiotemporal chaos. Overall, our framework removes the limitation to building an information processing device and opens a path to constructing a computational system using a wider variety of physical dynamics.

Figures (12)

• Figure 1: Conventional RC is a special case of GRC. a, Conventional and b, GRC frameworks. The reservoir receives the input $\boldsymbol{u}_t$ and updates the state $\boldsymbol{x}_t$. The outputs $\hat{\boldsymbol{y}}_t$ are calculated by a, a linear or nonlinear readout and b, a TI transformation, which would be realized through a nonlinear readout with memory $\boldsymbol{f}(\boldsymbol{x}_t)$. The green (red) ripples around the nodes represent the state with (without) node-wise ESP (See Supplementary Information S2 for further details). The conventional RC forms outputs with ESP using the reservoir states, thereby already fulfilling ESP, while the GRC can use not only states with ESP but also those without ESP, whose time-dependence is removed through TI transformation to make outputs equipped with ESP.
• Figure 2: TI transformation of time-variant reservoirs.a, b, Two analytical examples and c, a numerical transformation, are illustrated. a left, Removal of time-dependence from time-variant (TV, hatched bar) terms to form time-invariant (TI, non-hatched bar) terms. a middle, The trajectory of oscillatory dynamics, on which the state $(X_t, Y_t)^\top$ moves along a circular orbit with a fixed angular velocity and is perturbed by input $u_t$ in the radial direction. a right, Time-dependent elements in the state are canceled out with coordinate transformation to form the TI output. The TIPC decomposition $C_{\rm tot}$ is depicted by color bars where the non-hatched and hatched bars represent the TI and TV capacities, respectively. The color represents the degree of input: $0$ (purple), $1$ (green), $2$ (orange), and $3$ (red). b left, Amplification of TI terms by removal of TV terms. b middle, The trajectory of the Lissajous knot, on which the state $(X_t,Y_t,Z_t)^\top$ shows the periodic orbit perturbed by the input $u_t$ in the $Z$-direction. b right, The nonlinear readout enlarges the small TI term in the state by canceling out time-dependent functions (hatched purple). See Fig. S1 for further details of the TIPC decompositions in a and b. c, The numerical transformation using the ESN with TV states (its maximum conditional Lyapunov exponent was $\lambda_{\rm max}=1.5\times10^{-2}$) and 4-layer MLP. The ripple color represents whether the node holds the ESP (all the node-wise ESP indices $\bar{d}_i<0.3$, green) or not (red). The bar graphs represent the amount of TI and TV terms in the ESN and MLP layers. c right, Time series of the mean-field states $\bar{x}_t$ and outputs $\hat{y}_t$ with three different initial values (trial #1, blue; #2, red; #3, green) and target $y_t$ (black). The normalized mean square errors between $y_t$ and $\hat{y}_t$ were $0.066$ (#1), $0.068$ (#2), and $0.071$ (#3). Note that those with linear regression were $0.40$ (#1), $0.40$ (#2), and $0.41$ (#3).
• Figure 3: Memory in systems without ESP. a, Lorenz model, b, Rössler model, and c, real STO. a--c illustrate a 3D-trajectory driven by input (upper), the TIPC decomposition of state and outputs (middle), and memory functions of a dynamical system without ESP (lower). c upper, The time-multiplexing technique was applied to the STO reservoir to make $100$ virtual nodes, whose first three nodes were plotted. The TIPC decomposition of state $\boldsymbol{x}_t$ and outputs of a system that trained $\tau$-delayed input $u_{t-\tau}$, using the same colors for the degree of input as in Fig. \ref{['fig:TI_transformation']}. The memory functions $C(\tau)$ with linear (black) and nonlinear readout (red). The memory capacities ${\rm MC} = \sum_\tau C(\tau)$ with nonlinear readout are $0.97$ (Lorenz), $0.97$ (Rössler), and $4.3$ (STO). See Fig. S2 for the relationship between the memory function and TIPC.
• Figure 4: Application of GRC to the reservoirs without ESP. a, Procedure of embedding task. Three target systems of b, Rössler model, c, Four Lissajous curves, and d, KS model were emulated by the Lorenz 96 model. a, In the training phase, the system receives the current target $\boldsymbol{u}$ and predicts the target at the next step by training the nonlinear readout. In the test phase, the input is removed, and the output is fed back into the reservoir as input. b--d illustrate $100$-dimensional time-series of Lorenz 96 out of $500$--$5120$. b, Three variables $(u_1, u_2, u_3)$ of the Rössler model (upper left) are the targets, and $u_1$ is the input (lower). The target (blue) and output (red) trajectories are plotted in the 3D space (upper right). After training, the Lorenz 96 exhibited chaos [the maximum conditional Lyapunov exponent in the training phase was $\lambda_{\rm max}=1.18$, and the maximum Lyapunov exponent (MLE) in the test phase was $\lambda_{\rm max}=1.53$]. c The reservoir state (middle) and output $\boldsymbol{u}$ (upper) in $0\le t\le24$ are displayed. The feedback outputs were perturbed during $8\le t\le16$. In the test phase, the MLE of the Lorenz 96 reservoir was estimated to be $\lambda_{\rm max}=3.5$. Note that the MLE of the embedded signal is estimated to be $\lambda_{\rm max}=0.13$, implying that the embedded attractor is chaotic but has a similar shape as the original one kabayama2024designing. d illustrates the target $\boldsymbol{u}$, the prediction $\hat{\boldsymbol{u}}$, and their error $\boldsymbol{u}-\hat{\boldsymbol{u}}$. The MLEs of the target and output were estimated to be $\lambda_{\rm max}=0.13,0.12$, respectively. Note that $\lambda_{\rm max}$ in the horizontal axis represents the maximum Lyapunov exponent of the KS model and is the averaged time length where the initial value's error grows by a factor of $e$.
• Figure S1: Explanation of TIPC with examples of TI transformation in Fig. 2. The TI transformations of a, oscillatory dynamics and b, Lissajous knot converts the TV reservoir state $\boldsymbol{x}_t$ in Eqs. (\ref{['eqS:oscillatory_dynamics_state_orthogonal_expansion']}) and (\ref{['eqS:lissajous_knot_state_orthogonal_expansion']}) into the TI output $\hat{y}_t$ in Eqs. (\ref{['eqS:oscillatory_dynamics_output_orthogonal_expansion']}) and (\ref{['eqS:lissajous_knot_output_orthogonal_expansion']}), respectively. The state and output are expanded by the TI and TV terms. After normalizing the state and orthogonal bases, the squared norm of coefficient vector represents the TIPC. The TIPCs are summarized as the TIPC decomposition $C_{\rm tot}$, which is depicted by color bars, where the non-hatched and hatched bars represent the TI and TV capacities, respectively. The color of the TIPC represents the degree of input: $0$ (purple), $1$ (green), $2$ (orange), and $3$ (red). The orthogonal bases are underlined by the bar corresponding to that of the TIPC decomposition.