A Practical Guide to Applying Echo State Networks

Reservoir controllers design though robot-reservoir timescale alignment

Natural behavior emerging in nonlinear dynamical systems enables reservoir computers to control underactuated robots by approximating their inverse dynamics. Unlike other model-free approaches, the reservoir controllers are sample-efficient, meaning a weighted average of the reservoir output can be trained with a limited amount of pre-recorded data. However, developing and testing the reservoir controller relies on repetitive experiments that require researchers' proficiency in both robot and reservoir design. In this paper, we propose a design method for reliable reservoir controllers by synchronizing the timescales of the reservoir dynamics with those observed in the robot. The results demonstrate that our timescale alignment test filters out 99% of ineffective reservoirs. We further applied the selected reservoirs to computational tasks including short-term memory and parity checks, along with control tasks involving robot trajectory tracking. Our findings reveal that a higher computational capability reduces the control failure rate, though it concurrently increases the trajectory-tracking error.

Early prediction of CKD from time series data using adaptive PSO optimized echo state networks

Chronic Kidney Disease (CKD) is a significant problem in today's healthcare since it is challenging to detect until it has improved significantly, which increases medical expenses. If CKD was detected early, the patient might qualify for more effective treatment and prevent the disease from spreading further. Presently, existing methods that effectively detect CKD cannot detect symptoms early on. This problem motivates researchers to work on a predictive model that successfully detects disease symptoms in the early stages. This study introduces a novel Adaptive Particle Swarm Optimization (APSO)-optimized Echo State Network (ESN) model designed to overcome key limitations of existing methods. ESNs, while effective in processing temporal sequences, are highly sensitive to hyperparameter settings such as spectral radius, input scaling, and sparsity, which directly impact stability, memory retention, and predictive Classification Accuracy (CA). To address this, APSO optimizes these hyperparameters dynamically, ensuring a balanced trade-off between stability and computational efficiency. Moreover, Random Matrix Theory (RMT) is integrated into APSO to regulate the spectral radius, enhancing the ESN's capability to handle long-term dependencies while maintaining stability in training. This investigation exploited the Medical Information Mart for Intensive Care-III (MIMIC-III) dataset to train the model they developed. The proposed method employs this data collection to analyze the highly complex temporal sequences signifying CKD is present. The hyperparameters of the ESN, such as the range of the spectral region and the input data sizing, can be optimized in real-time with APSO by applying Random Matrix Theory (RMT). Compared with different recognized models, such as conventional ESN and standard M, the recommended APSO + ESN proved to have higher CA in medical investigations. The APSO + ESN improved the subsequent highest-performing model by 2% in recall and 3% in precision and attained a CA of 99.6%.

Simultaneous Measurement of Surface Tension and Viscosity Using a Liquid Dynamics Sensor

The dynamics of liquids upon impact with an object exhibit distinctive behaviors influenced by physical parameters such as surface tension and viscosity, which can be determined by analyzing a liquid's dynamic response. However, measuring these parameters typically requires different tools, a complicated setup, increased space, and higher costs. To streamline this process, a liquid dynamic sensor capable of simultaneously extracting surface tension and viscosity via a single-step measurement is proposed. The proposed measurement method uses a superhydrophobic sensor comprising three electrode pairs, which are fabricated using laser-induced graphene on polydimethylsiloxane. The sensor monitors time-series resistance changes triggered by liquid impact dynamics. The results show that time-series liquid dynamics on the sensor surface vary with the liquid's surface tension and viscosity, allowing for the differentiation of these properties. By implementing an echo state network algorithm, surface tension and viscosity are successfully estimated simultaneously. In addition, the system demonstrates reliable generalization capability, accurately estimating the properties of unknown liquids, which confirms the proposed sensor's robustness for simultaneous measurement of liquid physical parameters.

Simulating the impact of white matter connectivity on processing time scales using brain network models

The capacity of the brain to process input across temporal scales is exemplified in human narrative, which requires integration of information ranging from words, over sentences to long paragraphs. It has been shown that this processing is distributed in a hierarchy across multiple areas in the brain with areas close to the sensory cortex, processing on a faster time scale than areas in associative cortex. In this study we used reservoir computing with human derived connectivity to investigate the effect of the structural connectivity on time scales across brain regions during a narrative task paradigm. We systematically tested the effect of removal of selected fibre bundles (IFO, ILF, MLF, SLF I/II/III, UF, AF) on the processing time scales across brain regions. We show that long distance pathways such as the IFO provide a form of shortcut whereby input driven activation in the visual cortex can directly impact distant frontal areas. To validate our model we demonstrated significant correlation of our predicted time scale ordering with empirical results from the intact/scrambled narrative fMRI task paradigm. This study emphasizes structural connectivity's role in brain temporal processing hierarchies, providing a framework for future research on structure and neural dynamics across cognitive tasks.

Stability analysis of chaotic systems in latent spaces

Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a divide-and-conquer strategy: The solution is sought in a latent space, on which the temporal dynamics are inferred ("latent-space" approach). This is achieved by, first, compressing the data with an autoencoder, and, second, inferring the temporal dynamics with recurrent neural networks. The overarching goal of this paper is to show that a latent-space approach can not only infer the solution of a chaotic partial differential equation, but it can also predict the stability properties of the physical system. First, we employ the convolutional autoencoder echo state network (CAE-ESN) on the chaotic Kuramoto-Sivashinsky equation for various chaotic regimes. We show that the CAE-ESN (i) finds a low-dimensional latent-space representation of the observations and (ii) accurately infers the Lyapunov exponents and covariant Lyapunov vectors (CLVs) in this low-dimensional manifold for different attractors. Second, we extend the CAE-ESN to a turbulent flow, comparing the Lyapunov spectrum to estimates obtained from Jacobian-free methods. A latent-space approach based on the CAE-ESN effectively produces a latent space that preserves the key properties of the chaotic system, such as Lyapunov exponents and CLVs, thus retaining the geometric structure of the attractor. The latent-space approach based on the CAE-ESN is a reduced-order model that accurately predicts the dynamics of the chaotic system, or, alternatively, it can be used to infer stability properties of chaotic systems from data.

