Recurrent Neural Networks

We propose MATE, a simple yet effective memory architecture for solving Contextual Markov Decision Processes (CMDPs), a family of MDPs parameterized by an unobserved context. In CMDPs, an optimal agent can adapt online by maintaining the posterior belief over contexts. MATE replaces this intractable posterior with a sum-aggregated memory, leveraging the posterior's permutation invariance to retain provably sufficient expressiveness. Compared to prior memory architectures, MATE avoids the growing per-step rollout cost of Transformers and the gradient issues commonly associated with Recurrent Neural Networks (RNNs). Extensive evaluations across diverse benchmarks demonstrate that MATE provides clear computational advantages while achieving performance comparable to standard sequence-model baselines.

Transformers have become the dominant architecture for sequence modeling by using self-attention to enable expressive and highly parallel processing. However, the resulting quadratic time and memory costs limit efficiency in long-context settings. Recurrent models such as LSTMs provide explicit nonlinear state updates and strong state-tracking capabilities, yet their strictly sequential computation limits parallelism. We introduce the Parallel Recursive LSTM (PR-LSTM), a hierarchical recurrent architecture that replaces left-to-right recurrence with recursive nonlinear state composition over a balanced computation tree. Tokens are first mapped independently to latent states, which are then recursively merged by a learned gated composition block. This structure uses the reduction pattern underlying parallel scans as a fixed execution schedule, rather than assuming an associative recurrence. As a result, PR-LSTM retains nonlinear gated state representations while reducing recurrent parallel depth from linear to logarithmic. Empirically, PR-LSTM achieves strong sequence-length generalization on formal-language benchmarks, solving more tasks than standard RNN, LSTM, and Transformer baselines, while avoiding the quadratic scaling of attention. These results suggest that recurrent computation can be reorganized hierarchically to expose parallelism without restricting the transition dynamics to linear or associative forms.

The identification and modeling of time-varying systems is a fundamental challenge in signal processing and system identification. To address this challenge, we propose a class of time-varying state-space model (SSM) based neural networks in which the neurons' states are governed by time-varying dynamics. The proposed model provides the learnable time-varying dynamics through a dictionary of basis functions, where each basis function evolves differently over time. We evaluate the proposed approach on both synthetic data from switching systems and a speech denoising task where real audio is corrupted with switching dynamics noise. The results show that the proposed time-varying model consistently outperforms its time-invariant counterparts while maintaining comparable computational complexity. Our investigations also reveal which aspects of the time-varying dynamics of the data most need to be captured by the proposed time-invariant models, how the additional freedom provided by time-varying basis functions should be allocated across model components, and to what extent larger models can compensate for time-invariant limitations.

Modern pedestrian dead reckoning (PDR) systems rely on fusing noisy and biased estimates of position, velocity, and calibrated orientation derived from loosely coupled sensors to determine the current pose of a localized object. However, discrepancies in the sampling rates of sensor-specific estimation methods and unreliable transmission pose significant challenges. And traditional methods often fail to effectively fuse multimodal sensor data during dynamic movements characterized by high accelerations, velocities, and rapidly varying orientations. To address these limitations, we propose a simple recurrent neural network (RNN) architecture capable of implicitly forecasting asynchronous sensor data streams from diverse estimation methods along reference trajectories. The proposed approach introduces PDRNN, a modular hybrid AI-assisted PDR system that handles each component as an independent ensemble of machine learning (ML) models to estimate both key parameter means and variances. Separate ML-based models are employed to estimate orientation, (un)directed velocity or distance from acceleration and gyroscope data, with optional absolute positioning from synchronized radio systems such as 5G for stabilization. A final fusion model combines these outputs, position, velocity, and orientation, while using uncertainty estimates to enhance system robustness. The modular design allows individual components to be updated, fine-tuned, or replaced without affecting the entire system. Experiments on dynamic sports movement data show that PDRNN achieves superior accuracy and precision compared to classic and ML-based methods, effectively avoiding error accumulation common in black-box approaches. And PDRNN offers forecast capabilities and better component control despite increased system complexity.

History-dependent policies induced by recurrent neural networks (RNNs) rely on latent hidden state dynamics, making verification in partially observable reinforcement learning (RL) challenging. Existing RNN verification tools typically rely on restrictive modeling assumptions or coarse over-approximations of the hidden state space, which can lead to overly conservative or inconclusive results. We propose $\textbf{RNN}$ $\textbf{Pro}$babilistic $\textbf{Ve}$rification ($\texttt{RNN-ProVe}$), a probabilistic framework that $\textit{estimates the likelihood}$ of undesired behaviors in RNN-based policies. $\texttt{RNN-ProVe}$ uses policy-driven sampling to approximate the set of hidden states that are feasible under a trained policy, and derives statistical error bounds to produce bounded-error, high-confidence estimates of behavioral violations. Experiments on partially observable single-agent and cooperative multi-agent tasks show that $\texttt{RNN-ProVe}$ yields more quantitative, feasibility-aware probabilistic guarantees than existing tools, while scaling to recurrent and multi-agent settings.

