Sequence classification is essential in NLP for understanding and categorizing language patterns in tasks like sentiment analysis, intent detection, and topic classification. Transformer-based models, despite achieving state-of-the-art performance, have inherent limitations due to quadratic time and memory complexity, restricting their input length. Although extensive efforts have aimed at reducing computational demands, processing extensive contexts remains challenging. To overcome these limitations, we propose ResFormer, a novel neural network architecture designed to model varying context lengths efficiently through a cascaded methodology. ResFormer integrates an reservoir computing network featuring a nonlinear readout to effectively capture long-term contextual dependencies in linear time. Concurrently, short-term dependencies within sentences are modeled using a conventional Transformer architecture with fixed-length inputs. Experiments demonstrate that ResFormer significantly outperforms baseline models of DeepSeek-Qwen and ModernBERT, delivering an accuracy improvement of up to +22.3% on the EmoryNLP dataset and consistent gains on MultiWOZ, MELD, and IEMOCAP. In addition, ResFormer exhibits reduced memory consumption, underscoring its effectiveness and efficiency in modeling extensive contextual information.

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 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.

Reservoir computing (RC) is attracting attention as a machine-learning technique for edge computing. In time-series classification tasks, the number of features obtained using a reservoir depends on the length of the input series. Therefore, the features must be converted to a constant-length intermediate representation (IR), such that they can be processed by an output layer. Existing conversion methods involve computationally expensive matrix inversion that significantly increases the circuit size and requires processing power when implemented in hardware. In this article, we propose a simple but effective IR, namely, dot-product-based reservoir representation (DPRR), for RC based on the dot product of data features. Additionally, we propose a hardware-friendly delayed-feedback reservoir (DFR) consisting of a nonlinear element and delayed feedback loop with DPRR. The proposed DFR successfully classified multivariate time series data that has been considered particularly difficult to implement efficiently in hardware. In contrast to conventional DFR models that require analog circuits, the proposed model can be implemented in a fully digital manner suitable for high-level syntheses. A comparison with existing machine-learning methods via field-programmable gate array implementation using 12 multivariate time-series classification tasks confirmed the superior accuracy and small circuit size of the proposed method.

In the past three decades, a wide array of computational methodologies and simulation frameworks has emerged to address the complexities of modeling multi-phase flow and transport processes in fractured porous media. The conformal mesh approaches which explicitly align the computational grid with fracture surfaces are considered by many to be the most accurate. However, such methods require excessive fine-scale meshing, rendering them impractical for large or complex fracture networks. In this work, we propose to learn the complex multi-phase flow and transport dynamics in fractured porous media with graph neural networks (GNN). GNNs are well suited for this task due to the unstructured topology of the computation grid resulting from the Embedded Discrete Fracture Model (EDFM) discretization. We propose two deep learning architectures, a GNN and a recurrent GNN. Both networks follow a two-stage training strategy: an autoregressive one step roll-out, followed by a fine-tuning step where the model is supervised using the whole ground-truth sequence. We demonstrate that the two-stage training approach is effective in mitigating error accumulation during autoregressive model rollouts in the testing phase. Our findings indicate that both GNNs generalize well to unseen fracture realizations, with comparable performance in forecasting saturation sequences, and slightly better performance for the recurrent GNN in predicting pressure sequences. While the second stage of training proved to be beneficial for the GNN model, its impact on the recurrent GNN model was less pronounced. Finally, the performance of both GNNs for temporal extrapolation is tested. The recurrent GNN significantly outperformed the GNN in terms of accuracy, thereby underscoring its superior capability in predicting long sequences.

With a growing data privacy concern, federated learning has emerged as a promising framework to train machine learning models without sharing locally distributed data. In federated learning, local model training by multiple clients and model integration by a server are repeated only through model parameter sharing. Most existing federated learning methods assume training deep learning models, which are often computationally demanding. To deal with this issue, we propose federated learning methods with reservoir state analysis to seek computational efficiency and data privacy protection simultaneously. Specifically, our method relies on Mahalanobis Distance of Reservoir States (MD-RS) method targeting time series anomaly detection, which learns a distribution of reservoir states for normal inputs and detects anomalies based on a deviation from the learned distribution. Iterative updating of statistical parameters in the MD-RS enables incremental federated learning (IncFed MD-RS). We evaluate the performance of IncFed MD-RS using benchmark datasets for time series anomaly detection. The results show that IncFed MD-RS outperforms other federated learning methods with deep learning and reservoir computing models particularly when clients' data are relatively short and heterogeneous. We demonstrate that IncFed MD-RS is robust against reduced sample data compared to other methods. We also show that the computational cost of IncFed MD-RS can be reduced by subsampling from the reservoir states without performance degradation. The proposed method is beneficial especially in anomaly detection applications where computational efficiency, algorithm simplicity, and low communication cost are required.

