Recurrent Neural Networks

Also known as RNN

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Recurrent neural networks study how a model can process ordered inputs by repeatedly updating an internal state, which is useful for streaming data, long-term memory, and decisions under partial observability. Common approaches include gated LSTMs and GRUs, linear and state-space recurrences, continuous-time and spiking networks, and weight-shared loops for iterative reasoning; training ranges from backpropagation through time to online learning and parallel scans. Researchers test these models on state-tracking and recall tasks, language modeling, time-series forecasting, control, and physical-system prediction. A central tension is how to preserve useful information and stable learning over long sequences while keeping computation efficient and matching the accuracy and flexibility of Transformers.

Latest papers

303 papers47 papers in the last four weeks, up 15% on the four weeks before. 0.3% of all new papers.

May 4, 2026

ParaRNN: An Interpretable and Parallelizable Recurrent Neural Network for Time-Dependent Data

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.

Yuxi Cai, Lan Li, Feiqing Huang +1

Department of Statistics and Actuarial Science, University of Hong Kong

Interpretable ML Recurrent Neural Networks Nonparametric Regression Neural Network Approximation Theory

May 2, 2026

Benchmarking LightGBM and BiLSTM for Sentiment Analysis on Indonesian E-Commerce Reviews

This study presents a comparative analysis between two primary approaches in Natural Language Processing (NLP): Machine Learning (ML) utilizing the PyCaret AutoML framework, and Deep Learning (DL). The evaluation is conducted on a sentiment analysis task using an Indonesian e-commerce review dataset sourced from Hugging Face. The dataset, consisting of 15,000 samples, is partitioned into training, validation, and testing sets. The ML experiments compare LightGBM, Logistic Regression, and Support Vector Machine (SVM) algorithms, whereas the DL experiment implements a Bidirectional Long Short-Term Memory (BiLSTM) architecture. The experimental results demonstrate that the BiLSTM model outperforms all ML models, achieving an accuracy of 98.87% and an F1-Score of 98.87%. Meanwhile, LightGBM emerges as the best-performing ML model with an accuracy of 98.23% in a highly efficient training time. This research proves that the BiLSTM architecture is highly capable of capturing the sequential context of Indonesian review texts, making it the superior model for this specific classification task.

Lidia Natasyah Marpaung, Vania Claresta, Iqfina Haula Halika +3

Department of Data Science Institut Teknologi Sumatera · Institut Teknologi Sumatera · Department of Informatics Institut Teknologi Sumatera

Gradient-Boosted Decision Trees Sentiment Analysis Recurrent Neural Networks E-Commerce

May 1, 2026

Benchmarking local Hebbian learning rules for memory storage and prototype extraction

Associative memory or content-addressable memory is an important component function in computer science and information processing, and at the same time a key concept in cognitive and computational brain science. Many different neural network architectures and learning rules have been proposed to model the brain's associative memory while investigating key component functions like figure-ground segmentation, perceptual reconstruction and rivalry. A less investigated but equally important capability of associative memory is prototype extraction where the training set comprises distorted prototype instances and the task is to recall the correct generating prototype given a new distorted instance. In this paper we benchmark associative memory function of seven different Hebbian learning rules employed in non-modular and modular recurrent networks with winner-take-all dynamics operating on moderately sparse binary patterns. We measure pattern storage and weight information capacity, prototype extraction capabilities, and sensitivity to correlations in data. The original additive Hebb rule comes out with worst capacity, covariance learning proves to be robust but with moderate capacity, and the Bayesian-Hebbian learning rules show highest capacity in almost all different conditions tested.

Anders Lansner, Andreas Knoblauch, Naresh B Ravichandran +1

1KTH Royal Institute of Technology, EECS, Stockholm · 2Stockholm University, Dept. of Mathematics, Stockholm · 3KEIM-Institute, Albstadt-Sigmaringen University, Germany +1

Hebbian Learning Recurrent Neural Networks Associative Memory

May 1, 2026

Multi-frame Restoration for High-rate Lissajous Confocal Laser Endomicroscopy

Lissajous confocal laser endomicroscopy (CLE) is a promising solution for high speed in vivo optical biopsy for handheld scenarios. However, Lissajous scanning traces a resonant trajectory and samples only the visited pixels per frame; at high frame rates, many pixels remain unvisited, creating structured holes. In this work, we introduce the first benchmark for high-rate Lissajous CLE, consisting of low-quality video clips paired with high-quality reference images. The reference images are wide-FOV mosaics obtained by stitching stabilized, slow-scan frames of the same tissue, enabling temporally aligned supervision. Using this dataset, we propose MIRA, a lightweight recurrent framework for Lissajous CLE restoration that iteratively aggregates temporal context through feature reuse and displacement alignment. Our experiments demonstrate that MIRA outperforms both lightweight and high-complexity baselines in restoration quality while maintaining a favorable computational efficiency suitable for clinical deployment.