Reconstructing nonlinear dynamical systems (DS) from data (DSR) is a fundamental challenge in science and engineering, but it inherently relies on sequential models. Recent breakthroughs for sequential models have produced algorithms that parallelize computation along sequence length $T$, achieving logarithmic time complexity, $\mathcal{O}(\log T)$. Since sequence lengths have been practically limited due to the linear runtime complexity $\mathcal{O}(T)$ of classical backpropagation through time, this opens new avenues for DSR. This paper studies two prominent classes of parallel-in-time algorithms for this task, both of which leverage parallel associative scans as their core computational primitive. The first class comprises models with linear yet non-autonomous dynamics and a nonlinear readout, such as modern State Space Models (SSMs), while the second consists of general nonlinear models which can be parallelized using the DEER framework. We find that the linear training-time recurrence of the first class of models imposes limitations that often hinder learning of accurate nonlinear dynamics. To address this, we augment DEER with Generalized Teacher Forcing (GTF), a novel variant within the more general nonlinear framework that ensures stable and effective learning of nonlinear dynamics across arbitrary sequence lengths. Using GTF-DEER, we investigate the benefits of training on extremely long sequences ($T>10^4$) for DSR. Our results show that access to such long trajectories significantly improves DSR if the data features long time scales. This work establishes GTF-DEER as a robust tool for data-driven discovery and underscores the largely untapped potential of long-sequence learning in modeling complex DS.

Cortical neurons are complex, multi-timescale processors wired into recurrent circuits, shaped by long evolutionary pressure under stringent biological constraints. Mainstream machine learning, by contrast, predominantly builds models from extremely simple units, a default inherited from early neural-network theory. We treat this as a normative architectural question. How should one split a fixed parameter budget $P$ between the number of units $N$, per-unit effective complexity $k_e$, and per-unit connectivity $k_c$? What controls the optimal allocation? This calls for a model in which per-unit complexity can be tuned independently of width and connectivity. Accordingly, we introduce the ELM Network, whose recurrent layer is built from Expressive Leaky Memory (ELM) neurons, chosen to mirror functional components of cortical neurons. The architecture allows for individually adjusting $N$, $k_e$, and $k_c$ and trains stably across orders of magnitude in scale. We evaluate the model on two qualitatively different sequence benchmarks: the neuromorphic SHD-Adding task and Enwik8 character-level language modeling. Performance improves monotonically along each of the three axes individually. Under a fixed budget, a clear non-trivial optimum emerges in their tradeoff, and larger budgets favor both more and more complex neurons. A closed-form information-theoretic model captures these tradeoffs and attributes the diminishing returns at two ends to: per-neuron signal-to-noise saturation and across-neuron redundancy. A hyperparameter sweep spanning three orders of magnitude in trainable parameters traces a near-Pareto-frontier scaling law consistent with the framework. This suggests that the simple-unit default in ML is not obviously optimal once this tradeoff surface is probed, and offers a normative lens on cortex's reliance on complex spatio-temporal integrators.

Sequence learning is dominated by Transformers and parallelizable recurrent neural networks (RNNs) such as state-space models, yet learning long-term dependencies remains challenging, and state-of-the-art designs trade power consumption for performance. The Bistable Memory Recurrent Unit (BMRU) was introduced to enable hardware-software co-design of ultra-low power RNNs: quantized states with hysteresis provide persistent memory while mapping directly to analog primitives. However, BMRU performance lags behind parallelizable RNNs on complex sequential tasks. In this paper, we identify gradient blocking during state updates as a key limitation and propose a cumulative update formulation that restores gradient flow while preserving persistent memory, creating skip-connections through time. This leads to the Cumulative Memory Recurrent Unit (CMRU) and its relaxed variant, the $α$CMRU. Experiments show that the cumulative formulation dramatically improves convergence stability and reduces initialization sensitivity. The CMRU and $α$CMRU match or outperform Linear Recurrent Units (LRUs) and minimal Gated Recurrent Units (minGRUs) across diverse benchmarks at small model sizes, with particular advantages on tasks requiring discrete long-range retention, while the CMRU retains quantized states, persistent memory, and noise-resilient dynamics essential for analog implementation.

Brain-Computer Interfaces (BCIs) based on P300 event-related potentials offer promising applications in health, education, and assistive technologies. However, challenges related to inter- and intra-subject variability and the explainability of Deep Learning (DL) models limit their practical deployment. In this work, we present the Post-Recurrent Module (PRM), an additional layer designed to improve both performance and transparency, incorporated into a Recurrent Neural Network (RNN) architecture for classifying P300 signals from EEG data. Our approach enables a dual analysis of spatio-temporal signals through both global and local explainability techniques, allowing us not only to identify the most relevant brain regions and critical time intervals involved in classification, but also to interpret model decisions in terms of spatio-temporal EEG patterns consistent with well-stablished neurophysiological descriptions of the P300. Experimental results show a 9% improvement in performance over state of the art, while also revealing the importance of inter- and intra-subject variability, in alignment with established neuroscience literature. By making model decisions transparent and efficient, we present a framework for explainable EEG-based models. This framework is not limited to more efficient P300 detection, but can be generalized to a wide range of EEG-based tasks. Its ability to identify key spatial and temporal features makes it suitable for applications such as motor imagery, steady-state visual evoked potentials, and even cognitive workload assessment.