Patterned nanomagnet arrays (PNAs) have been shown to exhibit a strong geometrically frustrated dipole interaction. Some PNAs have also shown emergent domain wall dynamics. Previous works have demonstrated methods to physically probe these magnetization dynamics of PNAs to realize neuromorphic reservoir systems that exhibit chaotic dynamical behavior and high-dimensional nonlinearity. These PNA reservoir systems from prior works leverage echo state properties and linear/nonlinear short-term memory of component reservoir nodes to map and preserve the dynamical information of the input time-series data into nondelay spatial embeddings. Such mappings enable these PNA reservoir systems to imitate and predict/forecast the input time series data. However, these prior PNA reservoir systems are based solely on the nondelay spatial embeddings obtained at component reservoir nodes. As a result, they require a massive number of component reservoir nodes, or a very large spatial embedding (i.e., high-dimensional spatial embedding) per reservoir node, or both, to achieve acceptable imitation and prediction accuracy. These requirements reduce the practical feasibility of such PNA reservoir systems. To address this shortcoming, we present a mixed delay/nondelay embeddings-based PNA reservoir system. Our system uses a single PNA reservoir node with the ability to obtain a mixture of delay/nondelay embeddings of the dynamical information of the time-series data applied at the input of a single PNA reservoir node. Our analysis shows that when these mixed delay/nondelay embeddings are used to train a perceptron at the output layer, our reservoir system outperforms existing PNA-based reservoir systems for the imitation of NARMA 2, NARMA 5, NARMA 7, and NARMA 10 time series data, and for the short-term and long-term prediction of the Mackey Glass time series data.

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.

Reservoir computing is a brain-inspired machine learning framework for processing temporal data by mapping inputs into high-dimensional spaces. Physical reservoir computers (PRCs) leverage native fading memory and nonlinearity in physical substrates, including atomic switches, photonics, volatile memristors, and, recently, memcapacitors, to achieve efficient high-dimensional mapping. Traditional PRCs often consist of homogeneous device arrays, which rely on input encoding methods and large stochastic device-to-device variations for increased nonlinearity and high-dimensional mapping. These approaches incur high pre-processing costs and restrict real-time deployment. Here, we introduce a novel heterogeneous memcapacitor-based PRC that exploits internal voltage offsets to enable both monotonic and non-monotonic input-state correlations crucial for efficient high-dimensional transformations. We demonstrate our approach's efficacy by predicting a second-order nonlinear dynamical system with an extremely low prediction error (0.00018). Additionally, we predict a chaotic H\'enon map, achieving a low normalized root mean square error (0.080). Unlike previous PRCs, such errors are achieved without input encoding methods, underscoring the power of distinct input-state correlations. Most importantly, we generalize our approach to other neuromorphic devices that lack inherent voltage offsets using externally applied offsets to realize various input-state correlations. Our approach and unprecedented performance are a major milestone towards high-performance full in-materia PRCs.

In one calculation, adjoint sensitivity analysis provides the gradient of a quantity of interest with respect to all system's parameters. Conventionally, adjoint solvers need to be implemented by differentiating computational models, which can be a cumbersome task and is code-specific. To propose an adjoint solver that is not code-specific, we develop a data-driven strategy. We demonstrate its application on the computation of gradients of long-time averages of chaotic flows. First, we deploy a parameter-aware echo state network (ESN) to accurately forecast and simulate the dynamics of a dynamical system for a range of system's parameters. Second, we derive the adjoint of the parameter-aware ESN. Finally, we combine the parameter-aware ESN with its adjoint version to compute the sensitivities to the system parameters. We showcase the method on a prototypical chaotic system. Because adjoint sensitivities in chaotic regimes diverge for long integration times, we analyse the application of ensemble adjoint method to the ESN. We find that the adjoint sensitivities obtained from the ESN match closely with the original system. This work opens possibilities for sensitivity analysis without code-specific adjoint solvers.