Minhee Lee, Sangyoon Lee, Jiwook Lee +4

POSTECH, Pohang, Republic of Korea · VPIX Medical, Seoul, Republic of Korea

Endoscopy Medical Imaging Recurrent Neural Networks Image Restoration Video Restoration Medical Image Benchmarks+1

May 1, 2026

Scalable Learning in Structured Recurrent Spiking Neural Networks without Backpropagation

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.

Bo Tang, Weiwei Xie

Department of Electrical and Computer Engineering Worcester Polytechnic Institute Worcester, USA

Recurrent Neural Networks Spiking Neural Networks Neuromorphic Computing

Apr 30, 2026

State Stream Transformer (SST) V2: Parallel Training of Nonlinear Recurrence for Latent Space Reasoning

Current transformers discard their rich latent residual stream between positions, reconstructing latent reasoning context at each new position and leaving potential reasoning capacity untapped. The State Stream Transformer (SST) V2 enables parameter-efficient reasoning in continuous latent space through an FFN-driven nonlinear recurrence at each decoder layer, where latent states are streamed horizontally across the full sequence via a learned blend. This same mechanism supports continuous latent deliberation per position at inference time, dedicating additional FLOPs to exploring abstract reasoning before committing to a token. A two-pass parallel training procedure approximates the sequential recurrence, making co-training computationally practical. Hidden state analysis shows that the state stream facilitates reasoning through sharp, content-dependent reorganisations in continuous latent space; the LM head exposes the resulting latent belief states through the output distribution, while the state stream carries them forward to influence future positions. A learned probe shows that at the first generated token, the latent state already predicts whether the eventual answer will survive or break under additional latent computation for every subsequent position. Co-trained into an existing 27B backbone using only a small dataset of GSM8K examples and evaluated using an oracle to route each question to a depth of one to four recurrent forward passes per token, the SST achieves an architectural capacity bound of 61.11% on out-of-distribution GPQA-Diamond, a +15.15 point gain over a fine-tuning-matched baseline, and cuts that same baseline's remaining GSM8K errors by 46%. Together, these results provide a method for efficiently training a nonlinear recurrence and show that the state stream provides additional reasoning capacity beyond that of an otherwise training-matched transformer.

Thea Aviss

Fifth Dimension

Recurrent Neural Networks Recurrent Transformers Large Reasoning Models Latent Reasoning

Apr 30, 2026

Attractor FCM

In this paper an attractor FCM is created, tested, and analyzed. This FCM is neither a hebbian based nor agentic, nor a hybrid; it rather is a gradient descent based, physics constrained, Jacobian version of an FCM. Moreover, this model has several quirks; it uses residual memory, back propagation through time, and a fixed point anchor that is recursively implemented to update its weights. The residuals update the recursive part without losing the system memory. The model's anchor enables it to converge in a fixed point for which back propagation through time unrolls it and ensures that the error minimization is for an accurate gradient. Furthermore, a new learning algorithm is utilized. The Newton's method finds the system's fixed point attractor and then gradient descend is adaptively changing the landscape; an adaptive term is used to directly manipulate the weights through the attractor dynamics. As the adaptive term changes, the descent through the landscape is constantly adjusting according to sigmoid saturation, and that prevents premature convergence to a local minimum. Lastly, the updates are filtered by causal mask that informs the network about the physics, respecting the initial expert based opinions, for which model reduces the error to the target in an efficient way.

Alexis Kafantaris

Recurrent Neural Networks Physics-Informed ML Fuzzy Cognitive Maps Gradient Descent Dynamics

Apr 29, 2026

Benchmarking PyCaret AutoML Against BiLSTM for Fine-Grained Emotion Classification: A Comparative Study on 20-Class Emotion Detection

Fine-grained emotion classification, which identifies specific emotional states such as happiness, anger, sadness, and fear, remains a challenging task in natural language processing. This study benchmarks classical machine learning and deep learning approaches for 20-class emotion classification using the 20-Emotion Text Classification Dataset containing 79,595 English sentences. On the machine learning side, Logistic Regression, Multinomial Naive Bayes, and Support Vector Machine are evaluated using TF-IDF features. On the deep learning side, Bidirectional Long Short-Term Memory, Gated Recurrent Unit, and a lightweight Transformer implemented in PyTorch are compared. The results show that BiLSTM achieves the best overall performance with 89% accuracy and a weighted F1-score of 0.89, slightly outperforming the best machine learning model, SVM, which reaches 88.11% accuracy. The findings indicate that while traditional machine learning models remain competitive and computationally efficient, sequence-based deep learning models better capture contextual emotional cues in text.