An Analysis of Components and Enhancement Strategies for Advancing Memristive Neural Networks

Advancements in artificial intelligence (AI) and big data have highlighted the limitations of traditional von Neumann architectures, such as excessive power consumption and limited performance improvement with increasing parameter numbers. These challenges are significant for edge devices requiring higher energy and area efficiency. Recently, many reports on memristor-based neural networks (Mem-NN) using resistive switching memory have shown efficient computing performance with a low power requirement. Even further performance optimization can be made using engineering resistive switching mechanisms. Nevertheless, systematic reviews that address the circuit-to-material aspects of Mem-NNs, including their dedicated algorithms, remain limited. This review first categorizes the memristor-based neural networks into three components: pre-processing units, processing units, and learning algorithms. Then, the optimization methods to improve integration and operational reliability are discussed across materials, devices, circuits, and algorithms for each component. Furthermore, the review compares recent advancements in chip-level neuromorphic hardware with conventional systems, including graphic processing units. The ongoing challenges and future directions in the field are discussed, highlighting the research to enhance the functionality and reliability of Mem-NNs.

Structured Dynamics in the Algorithmic Agent

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether's theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent's constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain.

Dissipation Alters Modes of Information Encoding in Small Quantum Reservoirs near Criticality

Quantum reservoir computing (QRC) has emerged as a promising paradigm for harnessing near-term quantum devices to tackle temporal machine learning tasks. Yet, identifying the mechanisms that underlie enhanced performance remains challenging, particularly in many-body open systems where nonlinear interactions and dissipation intertwine in complex ways. Here, we investigate a minimal model of a driven-dissipative quantum reservoir described by two coupled Kerr-nonlinear oscillators, an experimentally realizable platform that features controllable coupling, intrinsic nonlinearity, and tunable photon loss. Using Partial Information Decomposition (PID), we examine how different dynamical regimes encode input drive signals in terms of redundancy (information shared by each oscillator) and synergy (information accessible only through their joint observation). Our key results show that, near a critical point marking a dynamical bifurcation, the system transitions from predominantly redundant to synergistic encoding. We further demonstrate that synergy amplifies short-term responsiveness, thereby enhancing immediate memory retention, whereas strong dissipation leads to more redundant encoding that supports long-term memory retention. These findings elucidate how the interplay of instability and dissipation shapes information processing in small quantum systems, providing a fine-grained, information-theoretic perspective for analyzing and designing QRC platforms.

Assessing predictability of environmental time series with statistical and machine learning models

The ever increasing popularity of machine learning methods in virtually all areas of science, engineering and beyond is poised to put established statistical modeling approaches into question. Environmental statistics is no exception, as popular constructs such as neural networks and decision trees are now routinely used to provide forecasts of physical processes ranging from air pollution to meteorology. This presents both challenges and opportunities to the statistical community, which could contribute to the machine learning literature with a model-based approach with formal uncertainty quantification. Should, however, classical statistical methodologies be discarded altogether in environmental statistics, and should our contribution be focused on formalizing machine learning constructs? This work aims at providing some answers to this thought-provoking question with two time series case studies where selected models from both the statistical and machine learning literature are compared in terms of forecasting skills, uncertainty quantification and computational time. Relative merits of both class of approaches are discussed, and broad open questions are formulated as a baseline for a discussion on the topic.

Echo state networks for modeling turbulent convection

Turbulent Rayleigh-Bénard convection (RBC) is one of the very prominent examples of chaos in fluid dynamics with significant relevance in nature. Meanwhile, Echo State Networks (ESN) are among the most fundamental machine learning algorithms suited for modeling sequential data. The current study conducts reduced order modeling of experimental RBC. The ESN successfully models the flow qualitatively. Even for this highly turbulent flow, it is challenging to distinguish predictions from the ground truth. The statistical convergence of the ESN goes beyond the velocity values and is represented in secondary aspects of the flow dynamics, such as spatial and temporal derivatives and vortices. Finally, ESN's main hyperparameters show values for best performance in strong relation to the flow dynamics. These findings from both the fluid dynamics and computer science perspective set the ground for future informed design of ESNs to tackle one of the most challenging problems in nature: turbulence.

Neural network based estimates of the climate impact on mortality in Germany: application to storyline climate simulations

The aim of this work is the prediction of heat-related mortality for Germany under future, i.e. hotter, climate conditions. The prediction is made based on 2m temperature data from climate storyline simulations using machine learning techniques. We use an echo state network for linking the outputs of storyline climate simulations to the target data. The target data are all-cause mortality rates of Germany for all ages. The network is trained with present day climate model outputs. Model outputs of future, i.e. 2K and 4K warmer, storylines are used to predict mortality rates under such climatic conditions. We find that we can train an echo state network with recent temperature data and mortality and make plausible predictions about expected developments of mortality in Germany based on future climate storylines. The trained network can successfully predict mortality rates for future climate conditions. We find increased mortality during the summer months which is attributed to the presence of more severe heat waves. The mortality decrease found during winter can be explained milder winters leading to fewer deaths caused by respiratory diseases. However, mortality in winter is largely influenced by other factors such as influenza waves or vaccination rate and explainability due to temperature is limited.