We present Key-Value Means ("KVM"), a novel block-recurrence for attention that can accommodate either fixed-size or growing state. Equipping a strong transformer baseline with fixed-size KVM attention layers yields a strong $O(N)$ chunked RNN, while adding only an insignificant number of new parameters. We train a transformer with a growable KVM cache and show it performs competitively on long-context tests with only subquadratic prefill time and sublinear state growth. KVM is implementable with standard operations and without custom kernels, and supports chunk-wise parallelizable training and prefill. It provides many of the benefits of both traditional transformers (expandable context memory, chunk-wise parallelizable training and prefill) and linear RNNs in a single unified package. It can be used on every layer, saving KV-cache memory, and allowing a continuous range of choices of prefill time complexity between $O(N)$ and $O(N^2)$. It can also be implemented in a hybrid solution in tandem with LRNN layers in place of traditional attention, to supplement the LRNN with improved sublinear memory growth context length usage and long context decoding. We release our code at https://github.com/featherless-ai/KVM-paper and trained models at https://huggingface.co/collections/featherless-ai/kvm-paper under the Apache 2.0 license.

This article presents a novel approach to keyframe detection in ultrasound videos, with a particular focus on fetal brain imaging. The proposed model is a composite neural network architecture that combines a Convolutional Neural Network (CNN) with a Recurrent Neural Network (RNN). The CNN extracts spatial features from individual video frames, while the RNN captures temporal dependencies between consecutive frames within each video sequence. The proposed model may improve the efficiency and accuracy of fetal brain ultrasound analysis, thereby supporting earlier detection, diagnosis, and treatment planning for selected fetal brain conditions.

Spiking Neural Networks (SNNs) have been proposed as biologically plausible and energy-efficient alternatives to conventional Artificial Neural Networks (ANNs). However, the training of SNN usually relies on surrogate gradients due to the non-differentiability of the spike function, introducing approximation errors that accumulate across layers. To address this challenge, we extend the work on convexification of parallel feedforward threshold networks to parallel recurrent threshold networks, which subsume parallel SNNs as a structured special case. Building on this theoretical framework, we propose a parameter reconstruction algorithm for SNN training that demonstrates consistent and significant advantages across various tasks, both as a standalone method and in combination with surrogate-gradient training. The ablations further demonstrate the data scalability and robustness to model configurations of our training algorithm, pointing toward its potential in large-scale SNN training.

The theory of state tracking in recurrent architectures has predominantly focused on expressive capacity: whether a fixed architecture can theoretically realize a set of symbolic transition rules. We argue that equally important is error control, the dynamics governing hidden-state drift along the directions that distinguish symbolic states. We prove that affine recurrent networks, a class of models encompassing State-Space Models and Linear Attention, cannot correct errors along state-separating subspaces once they preserve state representations. Consequently, practical affine trackers do not learn robust state tracking; rather, they learn finite horizon solutions governed by accumulated state-relevant error. We characterize the mechanics of this failure, showing that tracking remains readable only while the accumulating within-class spread remains small relative to the initial between-class separation. We demonstrate empirically on group state-tracking tasks that this breakdown is predictable: tracking collapses when the distinguishability ratio crosses the readability threshold of the trained decoder. Across trained models, the point of this crossing predicts the horizon at which downstream accuracy fails. These results establish that robust state tracking is determined not only by an architecture's theoretical expressivity but crucially by its error control.

We introduce MinMax Recurrent Neural Cascades (MinMax RNCs), a class of recurrent neural networks built from a novel form of recurrence over the MinMax algebra. We show that MinMax RNCs enjoy key properties that are difficult to obtain simultaneously: strong formal expressivity, efficient evaluation, stable dynamics, and non-vanishing state gradients. First, their formal expressivity corresponds to the regular languages, arguably the maximal expressivity for finite-memory systems. Second, in addition to evaluation in recurrent form, they also admit parallel-scan evaluation with logarithmic depth and linear work in the input length. Third, their states and activations are uniformly bounded for all sequence lengths. Fourth, their loss gradients exist almost everywhere and are uniformly bounded for all sequence lengths. Fifth, they do not exhibit vanishing state gradients: the gradient of a state with respect to a past state can retain norm one independently of the temporal distance between the states. Empirically, we find that these theoretical properties translate into strong practical performance. MinMax RNCs solve the considered synthetic tasks perfectly, generalise to long sequences, and outperform the recurrent baselines considered in our experiments. We also train a 112M-parameter MinMax RNC for next-token prediction, obtaining competitive performance for its size and providing initial evidence that MinMax recurrence can scale to real-world sequence-modelling tasks.

