Recurrent Quantum Feature Maps for Reservoir Computing
Department of Physics, University of Ottawa（渥太华大学物理系）；National Research Council of Canada（加拿大国家研究委员会）；Institute for Quantum Science and Technology, Department of Physics and Astronomy, University of Calgary（卡尔加里大学物理与天文学系量子科学与技术研究所）；Hotchkiss Brain Institute, University of Calgary（卡尔加里大学霍奇基斯脑研究所）
Utkarsh Singh, Aaron Z. Goldberg, Christoph Simon, Khabat Heshami
Reservoir computing promises a fast method for handling large amounts of temporal data. This hinges on constructing a good reservoir--a dynamical system capable of transforming inputs into a high-dimensional representation while remembering properties of earlier data. In this work, we introduce a reservoir based on recurrent quantum feature maps where a fixed quantum circuit is reused to encode both current inputs and a classical feedback signal derived from previous outputs. We evaluate the model on the Mackey-Glass time-series prediction task using our recently introduced CP feature map, and find that it achieves lower mean squared error than standard classical baselines, including echo state networks and multilayer perceptrons, while maintaining compact circuit depth and qubit requirements. We further analyze memory capacity and show that the model effectively retains temporal information, consistent with its forecasting accuracy. Finally, we study the impact of realistic noise and find that performance is robust to several noise channels but remains sensitive to two-qubit gate errors, identifying a key limitation for near-term implementations.
School of Computer Science University of St. Gallen（圣加尔登大学计算机科学学院）；Donostia International Physics Center（多尼亚国际物理中心）；Department of Mathematics, Imperial College（帝国理工学院数学系）；School of Physical and Mathematical Sciences Nanyang Technological University（南洋理工大学物理与数学科学学院）
Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；Collège de France（法兰西公学院）；LightOn（LightOn公司）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Reservoir Computing is a class of simple yet efficient Recurrent Neural Networks where internal weights are fixed at random and only a linear output layer is trained. In the large size limit, such random neural networks have a deep connection with kernel methods. Our contributions are threefold: a) We rigorously establish the recurrent kernel limit of Reservoir Computing and prove its convergence. b) We test our models on chaotic time series prediction, a classic but challenging benchmark in Reservoir Computing, and show how the Recurrent Kernel is competitive and computationally efficient whe…
Wells Fargo（富国银行）；IBM Quantum, IBM Research（IBM研究院IBM量子部门）
Recent developments in quantum computing and machine learning have propelled the interdisciplinary study of quantum machine learning. Sequential modeling is an important task with high scientific and commercial value. Existing VQC or QNN-based methods require significant computational resources to perform the gradient-based optimization of a larger number of quantum circuit parameters. The major drawback is that such quantum gradient calculation requires a large amount of circuit evaluation, posing challenges in current near-term quantum hardware and simulation software. In this work, we appro…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Yerevan State University（埃里温国立大学）
Reservoir Computing is a class of Recurrent Neural Networks with internal weights fixed at random. Stability relates to the sensitivity of the network state to perturbations. It is an important property in Reservoir Computing as it directly impacts performance. In practice, it is desirable to stay in a stable regime, where the effect of perturbations does not explode exponentially, but also close to the chaotic frontier where reservoir dynamics are rich. Open questions remain today regarding input regularization and discontinuous activation functions. In this work, we use the recurrent kernel …
Airbus Central Research & Technology（空客中央研究与技术中心）；Department of Computer Science（计算机科学系）；University of Pisa, Italy（意大利比萨大学）；Bielefeld University, Germany（德国比勒菲尔德大学）
Reservoir Computing (RC) is a well-known strategy for designing Recurrent Neural Networks featured by striking efficiency of training. The crucial aspect of RC is to properly instantiate the hidden recurrent layer that serves as dynamical memory to the system. In this respect, the common recipe is to create a pool of randomly and sparsely connected recurrent neurons. While the aspect of sparsity in the design of RC systems has been debated in the literature, it is nowadays understood mainly as a way to enhance the efficiency of computation, exploiting sparse matrix operations. In this paper, w…
Deep Echo State Networks (DeepESNs) recently extended the applicability of Reservoir Computing (RC) methods towards the field of deep learning. In this paper we study the impact of constrained reservoir topologies in the architectural design of deep reservoirs, through numerical experiments on several RC benchmarks. The major outcome of our investigation is to show the remarkable effect, in terms of predictive performance gain, achieved by the synergy between a deep reservoir construction and a structured organization of the recurrent units in each layer. Our results also indicate that a parti…
Recently, studies on deep Reservoir Computing (RC) highlighted the role of layering in deep recurrent neural networks (RNNs). In this paper, the use of linear recurrent units allows us to bring more evidence on the intrinsic hierarchical temporal representation in deep RNNs through frequency analysis applied to the state signals. The potentiality of our approach is assessed on the class of Multiple Superimposed Oscillator tasks. Furthermore, our investigation provides useful insights to open a discussion on the main aspects that characterize the deep learning framework in the temporal domain.
