Measurements acquired from distributed physical systems are often sparse and noisy. Therefore, signal processing and system identification tools are required to mitigate noise effects and reconstruct unobserved dynamics from limited sensor data. However, this process is particularly challenging because the fundamental equations governing the dynamics are largely unavailable in practice. Reservoir Computing (RC) techniques have shown promise in efficiently simulating dynamical systems through an unstructured and efficient computation graph comprising a set of neurons with random connectivity. However, the potential of RC to operate in noisy regimes and distinguish noise from the primary smooth or non-smooth deterministic dynamics of the system has not been fully explored. This paper presents a novel RC method for noise filtering and reconstructing unobserved nonlinear dynamics, offering a novel learning protocol associated with hyperparameter optimization. The performance of the RC in terms of noise intensity, noise frequency content, and drastic shifts in dynamical parameters is studied in two illustrative examples involving the nonlinear dynamics of the Lorenz attractor and the adaptive exponential integrate-and-fire system. It is demonstrated that denoising performance improves by truncating redundant nodes and edges of the reservoir, as well as by properly optimizing hyperparameters, such as the leakage rate, spectral radius, input connectivity, and ridge regression parameter. Furthermore, the presented framework shows good generalization behavior when tested for reconstructing unseen and qualitatively different attractors. Compared to the extended Kalman filter, the presented RC framework yields competitive accuracy at low signal-to-noise ratios and high-frequency ranges.

Artificial Intelligence has advanced significantly in recent years thanks to innovations in the design and training of artificial neural networks (ANNs). Despite these advancements, we still understand relatively little about how elementary forms of ANNs learn, fail to learn, and generate false information without the intent to deceive, a phenomenon known as `confabulation'. To provide some foundational insight, in this paper we analyse how confabulation occurs in reservoir computers (RCs): a dynamical system in the form of an ANN. RCs are particularly useful to study as they are known to confabulate in a well-defined way: when RCs are trained to reconstruct the dynamics of a given attractor, they sometimes construct an attractor that they were not trained to construct, a so-called `untrained attractor' (UA). This paper sheds light on the role played by UAs when reconstruction fails and their influence when modelling transitions between reconstructed attractors. Based on our results, we conclude that UAs are an intrinsic feature of learning systems whose state spaces are bounded, and that this means of confabulation may be present in systems beyond RCs.

We propose an online learning framework for forecasting nonlinear spatio-temporal signals (fields). The method integrates (i) dimensionality reduction, here, a simple proper orthogonal decomposition (POD) projection; (ii) a generalized autoregressive model to forecast reduced dynamics, here, a reservoir computer; (iii) online adaptation to update the reservoir computer (the model), here, ensemble sequential data assimilation. We demonstrate the framework on a wake past a cylinder governed by the Navier-Stokes equations, exploring the assimilation of full flow fields (projected onto POD modes) and sparse sensors. Three scenarios are examined: a naïve physical state estimation; a two-fold estimation of physical and reservoir states; and a three-fold estimation that also adjusts the model parameters. The two-fold strategy significantly improves ensemble convergence and reduces reconstruction error compared to the naïve approach. The three-fold approach enables robust online training of partially-trained reservoir computers, overcoming limitations of a priori training. By unifying data-driven reduced order modelling with Bayesian data assimilation, this work opens new opportunities for scalable online model learning for nonlinear time series forecasting.

We show that recurrent quantum reservoir computers (QRCs) and their recurrence-free architectures (RF-QRCs) are robust tools for learning and forecasting chaotic dynamics from time-series data. First, we formulate and interpret quantum reservoir computers as coupled dynamical systems, where the reservoir acts as a response system driven by training data; in other words, quantum reservoir computers are generalized-synchronization (GS) systems. Second, we show that quantum reservoir computers can learn chaotic dynamics and their invariant properties, such as Lyapunov spectra, attractor dimensions, and geometric properties such as the covariant Lyapunov vectors. This analysis is enabled by deriving the Jacobian of the quantum reservoir update. Third, by leveraging tools from generalized synchronization, we provide a method for designing robust quantum reservoir computers. We propose the criterion $GS=ESP$: GS implies the echo state property (ESP), and vice versa. We analytically show that RF-QRCs, by design, fulfill $GS=ESP$. Finally, we analyze the effect of simulated noise. We find that dissipation from noise enhances the robustness of quantum reservoir computers. Numerical verifications on systems of different dimensions support our conclusions. This work opens opportunities for designing robust quantum machines for chaotic time series forecasting on near-term quantum hardware.

We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers, focusing on how closely the model's nonlinearity should align with that of the data. By reducing minimal RCs to a single tunable nonlinearity parameter, we explore how the predictive performance varies with the degree of nonlinearity in the reservoir. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the reservoir's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we find that the correlation dimension of the predicted signal is reconstructed correctly when the smallest nonlinearity is matched. We use this observation to propose a method for estimating the minimal nonlinearity in unknown time series by sweeping the reservoir exponent and identifying the transition to a successful reconstruction. Applying this method to both synthetic and real-world datasets, including financial time series, we demonstrate its practical viability. Finally, we transfer these insights to classical RC by augmenting traditional architectures with fractional, generalized reservoir states. This yields performance gains, particularly in resource-constrained scenarios such as physical reservoirs, where increasing reservoir size is impractical or economically unviable. Our work provides a principled route toward tailoring RCs to the intrinsic complexity of the systems they aim to model.