Understanding how biological and artificial neural networks implement computation from connectivity is a central problem in neuroscience and machine learning. In neural systems, structural and functional connectivity are known to diverge, motivating approaches that move beyond direct connections alone. Here, we show that the spatial and temporal function of recurrent neural networks (RNNs) trained on hierarchically modular tasks can be recovered by modelling the network as a graph and analysing the multi-hop pathways between input and output units. In particular, decomposing these pathways by hop length reveals how the network temporally routes information. This perspective reframes regularisation: if function is implemented through multi-hop communication, then standard penalties such as L1 regularisation, which act only on individual weights, constrain single-hop structure rather than the multi-hop pathways that support computation. Motivated by this view, we introduce resolvent-RNNs (R-RNNs), which constrain multi-hop pathways and thereby induce temporal sparsity beyond that achieved by standard L1 regularisation. Compared with L1 regularisation, R-RNNs achieve improved performance by inducing temporal sparsity that matches the task structure, even when the task signal is sparse. Moreover, R-RNNs exhibit stronger sparsity-function alignment, reflected in their increased robustness under strong regularisation. Together, our results identify multi-hop communication as a key principle linking structure to function in recurrent networks, and suggest that sparsity should be defined over functional pathways rather than individual parameters.

Learning in neural systems arises from synaptic changes that reshape the representations underlying behavior. While low-rank recurrent neural networks (RNNs) have emerged as a powerful framework for linking connectivity to function, a theoretical understanding of their learning process remains elusive. Here, we extend the low-rank framework from activity to learning by deriving gradient-descent dynamics directly in a reduced overlap space. We formulate a closed-form, low-dimensional system of ODEs that governs learning in this space, exact for linear RNNs and asymptotically exact for nonlinear RNNs in the large-$N$ Gaussian limit. Central to our analysis is a distinction between two classes of overlaps: loss-visible overlaps, which fully determine network activity, output, and loss, and loss-invisible overlaps, which do not affect function but are required to describe learning. We illustrate the consequences of this decomposition through two phenomena. First, we show that learning can serve as a perturbation that exposes differences in connectivity between functionally equivalent networks. Second, we show that loss-invisible overlaps can act as memory variables that encode training history, and characterize the conditions under which this occurs. Finally, we present several testable predictions for biological learning experiments derived from our theory.

Continuous-memory recurrent networks must preserve phase, position, or orientation despite model imperfections and state noise. We develop a geometric framework that separates three questions: how many memory coordinates are neutrally transported, how deterministic perturbations alter their finite-horizon stability, and how ambient noise is decoded along them. Exact per-input equivariance transports analytical group tangents pathwise and, on a compact nondegenerate orbit stratum, yields at least $q=\dim(G/H)$ zero group-tangent Lyapunov exponents under stationary ergodic driving. For imperfect dynamics, a four-block tangent/normal decomposition gives local and finite-horizon bounds on tangent growth and subspace rotation, distinguishing first-order direct damage from second-order leakage through contracting normal directions. For noisy dynamics, a specified decoder maps ambient covariance $Q$ to coordinate covariance $ZQZ^\top$; under isotropic noise and fixed tangent energy, least-squares decoding and scaled-isometric action geometry minimize local diffusion. Local and finite-horizon evaluations include cases both within and outside the sufficient conditions. In a fresh twenty-seed $T^2$ replication, a decoder-covariance objective improves noisy horizon-256 memory in every pair while meeting a prespecified clean-error equivalence margin. Direct noise training also improves noisy memory but incurs a clean-error tradeoff. In coupled $T^4/T^8$ integrators, the advantage of anisotropic-covariance over isotropic regularization reverses when evaluation noise becomes isotropic. These results connect continuous symmetry to measurable limits and design choices for recurrent memory under explicit dynamical, decoder, and noise assumptions.

Building and maintaining state to learn policies and value functions is critical for deploying reinforcement learning (RL) agents in the real world. Recurrent neural networks (RNNs) have become a key point of interest for the state-building problem, and several large-scale reinforcement learning agents incorporate recurrent networks. While RNNs have become a mainstay in many RL applications, many key design choices and implementation details responsible for performance improvements are often not reported. In this work, we discuss one axis on which RNN architectures can be (and have been) modified for use in RL. Specifically, we look at how action information can be incorporated into the state update function of a recurrent cell. We discuss several choices in using action information and empirically evaluate the resulting architectures on a set of illustrative domains. Finally, we discuss future work in developing recurrent cells and discuss challenges specific to the RL setting.

The proliferation of large-scale and structurally complex data has spurred the integration of machine learning methods into statistical modeling. Recurrent neural networks (RNNs), a foundational class of models for time-dependent data, can be viewed as nonlinear extensions of classical autoregressive moving average models. Despite their flexibility and empirical success in machine learning, RNNs often suffer from limited interpretability and slow training, which hinders their use in statistics. This paper proposes the Parallelized RNN (ParaRNN), a novel model composed of multiple small recurrent units. ParaRNN admits an additive representation that decouples recurrent dynamics into interpretable components, whose behavior can be characterized through recurrence features. This interpretability enables its applications in nonparametric regression for time-dependent data, while the design also allows efficient parallelization. The approximation capacity and non-asymptotic prediction error bounds in a nonparametric regression setting are established for ParaRNN. Empirical results on three sequential modeling tasks further demonstrate that ParaRNN achieves performance comparable to vanilla RNNs while offering improved interpretability and efficiency.