Neuromorphic overparameterisation and few-shot learning in multilayer physical neural networks

Physical neuromorphic computing, exploiting the complex dynamics of physical systems, has seen rapid advancements in sophistication and performance. Physical reservoir computing, a subset of neuromorphic computing, faces limitations due to its reliance on single systems. This constrains output dimensionality and dynamic range, limiting performance to a narrow range of tasks. Here, we engineer a suite of nanomagnetic array physical reservoirs and interconnect them in parallel and series to create a multilayer neural network architecture. The output of one reservoir is recorded, scaled and virtually fed as input to the next reservoir. This networked approach increases output dimensionality, internal dynamics and computational performance. We demonstrate that a physical neuromorphic system can achieve an overparameterised state, facilitating meta-learning on small training sets and yielding strong performance across a wide range of tasks. Our approach's efficacy is further demonstrated through few-shot learning, where the system rapidly adapts to new tasks.

Learning noise-induced transitions by multi-scaling reservoir computing

Noise is usually regarded as adversarial to extracting effective dynamics from time series, such that conventional approaches usually aim at learning dynamics by mitigating the noisy effect. However, noise can have a functional role in driving transitions between stable states underlying many stochastic dynamics. We find that leveraging a machine learning model, reservoir computing, can learn noise-induced transitions. We propose a concise training protocol with a focus on a pivotal hyperparameter controlling the time scale. The approach is widely applicable, including a bistable system with white noise or colored noise, where it generates accurate statistics of transition time for white noise and specific transition time for colored noise. Instead, the conventional approaches such as SINDy and the recurrent neural network do not faithfully capture stochastic transitions even for the case of white noise. The present approach is also aware of asymmetry of the bistable potential, rotational dynamics caused by non-detailed balance, and transitions in multi-stable systems. For the experimental data of protein folding, it learns statistics of transition time between folded states, enabling us to characterize transition dynamics from a small dataset. The results portend the exploration of extending the prevailing approaches in learning dynamics from noisy time series.

Energy-Efficient Edge and Cloud Image Classification with Multi-Reservoir Echo State Network and Data Processing Units

In an era dominated by Internet of Things (IoT) devices, software-as-a-service (SaaS) platforms, and rapid advances in cloud and edge computing, the demand for efficient and lightweight models suitable for resource-constrained devices such as data processing units (DPUs) has surged. Traditional deep learning models, such as convolutional neural networks (CNNs), pose significant computational and memory challenges, limiting their use in resource-constrained environments. Echo State Networks (ESNs), based on reservoir computing principles, offer a promising alternative with reduced computational complexity and shorter training times. This study explores the applicability of ESN-based architectures in image classification and weather forecasting tasks, using benchmarks such as the MNIST, FashionMnist, and CloudCast datasets. Through comprehensive evaluations, the Multi-Reservoir ESN (MRESN) architecture emerges as a standout performer, demonstrating its potential for deployment on DPUs or home stations. In exploiting the dynamic adaptability of MRESN to changing input signals, such as weather forecasts, continuous on-device training becomes feasible, eliminating the need for static pre-trained models. Our results highlight the importance of lightweight models such as MRESN in cloud and edge computing applications where efficiency and sustainability are paramount. This study contributes to the advancement of efficient computing practices by providing novel insights into the performance and versatility of MRESN architectures. By facilitating the adoption of lightweight models in resource-constrained environments, our research provides a viable alternative for improved efficiency and scalability in modern computing paradigms.

Directed and acyclic synaptic connectivity in the human layer 2-3 cortical microcircuit

The computational capabilities of neuronal networks are fundamentally constrained by their specific connectivity. Previous studies of cortical connectivity have mostly been carried out in rodents; whether the principles established therein also apply to the evolutionarily expanded human cortex is unclear. We studied network properties within the human temporal cortex using samples obtained from brain surgery. We analyzed multineuron patch-clamp recordings in layer 2-3 pyramidal neurons and identified substantial differences compared with rodents. Reciprocity showed random distribution, synaptic strength was independent from connection probability, and connectivity of the supragranular temporal cortex followed a directed and mostly acyclic graph topology. Application of these principles in neuronal models increased dimensionality of network dynamics, suggesting a critical role for cortical computation.

Impact of time-history terms on reservoir dynamics and prediction accuracy in echo state networks

The echo state network (ESN) is an excellent machine learning model for processing time-series data. This model, utilising the response of a recurrent neural network, called a reservoir, to input signals, achieves high training efficiency. Introducing time-history terms into the neuron model of the reservoir is known to improve the time-series prediction performance of ESN, yet the reasons for this improvement have not been quantitatively explained in terms of reservoir dynamics characteristics. Therefore, we hypothesised that the performance enhancement brought about by time-history terms could be explained by delay capacity, a recently proposed metric for assessing the memory performance of reservoirs. To test this hypothesis, we conducted comparative experiments using ESN models with time-history terms, namely leaky integrator ESNs (LI-ESN) and chaotic echo state networks (ChESN). The results suggest that compared with ESNs without time-history terms, the reservoir dynamics of LI-ESN and ChESN can maintain diversity and stability while possessing higher delay capacity, leading to their superior performance. Explaining ESN performance through dynamical metrics are crucial for evaluating the numerous ESN architectures recently proposed from a general perspective and for the development of more sophisticated architectures, and this study contributes to such efforts.

Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction

Recently, machine learning methods, including reservoir computing (RC), have been tremendously successful in predicting complex dynamics in many fields. However, a present challenge lies in pushing for the limit of prediction accuracy while maintaining the low complexity of the model. Here, we design a data-driven, model-free framework named higher-order Granger reservoir computing (HoGRC), which owns two major missions: The first is to infer the higher-order structures incorporating the idea of Granger causality with the RC, and, simultaneously, the second is to realize multi-step prediction by feeding the time series and the inferred higher-order information into HoGRC. We demonstrate the efficacy and robustness of the HoGRC using several representative systems, including the classical chaotic systems, the network dynamical systems, and the UK power grid system. In the era of machine learning and complex systems, we anticipate a broad application of the HoGRC framework in structure inference and dynamics prediction.