An Analysis of Untrained Deep Reservoir Networks for Audio Surveillance
Department of Computer Science University of Pisa Pisa, Italy (ORCID: 0009-0002-6500-9578)；Department of Computer Science University of Pisa Pisa, Italy (ORCID: 0000-0001-8504-1503)
ETH Zürich（苏黎世联邦理工学院）；University of Maryland（马里兰大学）；Massachusetts Institute of Technology（麻省理工学院）
We examine the efficiency of Recurrent Neural Networks in forecasting the spatiotemporal dynamics of high dimensional and reduced order complex systems using Reservoir Computing (RC) and Backpropagation through time (BPTT) for gated network architectures. We highlight advantages and limitations of each method and discuss their implementation for parallel computing architectures. We quantify the relative prediction accuracy of these algorithms for the longterm forecasting of chaotic systems using as benchmarks the Lorenz-96 and the Kuramoto-Sivashinsky (KS) equations. We find that, when the ful…
Faculty of Electrical Engineering and Computer Science（电气工程与计算机科学学院）；VŠB-Technical University of Ostrava（俄斯特拉发技术大学）；INRIA Rennes – Bretagne Atlantique（法国国家信息与自动化研究所雷恩-布列塔尼大西洋中心）
The Echo State Network (ESN) is a class of Recurrent Neural Network with a large number of hidden-hidden weights (in the so-called reservoir). Canonical ESN and its variations have recently received significant attention due to their remarkable success in the modeling of non-linear dynamical systems. The reservoir is randomly connected with fixed weights that don't change in the learning process. Only the weights from reservoir to output are trained. Since the reservoir is fixed during the training procedure, we may wonder if the computational power of the recurrent structure is fully harnesse…
Reservoir computing (RC) offers efficient temporal data processing with a low training cost by separating recurrent neural networks into a fixed network with recurrent connections and a trainable linear network. The quality of the fixed network, called reservoir, is the most important factor that determines the performance of the RC system. In this paper, we investigate the influence of the hierarchical reservoir structure on the properties of the reservoir and the performance of the RC system. Analogous to deep neural networks, stacking sub-reservoirs in series is an efficient way to enhance …
Machine learning recently proved efficient in learning differential equations and dynamical systems from data. However, the data is commonly assumed to originate from a single never-changing system. In contrast, when modeling real-world dynamical processes, the data distribution often shifts due to changes in the underlying system dynamics. Continual learning of these processes aims to rapidly adapt to abrupt system changes without forgetting previous dynamical regimes. This work proposes an approach to continual learning based on reservoir computing, a state-of-the-art method for training rec…
Arizona State University（亚利桑那州立大学）；Shaanxi Normal University（陕西师范大学）
Reservoir computing systems, a class of recurrent neural networks, have recently been exploited for model-free, data-based prediction of the state evolution of a variety of chaotic dynamical systems. The prediction horizon demonstrated has been about half dozen Lyapunov time. Is it possible to significantly extend the prediction time beyond what has been achieved so far? We articulate a scheme incorporating time-dependent but sparse data inputs into reservoir computing and demonstrate that such rare "updates" of the actual state practically enable an arbitrarily long prediction horizon for a v…
Advanced Research Center, National Fusion Research Institute (NFRI)（国家聚变研究所高级研究中心）