Recently, Convolutional Neural Network (CNN) or Transformer architecture based models have been proposed to overcome the limitations of Recurrent Neural Network (RNN) based models in spatiotemporal prediction. These models prevent the inefficiency of parallelization limitation due to the sequential properties and stacked error due to the recursive method, and show high performance. Novertheless, there are still some challengies. First, CNN based models have difficulty considering global information due to the local properties of the kernel, and their performance is limited. In addition, information is mixed because the time axis is combined with the channel axis of the image for processing. Models based on Transformer architecture have high complexity due to the self-attention calcuation and take a long training time. In this paper, we propose a new structure model called CNN-based Multi-In-Multi-Out model for Efficient Spatiotemporal Prediction (MIMO-ESP) to overcome these limitations. MIMO-ESP considers global information and significantly improves complexity by configuring a Transformer architecture based on CNN. In addition, it treats the time axis as an independent axis without combining it, and effectively considers spatiotemporal information together by applying dilation. This structure makes MIMO-ESP efficient and high performance. Extensive experiment results on three promising benchmark datasets which including video, traffic, and precipitation prediction tasks demonstrate that the usefulness of MIMO-ESP due to the achieved competitive efficiency while outperforming existing models. Furthermore, the ablation study results demonstrate the usefulness of the components of MIMO-ESP, emphasizing the potential of the proposed approaches.

Spiking Neural Networks (SNNs) provide a promising framework for energy-efficient and biologically grounded computation; however, scalable learning in deep recurrent architectures with sparse connectivity remains a major challenge. In this work, we propose a structured multi-layer recurrent SNN architecture composed of locally dense recurrent layers augmented with sparse small-world long-range projections to a readout population. The long-range connectivity is largely fixed, preserving routing efficiency and hardware scalability, while synaptic adaptation is performed using strictly local plasticity mechanisms. To enable supervised learning without backpropagation or surrogate gradients, we introduce a biologically motivated learning framework that combines: (i) population-based winner-take-all (WTA) teaching signals at the output layer, (ii) fixed random broadcast alignment feedback pathways, and (iii) low-dimensional modulatory neuron populations that gate synaptic updates through three-factor learning rules with eligibility traces. This design supports deep recurrent computation with sparse global communication and purely local synaptic updates. We analyze the algorithmic properties, computational complexity, and hardware feasibility of the proposed approach, and demonstrate stable learning and competitive performance on benchmark classification tasks. The results highlight the potential of structured recurrence and neuromodulatory learning to enable scalable, hardware-compatible SNN training beyond gradient-based methods.

Deploying reinforcement learning policies in the real world requires adapting to time-varying environments. We study this problem in the contextual Markov Decision Process (cMDP) framework, where a family of environments is indexed by a low-dimensional context unknown at test time. The standard approach decomposes the problem: train a so-called "universal policy" which assumes knowledge of the true context, then pair it with a context estimator which approximates context using the observed trajectory. We identify a simple, counterintuitive trick that substantially improves the estimator: randomly delete a fraction of the training buffer after each round. This works because data is collected across multiple rounds using progressively better policies, and older trajectories come from a different distribution than what the estimator will face at deployment time; random deletion creates an implicit exponential decay on older data while preserving diversity without requiring any explicit identification of which samples are stale. This reduces robustness gap by 30% for MLPs and by 6% on average for recurrent networks. Strikingly, it allows a narrow MLP with 5x fewer parameters to outperform a wide MLP trained without deletion. To understand when and why deletion helps, we analyze regularized empirical risk minimization with a mismatch between the train distribution and the distribution at deployment; in this idealized setting, we prove that removing a single uniformly random training point decreases expected test loss in expectation under mild conditions. For ridge regression we make this quantitative: deletion helps when the regularization coefficient is moderate and the signal-to-noise ratio (SNR) is sufficiently low, and, crucially, this SNR threshold gives a direct measure of how large the distribution mismatch between training and deployment must be for deletion to be beneficial.

We propose a dual-channel reservoir-computing scheme for inferring the dynamics of two distinct chaotic systems with a single machine. By augmenting a standard reservoir with a system-label channel and a parameter-control channel, the machine can be trained from time series collected from a few sampled states of the two systems. We show that the trained machine not only predicts the short-time evolution of the sampled states, but also reproduces the long-term statistical properties of unseen states, thereby enabling reconstruction of the bifurcation diagrams of both systems from partial observations. The effectiveness of the scheme is demonstrated for the Lorenz and Rössler systems in numerical simulations and for the Chua and Rossler circuits in experiments. Functional-network analysis further shows that the two target systems are encoded by distinct dynamical patterns in the reservoir. These results extend multifunctional and parameter-aware reservoir computing, and provide a route to data-driven inference of multiple nonlinear systems using a single machine.

These lecture notes provide an introduction to the verification of neural networks from a theoretical perspective. We discuss feed-forward neural networks, recurrent neural networks, attention mechanisms, and transformers, together with specification languages and algorithmic verification techniques.