Reservoir Computing With Dynamic Reservoir using Cascaded DNA Memristors

This article proposes molecular and DNA memristors where the state is defined by a single output variable. In past molecular and DNA memristors, the state of the memristor was defined based on two output variables. These memristors cannot be cascaded because their input and output sizes are different. We introduce a different definition of state for the molecular and DNA memristors. This change allows cascading of memristors. The proposed memristors are used to build reservoir computing (RC) models that can process temporal inputs. An RC system consists of two parts: reservoir and readout layer. The first part projects the information from the input space into a high-dimensional feature space. We also study the input-state characteristics of the cascaded memristors and show that the cascaded memristors retain the memristive behavior. The cascade connections in a reservoir can change dynamically; this allows the synthesis of a dynamic reservoir as opposed to a static one in the prior work. This reduces the number of memristors significantly compared to a static reservoir. The inputs to the readout layer correspond to one molecule per state instead of two; this significantly reduces the number of molecular and DSD reactions for the readout layer. A DNA RC system consisting of DNA memristors and a DNA readout layer is used to detect seizures from intra-cranial electroencephalogram (iEEG). We also demonstrate that a DNA RC system consisting of three cascaded DNA memristors and a DNA readout layer can be used to solve the time-series prediction task. The proposed approach can reduce the number of DNA strand displacement (DSD) reactions by three to five times compared to prior approaches.

The spinal cord facilitates cerebellar upper limb motor learning and control; inputs from neuromusculoskeletal simulation

Complex interactions between brain regions and the spinal cord (SC) govern body motion, which is ultimately driven by muscle activation. Motor planning or learning are mainly conducted at higher brain regions, whilst the SC acts as a brain-muscle gateway and as a motor control centre providing fast reflexes and muscle activity regulation. Thus, higher brain areas need to cope with the SC as an inherent and evolutionary older part of the body dynamics. Here, we address the question of how SC dynamics affects motor learning within the cerebellum; in particular, does the SC facilitate cerebellar motor learning or constitute a biological constraint? We provide an exploratory framework by integrating biologically plausible cerebellar and SC computational models in a musculoskeletal upper limb control loop. The cerebellar model, equipped with the main form of cerebellar plasticity, provides motor adaptation; whilst the SC model implements stretch reflex and reciprocal inhibition between antagonist muscles. The resulting spino-cerebellar model is tested performing a set of upper limb motor tasks, including external perturbation studies. A cerebellar model, lacking the implemented SC model and directly controlling the simulated muscles, was also tested in the same. The performances of the spino-cerebellar and cerebellar models were then compared, thus allowing directly addressing the SC influence on cerebellar motor adaptation and learning, and on handling external motor perturbations. Performance was assessed in both joint and muscle space, and compared with kinematic and EMG recordings from healthy participants. The differences in cerebellar synaptic adaptation between both models were also studied. We conclude that the SC facilitates cerebellar motor learning; when the SC circuits are in the loop, faster convergence in motor learning is achieved with simpler cerebellar synaptic weight distributions. The SC is also found to improve robustness against external perturbations, by better reproducing and modulating muscle cocontraction patterns.

Computation harvesting from nature dynamics for predicting wind speed and direction

Natural phenomena generate complex dynamics because of nonlinear interactions among their components. The dynamics can be exploited as a kind of computational resource. For example, in the framework of natural computation, various natural phenomena such as quantum mechanics and cellular dynamics are used to realize general purpose calculations or logical operations. In recent years, simple collection of such nature dynamics has become possible in a sensor-rich society. For example, images of plant movement that have been captured indirectly by a surveillance camera can be regarded as sensor outputs reflecting the state of the wind striking the plant. Herein, based on ideas of physical reservoir computing, we present a methodology for wind speed and direction estimation from naturally occurring sensors in movies. Then we demonstrate its effectiveness through experimentation. Specifically using the proposed methodology, we investigate the computational capability of the nature dynamics, revealing its high robustness and generalization performance for computation.

fastESN: Fast Echo State Network

Echo state networks (ESNs) are reservoir computing-based recurrent neural networks widely used in pattern analysis and machine intelligence applications. In order to achieve high accuracy with large model capacity, ESNs usually contain a large-sized internal layer (reservoir), making the evaluation process too slow for some applications. In this work, we speed up the evaluation of ESN by building a reduced network called the fast ESN (fastESN) and achieve an ESN evaluation complexity independent of the original ESN size for the first time. FastESN is generated using three techniques. First, the high-dimensional state of the original ESN is approximated by a low-dimensional state through proper orthogonal decomposition (POD)-based projection. Second, the activation function evaluation number is reduced through the discrete empirical interpolation method (DEIM). Third, we show the directly generated fastESN has instability problems and provide a stabilization scheme as a solution. Through experiments on four popular benchmarks, we show that fastESN is able to accelerate the sparse storage-based ESN evaluation with a high parameter compression ratio and a fast evaluation speed.

Fast and adaptive dynamics-on-graphs to dynamics-of-graphs translation

Numerous networks in the real world change with time, producing dynamic graphs such as human mobility networks and brain networks. Typically, the "dynamics on graphs" (e.g., changing node attribute values) are visible, and they may be connected to and suggestive of the "dynamics of graphs" (e.g., evolution of the graph topology). Due to two fundamental obstacles, modeling and mapping between them have not been thoroughly explored: (1) the difficulty of developing a highly adaptable model without solid hypotheses and (2) the ineffectiveness and slowness of processing data with varying granularity. To solve these issues, we offer a novel scalable deep echo-state graph dynamics encoder for networks with significant temporal duration and dimensions. A novel neural architecture search (NAS) technique is then proposed and tailored for the deep echo-state encoder to ensure strong learnability. Extensive experiments on synthetic and actual application data illustrate the proposed method's exceptional effectiveness and efficiency.