Arya Muda Siregar, Arielva Simon Siahaan, Haikal Fransisko Simbolon +3

Institut Teknologi Sumatera Lampung Selatan, Indonesia

Recurrent Neural Networks Text Classification Emotion Recognition

Apr 28, 2026

PPG-Based Affect Recognition with Long-Range Deep Models: A Measurement-Driven Comparison of CNN, Transformer, and Mamba Architectures

Photoplethysmography (PPG) is increasingly used in wearable affective computing due to its low cost and ease of integration into consumer devices. Recent advances in deep learning have introduced long-range sequence models, such as Transformers, and state-space models, like Mamba, which have demonstrated strong performance on natural language and general time-series tasks. However, it remains unclear whether these architectures offer tangible benefits over widely used Convolutional Neural Networks (CNNs) and Long Short-Term Memory (LSTMs) for PPG-based affect recognition, given that datasets are typically small and noisy. This work presents a measurement-driven comparison of four deep learning architectures, CNN, CNN-LSTM hybrid, Transformers, and Mamba, for classifying arousal, valence, and relaxation states from wrist-based PPG signals. All models are evaluated under a subject-independent 5-fold cross-validation protocol using identical preprocessing, segmentation, and training pipelines. Our results show that the Transformer and Mamba models achieve performance comparable to that of a CNN baseline, but do not consistently outperform it across all tasks. CNNs remain the most effective overall, providing the highest accuracy with the smallest model size, whereas Transformers have a better balance of F1 scores for Arousal and Relaxation. The study provides the first evaluation of Transformer and Mamba models for PPG-based affect recognition, offering practical guidance on model selection for wearable affective monitoring systems.

Karim Alghoul, Hussein Al Osman, Abdulmotaleb El Saddik

School of Electrical Engineering and Computer Science University of Ottawa Ottawa, Canada

Selective SSMs Recurrent Neural Networks Convolutional Neural Networks Time Series Transformers Affective Computing

Apr 28, 2026

Teacher Forcing as Generalized Bayes: Optimization Geometry Mismatch in Switching Surrogates for Chaotic Dynamics

Identity teacher forcing (ITF) enables stable training of deterministic recurrent surrogates for chaotic dynamical systems and has been highly effective for dynamical systems reconstruction (DSR) with recurrent neural networks (RNNs), including interpretable almost-linear RNNs (AL-RNNs). However, as an intervention-based prediction loss (and thus a generalized Bayes update), teacher forcing need not match the free-running model's marginal likelihood geometry. We compare the objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs, estimating ambiguity-aware observed information via Louis' identity. In the switching setting studied here, conditioning on a single forced regime path (as ITF does) inflates curvature, while marginal likelihood curvature is reduced by a missing-information correction when multiple switching explanations remain plausible. In Lorenz-63 experiments, windowed evidence fine-tuning improves held-out evidence but can degrade dynamical quantities of interest (QoIs) relative to ITF-pretrained models.

Andre Herz, Daniel Durstewitz, Georgia Koppe

Interdisciplinary Center for Scientific Computing, Heidelberg University, Germany · Dept. of Theoretical Neuroscience, Central Institute of Mental Health (CIMH), Mannheim, Germany · Hector Institute for AI in Psychiatry and Dept. of Psychiatry and Psychotherapy, CIMH, Mannheim, Germany +1

Neural Surrogate Modeling Generalized Bayesian Inference Dynamical Systems Bayesian Inference Recurrent Neural Networks Chaotic Time-Series Prediction

Apr 28, 2026

Learning-Based Dynamics Modeling and Robust Control for Tendon-Driven Continuum Robots

Tendon-Driven Continuum Robots (TDCRs) pose significant modeling and control challenges due to complex nonlinearities, such as frictional hysteresis and transmission compliance. This paper proposes a differentiable learning framework that integrates high-fidelity dynamics modeling with robust neural control. We develop a GRU-based dynamics model featuring bidirectional multi-channel connectivity and residual prediction to effectively suppress compounding errors during long-horizon auto-regressive prediction. By treating this model as a gradient bridge, an end-to-end neural control policy is optimized through backpropagation, allowing it to implicitly internalize compensation for intricate nonlinearities. Experimental validation on a physical three-section TDCR demonstrates that our framework achieves accurate tracking and superior robustness against unseen payloads, outperforming Jacobian-based methods by eliminating self-excited oscillations. For implementation details and source code, please refer to https://github.com/ZiqingZou/ContinuumControl.