Inspired by recent theoretical arguments, physics-informed echo state network (ESN) is discussed on the attempt to train a reservoir model absolutely in physics-informed manner. As the plainest work on such a purpose, an ODE (ordinary differential equation) approximator is designed to replicate the solution in sequence with respect to the recurrent evaluations. On the principal invariance of differential equations, the constraint in recurrence just takes shape to secure a proper regression method for the ESN-based ODE approximator. After then, the actual training process is established on the …
Università Degli Studi Di Pavia（帕维亚大学）；STMicroelectronics（意法半导体）
Echo State Networks (ESN) are a class of Recurrent Neural Networks (RNN) that has gained substantial popularity due to their effectiveness, ease of use and potential for compact hardware implementation. An ESN contains the three network layers input, reservoir and readout where the reservoir is the truly recurrent network. The input and reservoir layers of an ESN are initialized at random and never trained afterwards and the training of the ESN is applied to the readout layer only. The alternative of Recursive Neural Gas (RNG) is one of the many proposals of fully-trainable reservoirs that can…
Inspired by the numerical solution of ordinary differential equations, in this paper we propose a novel Reservoir Computing (RC) model, called the Euler State Network (EuSN). The presented approach makes use of forward Euler discretization and antisymmetric recurrent matrices to design reservoir dynamics that are both stable and non-dissipative by construction. Our mathematical analysis shows that the resulting model is biased towards a unitary effective spectral radius and zero local Lyapunov exponents, intrinsically operating near to the edge of stability. Experiments on long-term memory tas…
Faculty of Electrical Engineering and Computer Science（电气与计算机科学学院）；VŠB-Technical University of Ostrava（俄斯特拉发VŠB技术大学）；Department of Computer Science and Engineering（计算机科学与工程系）；Shobhit University（肖布希特大学）
For many years, Evolutionary Algorithms (EAs) have been applied to improve Neural Networks (NNs) architectures. They have been used for solving different problems, such as training the networks (adjusting the weights), designing network topology, optimizing global parameters, and selecting features. Here, we provide a systematic brief survey about applications of the EAs on the specific domain of the recurrent NNs named Reservoir Computing (RC). At the beginning of the 2000s, the RC paradigm appeared as a good option for employing recurrent NNs without dealing with the inconveniences of the tr…
In the last decade, a new computational paradigm was introduced in the field of Machine Learning, under the name of Reservoir Computing (RC). RC models are neural networks which a recurrent part (the reservoir) that does not participate in the learning process, and the rest of the system where no recurrence (no neural circuit) occurs. This approach has grown rapidly due to its success in solving learning tasks and other computational applications. Some success was also observed with another recently proposed neural network designed using Queueing Theory, the Random Neural Network (RandNN). Bot…
Faculty of Engineering（工学院）；University of Tsukuba（筑波大学）；Toyota Motor Corporation（丰田汽车公司）
Reservoir computing derived from recurrent neural networks is more applicable to real world systems than deep learning because of its low computational cost and potential for physical implementation. Specifically, physical reservoir computing, which replaces the dynamics of reservoir units with physical phenomena, has recently received considerable attention. In this study, we propose a method of exploiting the dynamics of road traffic as a reservoir, and numerically confirm its feasibility by applying several prediction tasks based on a simple mathematical model of the traffic flow.
