Next-Generation Reservoir Computing

• NG-RC is a deterministic, feedforward framework that computes explicit nonlinear feature expansions from delay-embedded inputs, eliminating the randomness of traditional reservoir computing.
• It leverages polynomial and kernel-based methods with ridge regression to achieve state-of-the-art forecasting and classification, reducing training costs and energy consumption.
• NG-RC is implemented on diverse hardware such as resonant-tunnelling diodes and photonic platforms, demonstrating high accuracy and scalability for dynamic and quantum tasks.

Next-Generation Reservoir Computing (NG-RC) is a deterministic, feedforward computational framework that replaces the random, recurrent structure of classical reservoir computing (RC) with explicit nonlinear feature expansions of delay-embedded input signals. NG-RC is mathematically equivalent to polynomial nonlinear vector autoregression and can be physically realized in various substrates, including electronics (resonant-tunnelling diodes), silicon photonics, fiber-optics, and quantum algorithms. Its core design eliminates reservoir state initialization, warm-up transients, and random projections, yielding compact models with tunable expressivity, state-of-the-art forecasting and classification performance, low energy footprints, and hardware parallelizability.

1. Mathematical Foundations and Algorithms

NG-RC is rooted in direct construction of a nonlinear feature space on delay-embedded observations. At each time $t$, a vector of recent input states (delay depth $k$, spacing $s$) is formed:

$$O_{\rm lin}(t) = [x(t), x(t-s), \ldots, x(t-(k-1)s)]^\top \in \mathbb{R}^{d k}$$

where $x(t)\in\mathbb{R}^d$. Nonlinear features—usually all monomials up to a chosen order $p$—are computed:

$$O_{\rm nonlin}(t) = \{ \prod_{i=1}^{dk} O_{\rm lin,i}(t)^{\alpha_i} : 2\leq \sum_{i}\alpha_i \leq p \}$$

The complete feature is

$$O_{\rm total}(t) = [1; O_{\rm lin}(t); O_{\rm nonlin}(t)] \in \mathbb{R}^{N}$$

where $N$ includes the bias, linear, and nonlinear dimensions.

Learning proceeds via ridge regression: given training responses $Y$, the optimal readout $W_{\rm out}$ solves

$$W_{\rm out} = Y O^\top (O O^\top + \lambda I_N)^{-1}$$

with Tikhonov regularization $\lambda$ (Gauthier et al., 2021). For classification, $Y$ is a matrix of one-hot target labels; for forecasting, $Y$ collects next-step measurements or increments.

In infinite-dimensional NG-RC, polynomial expansions are replaced by Volterra-series kernels:

$$K_{\rm Volt}(\mathbf{z}, \mathbf{z}') = 1 + \sum_{\tau=1}^\infty \lambda^{2\tau} \prod_{t=0}^{\tau-1} (1-\theta^2\,\mathbf{z}_{-t}^\top\mathbf{z}'_{-t})^{-1}$$

allowing arbitrarily deep memory and nonlinearities without explicit feature enumeration (Grigoryeva et al., 13 Dec 2024).

2. Deterministic Nonlinearity, Memory, and Reservoir Elimination

Unlike RC, which relies on recurrent, random networks with tunable spectral radius and leak rate, NG-RC is fully deterministic. The nonlinear feature expansion (polynomial, kernel, or pseudorandom) directly enacts the high-dimensional projection:

$$\Phi: \mathbb{R}^{d k} \longrightarrow \mathbb{R}^N$$

There is no internal state update or echo effect. Memory is regulated by explicit choice of delays $k$ and spacing $s$. This removes the need for warm-up transients—autonomous prediction starts after $k$ samples—and enables high accuracy from sparse training data and minimal model parameters (Gauthier et al., 2022, Zhang et al., 2022).

Physical NG-RC implementations (RTDs, photonics) instantiate nonlinearity via device physics: negative differential resistance in RTDs or quadratic mixing in photodiodes automatically generates polynomial basis functions without random masks or virtual nodes (Abbas et al., 20 Jul 2025, Wang et al., 31 May 2024).