Constraining chaos: Enforcing dynamical invariants in the training of reservoir computers

Drawing on ergodic theory, we introduce a novel training method for machine learning based forecasting methods for chaotic dynamical systems. The training enforces dynamical invariants-such as the Lyapunov exponent spectrum and the fractal dimension-in the systems of interest, enabling longer and more stable forecasts when operating with limited data. The technique is demonstrated in detail using reservoir computing, a specific kind of recurrent neural network. Results are given for the Lorenz 1996 chaotic dynamical system and a spectral quasi-geostrophic model of the atmosphere, both typical test cases for numerical weather prediction.

Machine Learning Enables Prediction of Halide Perovskites' Optical Behavior with >90% Accuracy

The composition-dependent degradation of hybrid organic-inorganic perovskites (HOIPs) due to environmental stressors still precludes their commercialization. It is very difficult to quantify their behavior upon exposure to each stressor by exclusively using trial-and-error methods due to the high-dimensional parameter space involved. We implement machine learning (ML) models using high-throughput, in situ photoluminescence (PL) to predict the response of Cs _y FA_1-y Pb(Br _x I_1-x )₃ while exposed to relative humidity cycles. We quantitatively compare three ML models while generating forecasts of environment-dependent PL responses: linear regression, echo state network, and seasonal autoregressive integrated moving average with exogenous regressor algorithms. We achieve accuracy of >90% for the latter, while tracking PL changes over a 50 h window. Samples with 17% of Cs content consistently showed a PL increase as a function of cycle. Our precise time-series forecasts can be extended to other HOIP families, illustrating the potential of data-centric approaches to accelerate material development for clean-energy devices.

Stability analysis of chaotic systems from data

The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11071-023-08285-1.

Monkeypox: a model-free analysis

Monkeypox is a zoonotic disease caused by a virus that is a member of the orthopox genus, which has been causing an outbreak since May 2022 around the globe outside of its country of origin Democratic Republic of the Congo, Africa. Here we systematically analyze the data of cumulative infection per day adapting model-free analysis, in particular, statistically using the power law distribution, and then separately we use reservoir computing-based Echo state network (ESN) to predict and forecast the disease spread. We also use the power law to characterize the country-specific infection rate which will characterize the growth pattern of the disease spread such as whether the disease spread reached a saturation state or not. The results obtained from power law method were then compared with the outbreak of the smallpox virus in 1907 in Tokyo, Japan. The results from the machine learning-based method are also validated by the power law scaling exponent, and the correlation has been reported.

Growing Echo State Network With an Inverse-Free Weight Update Strategy

An echo state network (ESN) draws widespread attention and is applied in many scenarios. As the most typical approach for solving the ESN, the matrix inverse operation of high computational complexity is involved. However, in the modern big data era, addressing the heavy computational burden problem is necessary. In order to reduce the computational load, an inverse-free ESN (IFESN) is proposed for the first time in this article. Besides, an incremental IFESN is constructed to attain the network topology with theoretical proof on the training error's monotone decline property. Simulations and experiments are conducted on several numerical and real-world time-series benchmarks, and corresponding results indicate that the proposed model is superior to some existing models and possesses excellent practical application potential. The source code is publicly available at https://github.com/LongJin-lab/the-supplementary-file-for-CYB-E-2021-04-0944.

SpaRCe: Improved Learning of Reservoir Computing Systems Through Sparse Representations

"Sparse" neural networks, in which relatively few neurons or connections are active, are common in both machine learning and neuroscience. While, in machine learning, "sparsity" is related to a penalty term that leads to some connecting weights becoming small or zero, in biological brains, sparsity is often created when high spiking thresholds prevent neuronal activity. Here, we introduce sparsity into a reservoir computing network via neuron-specific learnable thresholds of activity, allowing neurons with low thresholds to contribute to decision-making but suppressing information from neurons with high thresholds. This approach, which we term "SpaRCe," optimizes the sparsity level of the reservoir without affecting the reservoir dynamics. The read-out weights and the thresholds are learned by an online gradient rule that minimizes an error function on the outputs of the network. Threshold learning occurs by the balance of two opposing forces: reducing interneuronal correlations in the reservoir by deactivating redundant neurons, while increasing the activity of neurons participating in correct decisions. We test SpaRCe on classification problems and find that threshold learning improves performance compared to standard reservoir computing. SpaRCe alleviates the problem of catastrophic forgetting, a problem most evident in standard echo state networks (ESNs) and recurrent neural networks in general, due to increasing the number of task-specialized neurons that are included in the network decisions.

Generalizability of reservoir computing for flux-driven two-dimensional convection

We explore the generalization properties of an echo state network applied as a reduced-order model to predict flux-driven two-dimensional turbulent convection. To this end, we consider a convection domain with constant height with a variable ratio of buoyancy fluxes at the top and bottom boundaries, which break the top-down symmetry in comparison to the standard Rayleigh-Bénard case, thus leading to highly asymmetric mean and fluctuation profiles across the layer. Our direct numerical simulation model describes a convective boundary layer in a simple way. The data are used to train and test a recurrent neural network in the form of an echo state network. The input of the echo state network is obtained in two different ways, either by a proper orthogonal decomposition or by a convolutional autoencoder. In both cases, the echo state network reproduces the turbulence dynamics and the statistical properties of the buoyancy flux, and is able to model unseen data records with different flux ratios.