Ziqing Zou, Ke Qiu, Fei Wang +3

Department of Control Science and Engineering, Zhejiang University, Hangzhou, China

Gated Recurrent Unit Robot Trajectory Tracking Robot Dynamics Learning Recurrent Neural Networks Robust Control Continuum Robot Control+2

Apr 28, 2026

Benchmarking Logistic Regression, SVM, and LightGBM Against BiLSTM with Attention for Sentiment Analysis on Indonesian Product Reviews

Sentiment analysis of product reviews on e-commerce platforms plays a critical role in automatically understanding customer satisfaction and providing actionable insights for sellers seeking to improve product quality. This paper presents a comprehensive benchmarking study comparing a Machine Learning (ML) approach via the PyCaret AutoML framework against a Deep Learning (DL) approach based on a Bidirectional Long Short-Term Memory (BiLSTM) architecture with an Attention mechanism for binary sentiment classification on Indonesian product reviews. The dataset comprises 19,728 samples balanced equally between positive and negative reviews. For the ML approach, three prominent algorithms were evaluated via 10-fold stratified cross-validation: Logistic Regression (LR), Support Vector Machine (SVM) with a linear kernel, and Light Gradient Boosting Machine (LightGBM). Logistic Regression achieved the best ML performance with an accuracy of 97.26% and an F1-score of 97.26%. The BiLSTM with Attention model, evaluated on 3,946 held-out test samples, achieved an accuracy of 97.24% and an F1-score of 97.24%. These comparative results demonstrate that traditional ML algorithms with proper preprocessing and feature extraction can compete closely with, and even marginally outperform, more complex sequential DL architectures on high-dimensional datasets, while simultaneously offering greater computational efficiency.

Razin Hafid Hamdi, Ivana Margareth Hutabarat, Hanna Gresia Sinaga +3

Institut Teknologi Sumatera Lampung Selatan, Indonesia

Sentiment Analysis Recurrent Neural Networks

Apr 27, 2026

Sentiment and Emotion Classification of Indonesian E-Commerce Reviews via Multi-Task BiLSTM and AutoML Benchmarking

Indonesian marketplace reviews mix standard vocabulary with slang, regional loanwords, numeric shorthands, and emoji, making lexicon-based sentiment tools unreliable in practice. This paper describes a two-track classification pipeline applied to the PRDECT-ID dataset, which contains 5,400 product reviews from 29 Indonesian e-commerce categories, each labeled for binary sentiment (Positive/Negative) and five-class emotion (Happy, Sad, Fear, Love, Anger). The first track applies TF-IDF vectorization with a PyCaret AutoML sweep across standard classifiers. The second track is a PyTorch Bidirectional Long Short-Term Memory (BiLSTM) network with a shared encoder and two task-specific output heads. A preprocessing module applies 14 sequential cleaning steps, including a 140-entry slang dictionary assembled from marketplace corpora. Four configurations are benchmarked: BiLSTM Baseline, BiLSTM Improved, BiLSTM Large, and TextCNN. Training uses class-weighted cross-entropy loss, ReduceLROnPlateau scheduling, and early stopping. Both tracks are deployed as Gradio applications on Hugging Face Spaces. Source code is publicly available at https://github.com/ikii-sd/pba2026-crazyrichteam.

Hermawan Manurung, Ibrahim Al-Kahfi, Ahmad Rizqi +1

Department of Data Science · Institut Teknologi Sumatera · Lampung Selatan, 35365, Indonesia +1

Sentiment Analysis Recurrent Neural Networks Multi-Task Learning Emotion Recognition

Apr 27, 2026

Primitive Recursion without Composition: Dynamical Characterizations, from Neural Networks to Polynomial ODEs