3. Physical Implementations and Hardware Scaling

NG-RC has been realized in diverse hardware:

• Resonant-Tunnelling Diodes (RTDs): Arrays of RTDs implement direct nonlinear current-voltage maps to realize feature expansion for image recognition. MNIST and Fruit360 benchmarks yield 92.5% and 99.1% accuracy, with μJ-scale energy per inference on compact monolithic circuits. RTD-based NG-RC eliminates random connectivity, feedback, and masks, supporting deterministic, low-memory, low-power classification (Abbas et al., 20 Jul 2025).

Substrate	Nonlinearity Mechanism	Task Benchmark	Accuracy/Energy/Speed
RTD array	$I(V)$ NDR, static kernel	MNIST	92.5% accuracy
Photonic star coupler	Quadratic via photodiode	Lorenz, COVID-Xray	NMSE 0.014, 92.1% accuracy, 60 Gbaud, 103 TOPS/mm²
Fiber-optic NGRC	Coherent Rayleigh mixing	Lorenz, KS, Rössler	NRMSE $\sim$0.02, low latency
FM-NGRC (freq. comb)	Dispersion, MZI nonlinear	Channel equalization	SER $2 \times 10^{-3}$ @ 5 GS/s

The photonic platforms exploit delay lines, phase encoding, and quadratic intensity detection for on-chip, ultrafast NG-RC computation, vastly exceeding conventional RC throughput and density (Wang et al., 31 May 2024, Cox et al., 14 Nov 2024, Wang et al., 11 Apr 2024, Cox et al., 10 Apr 2024).

4. Benchmark Results and Comparative Performance

NG-RC frequently surpasses or matches classical RC across prediction, classification, and control benchmarks:

• Human Activity Classification: Six-class accelerometer, NG-RC $75.4\%$ accuracy; ESN-based RC $74\!-\!75\%$ (Kitayama, 15 Dec 2025).
• Chaotic Forecast/Control: Lorenz, Rössler, Hénon, Mackey-Glass, NG-RC matches or exceeds classical RC, achieving longer valid prediction horizons with $10^2\!-\!10^3$ less data and $10^3\!-\!10^4\times$ lower training cost (Barbosa et al., 2022, Gauthier et al., 2022, Kent et al., 2023, Cheng et al., 14 May 2025).
• Quantum Dynamics: Quantum NG-RC (QNG-RC) block-encoding achieves exponential speedup for many-body propagation (Sornsaeng et al., 2023).
• Superconducting Qubit Readout: NG-RC yields $11\!-\!50\%$ error reductions and $2.5\times$ crosstalk suppression compared to matched filters, with $100\times$–$2.5\times$ fewer multiplies than neural networks (Kent et al., 18 Jun 2025).

NG-RC performance is sensitive to feature choice and data regime. For challenging maps or spatiotemporal chaos, locality-blending and translation symmetry can further enhance accuracy and efficiency (Gauthier et al., 30 Mar 2025, Barbosa et al., 2022).

5. Extensions: Locality, Hybridization, and Infinite-Dimensional Models

NG-RC generalizations include:

• Locality-Blended NG-RC (LB-NGRC): Divides phase space into local regions, learning independent low-order polynomials per region and blending via radial-basis weights. Empirically achieves $>5$ Lyapunov times prediction horizon on Ikeda map (Gauthier et al., 30 Mar 2025).
• Hybrid RC-NGRC: Combines small recurrent reservoirs with NG-RC feature blocks, rescuing performance when either paradigm is suboptimal alone (e.g., coarse sampling, small data, ill-chosen feature libraries). Hybrid models retain interpretability and computational savings (Chepuri et al., 4 Mar 2024).
• Infinite-Dimensional NG-RC (Volterra Kernel): Kernelizes feature expansion to eliminate explicit lag/depth and polynomial degree parameters. The Volterra kernel is universal on compact domains, achieving maximal forecasting horizons, favorable error metrics, and competitive climate reconstruction (Grigoryeva et al., 13 Dec 2024).

These variants are especially relevant for high-dimensional, non-polynomial, delay-rich dynamics, where classic polynomial NG-RC may face ill-conditioning and combinatorial feature explosion.