Predicting vertical ground reaction forces from 3D accelerometry using reservoir computers leads to accurate gait event detection

Accelerometers are low-cost measurement devices that can readily be used outside the lab. However, determining isolated gait events from accelerometer signals, especially foot-off events during running, is an open problem. We outline a two-step approach where machine learning serves to predict vertical ground reaction forces from accelerometer signals, followed by force-based event detection. We collected shank accelerometer signals and ground reaction forces from 21 adults during comfortable walking and running on an instrumented treadmill. We trained one common reservoir computer using segmented data using both walking and running data. Despite being trained on just a small number of strides, this reservoir computer predicted vertical ground reaction forces in continuous gait with high quality. The subsequent foot contact and foot off event detection proved highly accurate when compared to the gold standard based on co-registered ground reaction forces. Our proof-of-concept illustrates the capacity of combining accelerometry with machine learning for detecting isolated gait events irrespective of mode of locomotion.

Decomposing predictability to identify dominant causal drivers in complex ecosystems

Ecosystems are complex systems of various physical, biological, and chemical processes. Since ecosystem dynamics are composed of a mixture of different levels of stochasticity and nonlinearity, handling these data is a challenge for existing methods of time series-based causal inferences. Here, we show that, by harnessing contemporary machine learning approaches, the concept of Granger causality can be effectively extended to the analysis of complex ecosystem time series and bridge the gap between dynamical and statistical approaches. The central idea is to use an ensemble of fast and highly predictive artificial neural networks to select a minimal set of variables that maximizes the prediction of a given variable. It enables decomposition of the relationship among variables through quantifying the contribution of an individual variable to the overall predictive performance. We show how our approach, EcohNet, can improve interaction network inference for a mesocosm experiment and simulated ecosystems. The application of the method to a long-term lake monitoring dataset yielded interpretable results on the drivers causing cyanobacteria blooms, which is a serious threat to ecological integrity and ecosystem services. Since performance of EcohNet is enhanced by its predictive capabilities, it also provides an optimized forecasting of overall components in ecosystems. EcohNet could be used to analyze complex and hybrid multivariate time series in many scientific areas not limited to ecosystems.

Model-free prediction of multistability using echo state network

In the field of complex dynamics, multistable attractors have been gaining significant attention due to their unpredictability in occurrence and extreme sensitivity to initial conditions. Co-existing attractors are abundant in diverse systems ranging from climate to finance and ecological to social systems. In this article, we investigate a data-driven approach to infer different dynamics of a multistable system using an echo state network. We start with a parameter-aware reservoir and predict diverse dynamics for different parameter values. Interestingly, a machine is able to reproduce the dynamics almost perfectly even at distant parameters, which lie considerably far from the parameter values related to the training dynamics. In continuation, we can predict whole bifurcation diagram significant accuracy as well. We extend this study for exploring various dynamics of multistable attractors at an unknown parameter value. While we train the machine with the dynamics of only one attractor at parameter p, it can capture the dynamics of a co-existing attractor at a new parameter value p + Δ p. Continuing the simulation for a multiple set of initial conditions, we can identify the basins for different attractors. We generalize the results by applying the scheme on two distinct multistable systems.

Detecting Gait Events from Accelerations Using Reservoir Computing

Segmenting the gait cycle into multiple phases using gait event detection (GED) is a well-researched subject with many accurate algorithms. However, the algorithms that are able to perform accurate and robust GED for real-life environments and physical diseases tend to be too complex for their implementation on simple hardware systems limited in computing power and memory, such as those used in wearable devices. This study focuses on a numerical implementation of a reservoir computing (RC) algorithm called the echo state network (ESN) that is based on simple computational steps that are easy to implement on portable hardware systems for real-time detection. RC is a neural network method that is widely used for signal processing applications and uses a fast-training method based on a ridge regression adapted to the large quantity and variety of IMU data needed to use RC in various real-life environment GED. In this study, an ESN was used to perform offline GED with gait data from IMU and ground force sensors retrieved from three databases for a total of 28 healthy adults and 15 walking conditions. Our main finding is that despite its low complexity, ESN is robust for GED, with performance comparable to other state-of-the-art algorithms. Our results show the ESN is robust enough to obtain good detection results in all conditions if the algorithm is trained with variable data that match those conditions. The distribution of the mean absolute errors (MAE) between the detection times from the ESN and the force sensors were between 40 and 120 ms for 6 defined gait events (95th percentile). We compared our ESN with four different state-of-the-art algorithms from the literature. The ESN obtained a MAE not more than 10 ms above three other reference algorithms for normal walking indoor and outdoor conditions and yielded the 2nd lowest MAE and the 2nd highest true positive rate and specificity when applied to outdoor walking and running conditions. Our work opens the door to using the ESN as a GED for applications in wearable sensors for long-term patient monitoring.

Reconfigurable electro-optical logic gates using a 2-layer multilayer perceptron

In this work, we aim to use the optical amplifiers, directional couplers and phase modulators to build the electro-optical gates. Thanks to the 2-layer-multilayer-perceptron structure, the inversion of matrix is performed to obtain the coupling ratio of the directional couplers and the phase delay of the phase modulators. The electro-optical OR, AND, XOR, NAND, NOR and XNOR gates are demonstrated. Moreover, we not only study the results under the ideal condition of device, but also discuss the imperfect situation with 1% error of fabrication or operation to study the tolerance of this system. Through our simulation results, the visibility of the gate output can be higher than 0.83. The gates can be fabricated in a silicon-based chip to develop the integrated optics computing system.

A RUL Estimation System from Clustered Run-to-Failure Degradation Signals

The prognostics and health management disciplines provide an efficient solution to improve a system's durability, taking advantage of its lifespan in functionality before a failure appears. Prognostics are performed to estimate the system or subsystem's remaining useful life (RUL). This estimation can be used as a supply in decision-making within maintenance plans and procedures. This work focuses on prognostics by developing a recurrent neural network and a forecasting method called Prophet to measure the performance quality in RUL estimation. We apply this approach to degradation signals, which do not need to be monotonical. Finally, we test our system using data from new generation telescopes in real-world applications.