Continuous estimation of high-dimensional finger kinematics from forearm surface electromyography (EMG) could enable natural control for hand prostheses, AR/XR interfaces, and teleoperation. However, the complexity of human hand gestures and the entanglement of forearm muscles make accurate recognition intrinsically challenging. Existing approaches typically reduce task complexity by relying on classification-based machine learning, limiting the controllable degrees of freedom and compromising on natural interaction. We present an end-to-end framework for continuous EMG-to-kinematics regression using only consumer-grade hardware. The framework combines an 8-channel EMG armband, a single webcam, and an automatic synchronization procedure, enabling the collection of the EMG Finger-Kinematics dataset (EMG-FK), a 10-h dataset of synchronized EMG and 15 finger joint angles from 20 participants performing rich, unconstrained right-hand motions. We also introduce the Temporal Riemannian Regressor (TRR), a lightweight GRU-based model that uses sequences of multi-band Riemannian covariance features to decode finger motion. Across EMG-FK and the public emg2pose benchmark, TRR outperforms state-of-the-art methods in both intra- and cross-subject evaluation. On EMG-FK, it reaches an average absolute error of $9.79 °\pm 1.48$ in intra-subject and $16.71 °\pm 3.97$ in cross-subject. Finally, we demonstrate real-time deployment on a Raspberry Pi 5 and intuitive control of a robotic hand; TRR runs at nearly 10 predictions/s and is roughly an order of magnitude faster than state-of-the-art approaches. Together, these contributions lower the barrier to reproducible, real-time EMG-based decoding of high-dimensional finger motion, and pave the way toward more natural and intuitive control of embedded EMG-based systems.

Reservoir computing (RC) is an emerging recurrent neural network architecture that has attracted growing attention for its low training cost and modest hardware requirements. Memristor-based circuits are particularly promising for RC, as their intrinsic dynamics can reduce network size and parameter overhead in tasks such as time-series prediction and image recognition. Although RC has been demonstrated with several memristive devices, a comprehensive evaluation of device-level requirements remains limited. In this paper, we analyze and explain the operation of a parallel delayed feedback network (PDFN) RC architecture with volatile memristors, focusing on how device characteristics -- such as decay rate, quantization, and variability -- affect reservoir performance. We further discuss strategies to improve data representation in the reservoir using preprocessing methods and suggest potential improvements. The proposed approach achieves 95.89% classification accuracy on MNIST, comparable with the best reported memristor-based RC implementations. Furthermore, the method maintains high robustness under 20% device variability, achieving an accuracy of up to 94.2%. These results demonstrate that volatile memristors can support reliable spatio-temporal information processing and reinforce their potential as key building blocks for compact, high-speed, and energy-efficient neuromorphic computing systems.

Language models trained on natural text learn to represent numbers using periodic features with dominant periods at $T=2, 5, 10$. In this paper, we identify a two-tiered hierarchy of these features: while Transformers, Linear RNNs, LSTMs, and classical word embeddings trained in different ways all learn features that have period-$T$ spikes in the Fourier domain, only some learn geometrically separable features that can be used to linearly classify a number mod-$T$. To explain this incongruity, we prove that Fourier domain sparsity is necessary but not sufficient for mod-$T$ geometric separability. Empirically, we investigate when model training yields geometrically separable features, finding that the data, architecture, optimizer, and tokenizer all play key roles. In particular, we identify two different routes through which models can acquire geometrically separable features: they can learn them from complementary co-occurrence signals in general language data, including text-number co-occurrence and cross-number interaction, or from multi-token (but not single-token) addition problems. Overall, our results highlight the phenomenon of convergent evolution in feature learning: A diverse range of models learn similar features from different training signals.

Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.

Time series forecasting is traditionally dominated by sequence-based architectures such as recurrent neural networks and attention mechanisms, which process all time steps uniformly and often incur substantial computational cost. However, real-world temporal signals typically exhibit heterogeneous structure, where informative patterns are sparsely distributed and interspersed with redundant observations. This work introduces \textbf{SPaRSe-TIME}, a structured and computationally efficient framework that models time series through a decomposition into three complementary components: saliency, memory, and trend. The proposed approach reformulates temporal modeling as a projection onto informative subspaces, where saliency acts as a data-dependent sparsification operator, memory captures dominant low-rank temporal patterns, and trend encodes low-frequency dynamics. These components are integrated through a lightweight, adaptive mapping that enables simplified, selective, and interpretable temporal reasoning. Extensive experiments on diverse real-world datasets demonstrate that SPaRSe-TIME achieves competitive predictive performance compared to recurrent and attention-based architectures, while significantly reducing computational complexity. The model is particularly effective in structured time series with clear temporal components and provides explicit interpretability through component-wise contributions. Furthermore, analysis reveals both the strengths and limitations of decomposition-based modeling, highlighting challenges in highly stochastic and complex multivariate settings. Overall, SPaRSe-TIME offers a principled alternative to monolithic sequence models, bridging efficiency, interpretability, and performance, and providing a scalable framework for time series learning.