Università della Svizzera Italiana（瑞士意大利语大学）；Politecnico di Milano（米兰理工大学）；University of Manitoba（曼尼托巴大学）；University of Exeter（埃克塞特大学）；University of Birmingham（伯明翰大学）
Reservoir computing is a popular approach to design recurrent neural networks, due to its training simplicity and approximation performance. The recurrent part of these networks is not trained (e.g., via gradient descent), making them appealing for analytical studies by a large community of researchers with backgrounds spanning from dynamical systems to neuroscience. However, even in the simple linear case, the working principle of these networks is not fully understood and their design is usually driven by heuristics. A novel analysis of the dynamics of such networks is proposed, which allows…
A common difficulty in applications of machine learning is the lack of any general principle for guiding the choices of key parameters of the underlying neural network. Focusing on a class of recurrent neural networks - reservoir computing systems that have recently been exploited for model-free prediction of nonlinear dynamical systems, we uncover a surprising phenomenon: the emergence of an interval in the spectral radius of the neural network in which the prediction error is minimized. In a three-dimensional representation of the error versus time and spectral radius, the interval correspon…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）
There is a wave of interest in using unsupervised neural networks for solving differential equations. The existing methods are based on feed-forward networks, {while} recurrent neural network differential equation solvers have not yet been reported. We introduce an unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs). We suggest an approach to calculate time derivatives of recurrent neural network outputs without using backpropagation. The internal weights of an RC are fixe…
Connectome-to-Function: Conditional Generative Latent Representations for Reservoir Computing
National Key Laboratory for Multimedia Information Processing, School of Computer Science, Peking University（北京大学计算机学院多媒体信息处理全国重点实验室）；School of Electronics Engineering and Computer Science, Peking University（北京大学电子工程与计算机科学学院）；School of Computer Science, Beijing University of Posts and Telecommunications（北京邮电大学计算机学院）；College of Engineering, Peking University（北京大学工学院）；Yuanpei College, Peking University（北京大学元培学院）
The Ohio State University（俄亥俄州立大学）；ResCon Technologies, LLC（ResCon Technologies公司）；Clarkson University（克拉克森大学）
Reservoir computing is a best-in-class machine learning algorithm for processing information generated by dynamical systems using observed time-series data. Importantly, it requires very small training data sets, uses linear optimization, and thus requires minimal computing resources. However, the algorithm uses randomly sampled matrices to define the underlying recurrent neural network and has a multitude of metaparameters that must be optimized. Recent results demonstrate the equivalence of reservoir computing to nonlinear vector autoregression, which requires no random matrices, fewer metap…
Reservoir Computing (RC) refers to a Recurrent Neural Networks (RNNs) framework, frequently used for sequence learning and time series prediction. The RC system consists of a random fixed-weight RNN (the input-hidden reservoir layer) and a classifier (the hidden-output readout layer). Here we focus on the sequence learning problem, and we explore a different approach to RC. More specifically, we remove the non-linear neural activation function, and we consider an orthogonal reservoir acting on normalized states on the unit hypersphere. Surprisingly, our numerical results show that the system's…
Deep Recurrent Stochastic Configuration Networks for Modelling Nonlinear Dynamic Systems
Northeastern University（东北大学）；China University of Mining and Technology（中国矿业大学）；State Key Laboratory of Synthetical Automation for Process Industries（过程综合自动化国家重点实验室）；Research Center for Stochastic Configuration Machines（随机配置机器研究中心）
Reservoir Computing (RC) provides an efficient way for designing dynamical recurrent neural models. While training is restricted to a simple output component, the recurrent connections are left untrained after initialization, subject to stability constraints specified by the Echo State Property (ESP). Literature conditions for the ESP typically fail to properly account for the effects of driving input signals, often limiting the potentialities of the RC approach. In this paper, we study the fundamental aspect of asymptotic stability of RC models in presence of driving input, introducing an emp…
Reservoir computing is a recently introduced machine learning paradigm that has been shown to be well-suited for the processing of spatiotemporal data. Rather than training the network node connections and weights via backpropagation in traditional recurrent neural networks, reservoirs instead have fixed connections and weights among the `hidden layer' nodes, and traditionally only the weights to the output layer of neurons are trained using linear regression. We claim that for signal classification tasks one may forgo the weight training step entirely and instead use a simple supervised clust…
Federal University of Minas Gerais（米纳斯吉拉斯联邦大学）；Federal Institute of Education Science and Technology of Northern Minas Gerais（北米纳斯吉拉斯联邦教育科技学院）；Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）