We propose an innovative design for an all-optical Echo State Network (ESN), an advanced type of reservoir computer known for its universal computational capabilities. Our design enables fully optical implementation of arbitrary ESNs, featuring complete flexibility in optical matrix multiplication and nonlinear activation. Leveraging the nonlinear characteristics of stimulated Brillouin scattering (SBS), the architecture efficiently realizes measurement-free operations crucial for reservoir computing. The approach significantly reduces computational overhead and energy consumption compared to traditional software-based methods. Comprehensive simulations validate the system's memory capacity, nonlinear processing strength, and polynomial algebra capabilities, showcasing performance comparable to software ESNs across key benchmark tasks. Our design establishes a feasible, scalable, and universally applicable framework for optical reservoir computing, suitable for diverse machine learning applications.

Traditional artificial neural networks take inspiration from biological networks, using layers of neuron-like nodes to pass information for processing. More realistic models include spiking in the neural network, capturing the electrical characteristics more closely. However, a large proportion of brain cells are of the glial cell type, in particular astrocytes which have been suggested to play a role in performing computations. Here, we introduce a modified spiking neural network model with added astrocyte-like units in a neural network and asses their impact on learning. We implement the network as a liquid state machine and task the network with performing a chaotic time-series prediction task. We varied the number and ratio of neuron-like and astrocyte-like units in the network to examine the latter units effect on learning. We show that the combination of neurons and astrocytes together, as opposed to neural- and astrocyte-only networks, are critical for driving learning. Interestingly, we found that the highest learning rate was achieved when the ratio between astrocyte-like and neuron-like units was roughly 2 to 1, mirroring some estimates of the ratio of biological astrocytes to neurons. Our results demonstrate that incorporating astrocyte-like units which represent information across longer timescales can alter the learning rates of neural networks, and the proportion of astrocytes to neurons should be tuned appropriately to a given task.

For anticipating critical transitions in complex dynamical systems, the recent approach of parameter-driven reservoir computing requires explicit knowledge of the bifurcation parameter. We articulate a framework combining a variational autoencoder (VAE) and reservoir computing to address this challenge. In particular, the driving factor is detected from time series using the VAE in an unsupervised-learning fashion and the extracted information is then used as the parameter input to the reservoir computer for anticipating the critical transition. We demonstrate the power of the unsupervised learning scheme using prototypical dynamical systems including the spatiotemporal Kuramoto-Sivashinsky system. The scheme can also be extended to scenarios where the target system is driven by several independent parameters or with partial state observations.

In the current Noisy Intermediate Scale Quantum (NISQ) era, the presence of noise deteriorates the performance of quantum computing algorithms. Quantum Reservoir Computing (QRC) is a type of Quantum Machine Learning algorithm, which, however, can benefit from different types of tuned noise. In this paper, we analyse the effect that finite-sampling noise has on the chaotic time-series prediction capabilities of QRC and Recurrence-free Quantum Reservoir Computing (RF-QRC). First, we show that, even without a recurrent loop, RF-QRC contains temporal information about previous reservoir states using leaky integrated neurons. This makes RF-QRC different from Quantum Extreme Learning Machines (QELM). Second, we show that finite sampling noise degrades the prediction capabilities of both QRC and RF-QRC while affecting QRC more due to the propagation of noise. Third, we optimize the training of the finite-sampled quantum reservoir computing framework using two methods: (a) Singular Value Decomposition (SVD) applied to the data matrix containing noisy reservoir activation states; and (b) data-filtering techniques to remove the high-frequencies from the noisy reservoir activation states. We show that denoising reservoir activation states improve the signal-to-noise ratios with smaller training loss. Finally, we demonstrate that the training and denoising of the noisy reservoir activation signals in RF-QRC are highly parallelizable on multiple Quantum Processing Units (QPUs) as compared to the QRC architecture with recurrent connections. The analyses are numerically showcased on prototypical chaotic dynamical systems with relevance to turbulence. This work opens opportunities for using quantum reservoir computing with finite samples for time-series forecasting on near-term quantum hardware.

We tested the performance of reservoir computing (RC) in predicting the dynamics of a certain non-autonomous dynamical system. Specifically, we considered a van del Pol oscillator subjected to periodic external force with frequent phase shifts. The reservoir computer, which was trained and optimized with simulation data generated for a particular phase shift, was designed to predict the oscillation dynamics under periodic external forces with different phase shifts. The results suggest that if the training data have some complexity, it is possible to quantitatively predict the oscillation dynamics exposed to different phase shifts. The setting of this study was motivated by the problem of predicting the state of the circadian rhythm of shift workers and designing a better shift work schedule for each individual. Our results suggest that RC could be exploited for such applications.