Physical neural networks offer a transformative route to edge intelligence, providing superior inference speed and energy efficiency compared to conventional digital architectures. However, realizing scalable, end-to-end, fully analog recurrent neural networks for temporal information processing remains challenging due to the difficulty of faithfully mapping trained network models onto physical hardware. Here we present a fully analog resonant recurrent neural network (R$^2$NN) implemented via a metacircuit architecture composed of coupled electrical local resonators. A reformulated mechanical-electrical analogy establishes a direct mapping between the R$^2$NN model and metacircuit elements, enabling accurate physical implementation of trained neural network parameters. By integrating jointly trainable global resistive coupling and local resonances, which generate effective frequency-dependent negative resistances, the architecture shapes an impedance landscape that steers currents along frequency-selective pathways. This mechanism enables direct extraction of discriminative spectral features, facilitating real-time temporal classification of raw analog inputs while bypassing analog-to-digital conversion. We demonstrate the cross-domain versatility of this framework using integrated hardware for tactile perception, speech recognition, and condition monitoring. This work establishes a scalable, fully analog paradigm for intelligent temporal processing and paves the way for low-latency, resource-efficient physical neural hardware for edge intelligence.

Spiking neural networks (SNNs) are rapidly gaining momentum as an alternative to conventional artificial neural networks in resource constrained edge systems. In this work, we continue a recent research line on recurrent SNNs where axonal delays are learned at runtime along with the other network parameters. The first proposed approach, dubbed DelRec, demonstrated the benefit of recurrent delay learning in SNNs. Here, we extend it by advocating the use of convolutional recurrent connections in conjunction with the DelRec delay learning mechanism. According to our tests on an audio classification task, this leads to a streamlined architecture with smaller memory footprint (around 99% savings in terms of number of recurrent parameters) and a much faster (52x) inference time, while retaining DelRec's accuracy. Our code is available at: https://github.com/luciozebendo/delrec_snn/tree/conv_delays

Experimental evidence indicates that intrinsic temporal dynamics operating across multiple time scales are closely associated with the emergence of periodic spatial activity of increasing complexity. However, how information encoded in grid-like firing patterns for path integration is processed across these intrinsic time scales remains unclear. To address this question, we introduce adaptive time scales through a leak term in recurrent neural networks (RNNs), forming leaky RNNs discretized from the continuous attractors of firing rate models. Our results demonstrate that leaky RNNs substantially enhance the emergence of well-defined and highly regular hexagonal firing patterns. Compared with vanilla RNNs lacking a leak term, the trained leaky RNNs produce more accurate position estimates while generating reliable grid-cell-like representations. Furthermore, under identical noise conditions, leaky RNNs consistently exhibit more stable dynamics and better-defined grid structures. The learned dynamics also give rise to stable torus attractors with a clear central hole, supporting robust and regular grid-like activity. Overall, the dynamic leak acts as a low-pass filtering mechanism that protects recurrent neural circuitry from noise, stabilizes network dynamics, and improves path-integration accuracy in recurrent neural networks.

Understanding how training shapes the geometry of recurrent network dynamics is a central problem in time-series modeling. We study the emergence of low-dimensional dominant manifolds in the training of Reservoir Computing (RC) networks for temporal forecasting tasks. For a general linear continuous-time reservoir in the infinite-data limit, we show that the training data generate an invariant subspace of the trained reservoir, whose dimension equals the number of dominant modes. We then specialize to a simplified diagonal linear reservoir, where we link the dominant eigenvalues and eigenvectors to the spectrum of a backward Dynamic Mode Decomposition matrix, which yields a finite-dimensional approximation of the backward-time Koopman operator of the system generating the training data. We illustrate the emergence of these dominant modes during training in simulation, and discuss how the analysis may be extended to nonlinear RC via tangent dynamics and differential p-dominance.

Over the last years, state-tracking tasks, particularly permutation composition, have become a testbed to understand the limits of sequence models architectures like Transformers and RNNs (linear and non-linear). However, these are often sequence-to-sequence tasks: learning to map actions (permutations) to states, which is incompatible with the next-token prediction setting commonly used to train language models. We address this gap by converting permutation composition into code via REPL traces that interleave state-reveals through prints and variable transformations. We show that linear RNNs capable of state-tracking excel also in this setting, while Transformers still fail. Motivated by this representation, we investigate why tracking states in code is generally difficult: actions are not always fully observable. We frame this as tracking the state of a probabilistic finite-state automaton with deterministic state reveals and show that linear RNNs can be worse than non-linear RNNs at tracking states in this setup.

We develop a statistical theory of temporal learnability in recurrent neural networks, quantifying the maximal temporal horizon $\mathcal{H}_N$ over which gradient-based learning can recover lag-dependent structure at finite sample size $N$. The theory is built on the effective learning rate envelope $f(\ell)$, a function that captures how gating mechanisms and adaptive optimizers jointly shape the coupling between state-space dynamics and parameter updates during Backpropagation Through Time. Under heavy-tailed ($α$-stable) fluctuations, where empirical averages concentrate at rate $N^{-1/κ_α}$ with $κ_α= α/(α-1)$, the interplay between envelope decay and statistical concentration yields explicit scaling laws for the growth of $\mathcal{H}_N$: logarithmic, polynomial, and exponential temporal learning regimes emerge according to the decay law of $f(\ell)$. These results identify envelope decay as the key determinant of temporal learnability. Slower attenuation of $f(\ell)$ enlarges $\mathcal{H}_N$, while heavy-tailed fluctuations compress it by weakening statistical concentration. Moreover, envelope geometry outweighs dataset size: slowing the envelope's decay enlarges $\mathcal{H}_N$ more than adding data, so more complex architectures that realize slower-decaying envelopes can be more data-efficient than simpler ones. Experiments across multiple gated architectures and optimizers corroborate these structural predictions.