6. Numerical Stability, Conditioning, and Implementation Guidelines

NG-RC’s stability is controlled by the conditioning of the feature matrix $\Phi$:

• Conditioning grows exponentially with polynomial degree and shrinks with increased time lag $\tau$. Large feature dimensions require adequate training samples, delay decorrelation, and careful regularization.
• Solver choice: SVD-based ridge regression is robust to ill-conditioning; Cholesky/LU may fail for small $\lambda$ or highly correlated data (Santos et al., 1 May 2025).
• Dimensionality reduction: Sparse feature selection, column-normalization, or switching to orthogonal (Chebyshev) polynomials can mitigate numerical instability.
• Physical implementations (RTDs, photonics) avoid explicit large matrix formation by leveraging native device physics for nonlinear projection.

7. Limitations, Critical Discussion, and Future Prospects

Despite its advantages, NG-RC is subject to several constraints:

• Feature selection: Optimal performance requires domain-informed feature/block selection. Small errors in nonlinear feature construction can degrade prediction accuracy to chance, especially for basin classification in multistable systems (Zhang et al., 2022).
• High-dimensional PDEs: Combinatorial growth in feature set can cause computational and numerical challenges, though kernelized or randomized projections offer partial relief (Cestnik et al., 14 Sep 2025).
• Dynamic memory: Static feature maps lack feedback, limiting suitability for purely temporal (memory-rich) tasks unless memory is explicitly engineered via delays or feedback extension (Abbas et al., 20 Jul 2025, Wang et al., 31 May 2024, Cox et al., 14 Nov 2024).

Current research directions include quantum reservoir extensions, increased nonlinearity via quantum-dot tunnelling, on-chip photonic integration, adaptive locality blending, recursive/online training, and further exploration of kernel-based universality.

• "Resonant-Tunnelling Diode Reservoir Computing System for Image Recognition" (Abbas et al., 20 Jul 2025)
• "Next-generation reservoir computing validated by classification task" (Kitayama, 15 Dec 2025)
• "Optical next generation reservoir computing" (Wang et al., 11 Apr 2024)
• "Locality Blended Next Generation Reservoir Computing For Attention Accuracy" (Gauthier et al., 30 Mar 2025)
• "Adaptive control for multi-scale stochastic dynamical systems with stochastic next generation reservoir computing" (Cheng et al., 14 May 2025)
• "On the emergence of numerical instabilities in Next Generation Reservoir Computing" (Santos et al., 1 May 2025)
• "A 103-TOPS/mm$^2$ Integrated Photonic Computing Engine Enabling Next-Generation Reservoir Computing" (Wang et al., 31 May 2024)
• "Photonic next-generation reservoir computer based on distributed feedback in optical fiber" (Cox et al., 10 Apr 2024)
• "Photonic frequency multiplexed next-generation reservoir computer" (Cox et al., 14 Nov 2024)
• "Quantum Next Generation Reservoir Computing: An Efficient Quantum Algorithm for Forecasting Quantum Dynamics" (Sornsaeng et al., 2023)
• "Learning unseen coexisting attractors" (Gauthier et al., 2022)
• "Catch-22s of reservoir computing" (Zhang et al., 2022)
• "Learning Spatiotemporal Chaos Using Next-Generation Reservoir Computing" (Barbosa et al., 2022)
• "Controlling Chaotic Maps using Next-Generation Reservoir Computing" (Kent et al., 2023)
• "Controlling dynamical systems to complex target states using machine learning: next-generation vs. classical reservoir computing" (Haluszczynski et al., 2023)
• "Infinite-dimensional next-generation reservoir computing" (Grigoryeva et al., 13 Dec 2024)
• "Hybridizing Traditional and Next-Generation Reservoir Computing to Accurately and Efficiently Forecast Dynamical Systems" (Chepuri et al., 4 Mar 2024)
• "Superconducting Qubit Readout Using Next-Generation Reservoir Computing" (Kent et al., 18 Jun 2025)
• "Next Generation Reservoir Computing" (Gauthier et al., 2021)
• "Next-Generation Reservoir Computing for Dynamical Inference" (Cestnik et al., 14 Sep 2025)