Reducing echo state network size with controllability matrices

Echo state networks are a fast training variant of recurrent neural networks excelling at approximating nonlinear dynamical systems and time series prediction. These machine learning models act as nonlinear fading memory filters. While these models benefit from quick training and low complexity, computation demands from a large reservoir matrix are a bottleneck. Using control theory, a reduced size replacement reservoir matrix is found. Starting from a large, task-effective reservoir matrix, we form a controllability matrix whose rank indicates the active sub-manifold and candidate replacement reservoir size. Resulting time speed-ups and reduced memory usage come with minimal error increase to chaotic climate reconstruction or short term prediction. Experiments are performed on simple time series signals and the Lorenz-1963 and Mackey-Glass complex chaotic signals. Observing low error models shows variation of active rank and memory along a sequence of predictions.

A Multitasking Flexible Sensor via Reservoir Computing

Natural disasters are reported globally, and one source of severe damage to cities is flooding caused by locally heavy rain. Sharing of local weather information can save lives. However, it is difficult to collect local weather information in real-time because such data collection requires bulky, expensive sensors. For local, real-time monitoring of heavy rain and wind, a sensor system should be simple and low-cost so that it can be attached to a variety of surfaces, including roofs, vehicles, and umbrellas. To develop simple, low-cost multitasking sensors located on nonplanar surfaces, a flexible rain sensor to monitor waterdrop volume and wind velocity is devised. To monitor both simultaneously, a laser-induced graphene-based superhydrophobic conductive film is introduced. Using the superhydrophobic surface, water dynamics are measured when waterdrops collide with the sensor surface, and obtained time-series data are processed using "reservoir computing" to extract the volume and velocity from a single sensor as multitasking electronics. As a proof-of-concept, it is shown that the sensor measures continuous, long-term volume and wind-change dynamics. The results demonstrate feasibility of multitasking electronics with reservoir computing to reduce sensor integration complexity with low power consumption for both sensor and signal processing.

Simulation platform for pattern recognition based on reservoir computing with memristor networks

Memristive systems and devices are potentially available for implementing reservoir computing (RC) systems applied to pattern recognition. However, the computational ability of memristive RC systems depends on intertwined factors such as system architectures and physical properties of memristive elements, which complicates identifying the key factor for system performance. Here we develop a simulation platform for RC with memristor device networks, which enables testing different system designs for performance improvement. Numerical simulations show that the memristor-network-based RC systems can yield high computational performance comparable to that of state-of-the-art methods in three time series classification tasks. We demonstrate that the excellent and robust computation under device-to-device variability can be achieved by appropriately setting network structures, nonlinearity of memristors, and pre/post-processing, which increases the potential for reliable computation with unreliable component devices. Our results contribute to an establishment of a design guide for memristive reservoirs toward the realization of energy-efficient machine learning hardware.

Cellular Automata Can Reduce Memory Requirements of Collective-State Computing

Various nonclassical approaches of distributed information processing, such as neural networks, reservoir computing (RC), vector symbolic architectures (VSAs), and others, employ the principle of collective-state computing. In this type of computing, the variables relevant in computation are superimposed into a single high-dimensional state vector, the collective state. The variable encoding uses a fixed set of random patterns, which has to be stored and kept available during the computation. In this article, we show that an elementary cellular automaton with rule 90 (CA90) enables the space-time tradeoff for collective-state computing models that use random dense binary representations, i.e., memory requirements can be traded off with computation running CA90. We investigate the randomization behavior of CA90, in particular, the relation between the length of the randomization period and the size of the grid, and how CA90 preserves similarity in the presence of the initialization noise. Based on these analyses, we discuss how to optimize a collective-state computing model, in which CA90 expands representations on the fly from short seed patterns-rather than storing the full set of random patterns. The CA90 expansion is applied and tested in concrete scenarios using RC and VSAs. Our experimental results show that collective-state computing with CA90 expansion performs similarly compared to traditional collective-state models, in which random patterns are generated initially by a pseudorandom number generator and then stored in a large memory.

Real-Time and Adaptive Reservoir Computing With Application to Profile Prediction in Fusion Plasma

Nuclear fusion is a promising alternative to address the problem of sustainable energy production. The tokamak is an approach to fusion based on magnetic plasma confinement, constituting a complex physical system with many control challenges. We study the characteristics and optimization of reservoir computing (RC) for real-time and adaptive prediction of plasma profiles in the DIII-D tokamak. Our experiments demonstrate that RC achieves comparable results to state-of-the-art (deep) convolutional neural networks (CNNs) and long short-term memory (LSTM) models, with a significantly easier and faster training procedure. This efficient approach allows for fast and frequent adaptation of the model to new situations, such as changing plasma conditions or different fusion devices.

Adapting reservoir computing to solve the Schrödinger equation

Reservoir computing is a machine learning algorithm that excels at predicting the evolution of time series, in particular, dynamical systems. Moreover, it has also shown superb performance at solving partial differential equations. In this work, we adapt this methodology to integrate the time-dependent Schrödinger equation, propagating an initial wavefunction in time. Since such wavefunctions are complex-valued high-dimensional arrays, the reservoir computing formalism needs to be extended to cope with complex-valued data. Furthermore, we propose a multi-step learning strategy that avoids overfitting the training data. We illustrate the performance of our adapted reservoir computing method by application to four standard problems in molecular vibrational dynamics.

High-Performance Reservoir Computing With Fluctuations in Linear Networks

Reservoir computing has emerged as a powerful machine learning paradigm for harvesting nontrivial information processing out of disordered physical systems driven by sequential inputs. To this end, the system observables must become nonlinear functions of the input history. We show that encoding the input to quantum or classical fluctuations of a network of interacting harmonic oscillators can lead to a high performance comparable to that of a standard echo state network in several nonlinear benchmark tasks. This equivalence in performance holds even with a linear Hamiltonian and a readout linear in the system observables. Furthermore, we find that the performance of the network of harmonic oscillators in nonlinear tasks is robust to errors both in input and reservoir observables caused by external noise. For any reservoir computing system with a linear readout, the magnitude of trained weights can either amplify or suppress noise added to reservoir observables. We use this general result to explain why the oscillators are robust to noise and why having precise control over reservoir memory is important for noise robustness in general. Our results pave the way toward reservoir computing harnessing fluctuations in disordered linear systems.