This paper presents a framework to prompt multi-robot teams with high-level tasks using natural language expressions. Our objective is to use the reasoning capabilities of language models in understanding and decomposing multi-robot collaboration and decision-making tasks, but in settings where such models cannot be called at deployment time. However, it is hard to specify the behavior of an individual robot from a team instruction, and have it continuously adapt to actions from other robots. This necessitates a framework with the representational capacity required by the logic and semantics of a task, and yet supports decentralized, real-time operation. We solve this dilemma by recognizing that a task can be represented as a deterministic finite automaton, and that recurrent neural networks (RNNs) can encode numerous automata. This allows us to distill the logic and sequential decompositions of sub-tasks obtained from a language model into an RNN, and align its internal states with the semantics of a given task. This leads to a tiny model that encapsulates the reasoning of the language model and can be implemented onboard. To interpret the internal state of the RNN for a decentralized execution, we train a graph neural network control policy conditioned on the hidden states of the RNN and the language embeddings. We present evaluations on simulated and real-world multi-robot tasks that require sequential and collaborative behavior by the team, demonstrating scalable, robust, real-time performance -- sites.google.com/view/prompting-teams.

Virtual analog (VA) modeling methods seek to emulate analog audio hardware using digital signal processing (DSP). Modeling approaches fall into three broad categories: white-box methods, which use detailed device knowledge for accurate simulation; gray-box methods that use generic DSP blocks to model the system; and black-box methods, which rely solely on opaque models learned from input-output data. A category of architectures used widely in black-box modeling are recurrent neural networks (RNNs). To model device controls, the control values can be provided as conditioning input to the network. However, when the conditioning is time-varied, the models are susceptible to producing noise artifacts. Regularization of the RNN dynamics significantly reduces these artifacts, though at a loss in modeling accuracy. This paper closes the dynamics regularization quality gap by introducing deep control-conditioned LSTMs and a gammatone filterband (GFB) loss. Experiments indicate that the proposed method achieves comparable modeling performance as unregularized baselines while avoiding the noise artifacts caused by time-varying control inputs.

Recurrent neural networks are canonical models of biological memory. In these models, memories are represented by distributed patterns of neural activity that are stored in the recurrent connections between neurons, such that they become attractors of the network's dynamics. During memory recall, network dynamics thus converge toward one of these memory patterns when started from a noisy or partial cue. Therefore, memory performance critically hinges on the dynamical stability of the stored patterns. However, previous theoretical approaches only studied dynamical stability under highly restrictive conditions that do not readily apply to biological neural circuits. Here, we develop a theory of the local stability of discrete fixed points in a broad class of networks with graded neural activities and in the presence of noise. Using methods from random matrix theory, we analyze the bulk and outliers of the eigenvalue spectra of the Jacobians that characterize network dynamics around fixed points. We show that either all fixed points are stable or all of them are unstable, depending on whether their number is below a ``critical load for stability'', which is distinct from the classical critical capacity that measures the maximal number of achievable fixed points regardless of their stability. We further analyze the dependence of this critical load for stability on experimentally measurable quantities characterizing the statistics of memory patterns and the activation functions of neurons. Our analysis highlights the computational benefits of sparse-like patterns and threshold-linear activation functions and offers testable predictions for neural circuits supporting memory.

This paper presents the Robust Recurrent Deep Network (R2DN), a scalable parameterization of stable and robust recurrent neural networks for machine learning and data-driven control. We construct R2DNs as the feedback interconnection of a linear time-invariant system and a 1-Lipschitz deep feedforward network, and directly parameterize the weights so that our models are stable (contracting) and robust to input perturbations (Lipschitz) by design. Our parameterization uses a structure similar to the recurrent equilibrium network (REN), but without having to iteratively solve an equilibrium layer at each time-step. This speeds up model inference and training on GPUs, and makes it computationally feasible to scale up the network size and input sequence length in comparison to RENs. We compare R2DNs to RENs on representative problems in nonlinear system identification, observer design, learning-based feedback control, and sequential image classification. We find that training and inference are up to an order of magnitude faster with similar performance, and that they scale more favorably with respect to model expressivity.

Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. In the context of dynamical systems, training with backpropagation through time can lead to challenges arising from exploding or vanishing gradients. In this contribution, we introduce Koopman-informed recurrent neural networks, a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates some problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, control problems, and on real-world data, we observe that with comparable forecasting accuracy, the training time of the Koopman-informed recurrent neural networks is significantly improved when compared to models trained with commonly used gradient-based methods.