Fuzzy Cognitive Maps (FCMs) have emerged as an interpretable signed weighted digraph method consisting of nodes (concepts) and weights which represent the dependencies among the concepts. Although FCMs have attained considerable achievements in various time series prediction applications, designing an FCM model with time-efficient training method is still an open challenge. Thus, this paper introduces a novel univariate time series forecasting technique, which is composed of a group of randomized high order FCM models labeled R-HFCM. The novelty of the proposed R-HFCM model is relevant to merg…
Reservoir computing is a recently introduced, highly efficient bio-inspired approach for processing time dependent data. The basic scheme of reservoir computing consists of a non linear recurrent dynamical system coupled to a single input layer and a single output layer. Within these constraints many implementations are possible. Here we report an opto-electronic implementation of reservoir computing based on a recently proposed architecture consisting of a single non linear node and a delay line. Our implementation is sufficiently fast for real time information processing. We illustrate its p…
The extension of deep learning towards temporal data processing is gaining an increasing research interest. In this paper we investigate the properties of state dynamics developed in successive levels of deep recurrent neural networks (RNNs) in terms of short-term memory abilities. Our results reveal interesting insights that shed light on the nature of layering as a factor of RNN design. Noticeably, higher layers in a hierarchically organized RNN architecture results to be inherently biased towards longer memory spans even prior to training of the recurrent connections. Moreover, in the conte…
University of Vienna（维也纳大学）；Ludwig-Maximilians-Universität München（路德维希-马克西米利安慕尼黑大学）；Universität Konstanz（康斯坦茨大学）；Universität Sankt Gallen（圣加仑大学）；Centre National de la Recherche Scientifique (CNRS)（法国国家科学研究中心）；ETH Zürich（苏黎世联邦理工学院）
A new explanation of geometric nature of the reservoir computing phenomenon is presented. Reservoir computing is understood in the literature as the possibility of approximating input/output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logar…
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿科学研究所）；Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；Department of Physics, Humboldt University Berlin（柏林洪堡大学物理系）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究所）；School of Physics, University of Electronic Science and Technology of China（电子科技大学物理学院）；Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China（电子科技大学教育部量子物理与光子量子信息重点实验室）；Non-classical Information Science Basic Discipline Research Center of Sichuan Province, University of Electronic Science and Technology of China（电子科技大学四川省非经典信息科学基础学科研究中心）
University of New Mexico（新墨西哥大学）；Portland State University（波特兰州立大学）
Reservoir computing (RC) is a novel approach to time series prediction using recurrent neural networks. In RC, an input signal perturbs the intrinsic dynamics of a medium called a reservoir. A readout layer is then trained to reconstruct a target output from the reservoir's state. The multitude of RC architectures and evaluation metrics poses a challenge to both practitioners and theorists who study the task-solving performance and computational power of RC. In addition, in contrast to traditional computation models, the reservoir is a dynamical system in which computation and memory are insep…
Recurrent neural network (RNN) based reinforcement learning (RL) is used for learning context-dependent tasks and has also attracted attention as a method with remarkable learning performance in recent research. However, RNN-based RL has some issues that the learning procedures tend to be more computationally expensive, and training with backpropagation through time (BPTT) is unstable because of vanishing/exploding gradients problem. An approach with replay memory introducing reservoir computing has been proposed, which trains an agent without BPTT and avoids these issues. The basic idea of th…
Embedding Theory of Reservoir Computing and Reducing Reservoir Network Using Time Delays
Soochow University（苏州大学）；Fudan University（复旦大学）；Potsdam Institute for Climate Impact Research (PIK)（波茨坦气候影响研究所（PIK））；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）
Università di Pisa（比萨大学）；Biobeats Ltd（Biobeats 有限公司）
The paper presents a novel, principled approach to train recurrent neural networks from the Reservoir Computing family that are robust to missing part of the input features at prediction time. By building on the ensembling properties of Dropout regularization, we propose a methodology, named DropIn, which efficiently trains a neural model as a committee machine of subnetworks, each capable of predicting with a subset of the original input features. We discuss the application of the DropIn methodology in the context of Reservoir Computing models and targeting applications characterized by input…
University of California San Diego（加州大学圣迭戈分校）；Cooperative Institute for Research in Environmental Sciences at the University of Colorado Boulder（科罗拉多大学博尔德分校环境科学合作研究所）；NOAA Physical Sciences Laboratory（美国国家海洋和大气管理局物理科学实验室）；Scripps Institution of Oceanography, University of California San Diego（加州大学圣迭戈分校斯克里普斯海洋研究所）
Reservoir computers (RC) are a form of recurrent neural network (RNN) used for forecasting time series data. As with all RNNs, selecting the hyperparameters presents a challenge when training on new inputs. We present a method based on generalized synchronization (GS) that gives direction in designing and evaluating the architecture and hyperparameters of a RC. The 'auxiliary method' for detecting GS provides a pre-training test that guides hyperparameter selection. Furthermore, we provide a metric for a "well trained" RC using the reproduction of the input system's Lyapunov exponents.