Adaptive Practical Nonlinear Model Predictive Control for Echo State Network Models

This article proposes an adaptive practical nonlinear model predictive (NMPC) control algorithm which uses an echo state network (ESN) estimated online as a process model. In the proposed control algorithm, the ESN readout parameters are estimated online using a recursive least-squares method that considers an adaptive directional forgetting factor. The ESN model is used to obtain online a nonlinear prediction of the system free response, and a linearized version of the neural model is obtained at each sampling time to get a local approximation of the system step response, which is used to build the dynamic matrix of the system. The proposed controller was evaluated in a benchmark conical tank level control problem, and the results were compared with three baseline controllers. The proposed approach achieved similar results as the ones obtained by its nonadaptive baseline version in a scenario with the process operating with the nominal parameters, and outperformed all baseline algorithms in a scenario with process parameter changes. Additionally, the computational time required by the proposed algorithm was one-tenth of that required by the baseline NMPC, which shows that the proposed algorithm is suitable to implement state-of-the-art adaptive NMPC in a computationally affordable manner.

Information Processing Capacity of a Single-Node Reservoir Computer: An Experimental Evaluation

Physical dynamical systems are able to process information in a nontrivial manner. The machine learning paradigm of reservoir computing (RC) provides a suitable framework for information processing in (analog) dynamical systems. The potential of dynamical systems for RC can be quantitatively characterized by the information processing capacity (IPC) measure. Here, we evaluate the IPC measure of a reservoir computer based on a single-analog nonlinear node coupled with delay. We link the extracted IPC measures to the dynamical regime of the reservoir, reporting an experimentally measured nonlinear memory of up to seventh order. In addition, we find a nonhomogeneous distribution of the linear and nonlinear contributions to the IPC as a function of the system operating conditions. Finally, we unveil the role of noise in the IPC of the analog implementation by performing ad hoc numerical simulations. In this manner, we identify the so-called edge of stability as being the most promising operating condition of the experimental implementation for RC purposes in terms of computational power and noise robustness. Similarly, a strong input drive is shown to have beneficial properties, albeit with a reduced memory depth.

A machine-learning approach for long-term prediction of experimental cardiac action potential time series using an autoencoder and echo state networks

Computational modeling and experimental/clinical prediction of the complex signals during cardiac arrhythmias have the potential to lead to new approaches for prevention and treatment. Machine-learning (ML) and deep-learning approaches can be used for time-series forecasting and have recently been applied to cardiac electrophysiology. While the high spatiotemporal nonlinearity of cardiac electrical dynamics has hindered application of these approaches, the fact that cardiac voltage time series are not random suggests that reliable and efficient ML methods have the potential to predict future action potentials. This work introduces and evaluates an integrated architecture in which a long short-term memory autoencoder (AE) is integrated into the echo state network (ESN) framework. In this approach, the AE learns a compressed representation of the input nonlinear time series. Then, the trained encoder serves as a feature-extraction component, feeding the learned features into the recurrent ESN reservoir. The proposed AE-ESN approach is evaluated using synthetic and experimental voltage time series from cardiac cells, which exhibit nonlinear and chaotic behavior. Compared to the baseline and physics-informed ESN approaches, the AE-ESN yields mean absolute errors in predicted voltage 6-14 times smaller when forecasting approximately 20 future action potentials for the datasets considered. The AE-ESN also demonstrates less sensitivity to algorithmic parameter settings. Furthermore, the representation provided by the feature-extraction component removes the requirement in previous work for explicitly introducing external stimulus currents, which may not be easily extracted from real-world datasets, as additional time series, thereby making the AE-ESN easier to apply to clinical data.

Chain-Structure Echo State Network With Stochastic Optimization: Methodology and Application

In this article, a chain-structure echo state network (CESN) with stacked subnetwork modules is newly proposed as a new kind of deep recurrent neural network for multivariate time series prediction. Motivated by the philosophy of "divide and conquer," the related input vectors are first divided into clusters, and the final output results of CESN are then integrated by successively learning the predicted values of each clustered variable. Network structure, mathematical model, training mechanism, and stability analysis are, respectively, studied for the proposed CESN. In the training stage, least-squares regression is first used to pretrain the output weights in a module-by-module way, and stochastic local search (SLS) is developed to fine-tune network weights toward global optima. The loss function of CESN can be effectively reduced by SLS. To avoid overfitting, the optimization process is stopped when the validation error starts to increase. Finally, SLS-CESN is evaluated in chaos prediction benchmarks and real applications. Four different examples are given to verify the effectiveness and robustness of CESN and SLS-CESN.

Integer Echo State Networks: Efficient Reservoir Computing for Digital Hardware

We propose an approximation of echo state networks (ESNs) that can be efficiently implemented on digital hardware based on the mathematics of hyperdimensional computing. The reservoir of the proposed integer ESN (intESN) is a vector containing only n -bits integers (where is normally sufficient for a satisfactory performance). The recurrent matrix multiplication is replaced with an efficient cyclic shift operation. The proposed intESN approach is verified with typical tasks in reservoir computing: memorizing of a sequence of inputs, classifying time series, and learning dynamic processes. Such architecture results in dramatic improvements in memory footprint and computational efficiency, with minimal performance loss. The experiments on a field-programmable gate array confirm that the proposed intESN approach is much more energy efficient than the conventional ESN.