What do recurrent neural networks, polynomial ODEs, and discrete polynomial maps each bring to computation, and what do they lack? All three operate over the continuum--real-valued states evolved by real-valued dynamics--even when the target functions are discrete. We study them through primitive recursion. We prove that primitive recursion admits equivalent characterizations in all three frameworks: bounded iteration of a fixed recurrent ReLU network, robust computation by a fixed polynomial ODE, and iteration of a fixed polynomial map with an externally supplied step-size parameter. In each, the time bound is itself primitive recursive, composition emerges from the dynamics rather than as a closure rule, and inputs are raw integer vectors. Every primitive recursive function is first compiled into bounded iteration of a single threshold-affine normal form, then interpreted as a ReLU computation and as a polynomial ODE. The equivalences expose a structural asymmetry: no fixed polynomial map can round uniformly to the nearest integer or realize exact phase selection--operations polynomial ODEs perform robustly via continuous-time flow. Each formalism compensates for a limitation the others lack: the ReLU gate provides exact branching, continuous time provides autonomous rounding and control, and the step-size parameter recovers both at the cost of discretization precision. This opens dynamical characterizations of subrecursive hierarchies and complexity classes by restricting time bounds, polynomial degrees, or discretization resources within one framework. More broadly, these models do not compute by composing subroutines: they shape the trajectory of a dynamical system through clocks, phase selectors, and error correction built into the dynamics. This differs structurally from symbolic programming, and our theorem gives a precise framework to study the difference.

Olivier Bournez

Institut Polytechnique de Paris, Ecole Polytechnique, LIX, Palaiseau, France

Dynamical Systems ReLU Neural Networks Recurrent Neural Networks

Apr 27, 2026

New non-Euclidean neural quantum states from additional types of hyperbolic recurrent neural networks

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincare hyperbolic GRU, to new variants including Poincare RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing the new non-Euclidean hyperbolic NQS ansatzes, we generalize the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincare GRU NQS when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the Heisenberg

J_1J_2

and

J_1J_2J_3

models. Here, using larger systems consisting of 100 spins, we find that all four hyperbolic RNN/GRU NQS variants always outperform their respective Euclidean counterpart with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different

J_2

and (

J_2, J_3

) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatzes are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Poincare RNN/GRU and Lorentz RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.

H. L. Dao

Neural Quantum States Recurrent Neural Networks Quantum Many-Body Physics

Apr 23, 2026

Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning

Simulations of crystal growth are performed by using Convolutional Recurrent Neural Network surrogate models, trained on a dataset of time sequences computed by numerical integration of Allen-Cahn dynamics including faceting via kinetic anisotropy. Two network architectures are developed to take into account the effects of a variable supersaturation value. The first infers it implicitly by processing an input mini-sequence of a few evolution frames and then returns a consistent continuation of the evolution. The second takes the supersaturation parameter as an explicit input along with a single initial frame and predicts the entire sequence. The two models are systematically tested to establish strengths and weaknesses, comparing the prediction performance for models trained on datasets of different size and, in the first architecture, different lengths of input mini-sequence. The analysis of point-wise and mean absolute errors shows how the explicit parameter conditioning guarantees the best results, reproducing with high-fidelity the ground-truth profiles. Comparable results are achievable by the mini-sequence approach only when using larger training datasets. The trained models show strong conditioning by the supersaturation parameter, consistently reproducing its overall impact on growth rates as well as its local effect on the faceted morphology. Moreover, they are perfectly scalable even on 256 times larger domains and can be successfully extended to more than 10 times longer sequences with limited error accumulation. The analysis highlights the potential and limits of these approaches in view of their general exploitation for crystal growth simulations.

Matteo Rigoni, Daniele Lanzoni, Francesco Montalenti +1

Dept. of Materials Science, University of Milano-Bicocca, 20125 Milano, Italy · Dept. of Physics, University of Genova, 16146 Genova, Italy

Neural Surrogate Modeling Convolutional RNNs Surrogate Modeling PDE Surrogate Modeling Recurrent Neural Networks

Apr 22, 2026

How Much Is One Recurrence Worth? Iso-Depth Scaling Laws for Looped Language Models

We measure how much one recurrence is worth to a looped (depth-recurrent) transformer, in equivalent unique parameters. From an iso-depth pretraining sweep across recurrence counts

r \in \{1, 2, 4, 8\}

spanning

{\sim}50\times

in training compute, we fit a joint scaling law

L = E + A\,(N_\text{once} + r^{\varphi} N_\text{rec})^{-α} + B\,D^{-β}

and measure a recurrence-equivalence exponent

\varphi = 0.46

. Intuitively,

\varphi

tells us whether looping a block

r

times is equivalent in validation loss to

r

unique blocks of a non-looped model (full equivalence,

\varphi{=}1

) or to a single block run repeatedly with no capacity gain (

\varphi{=}0

). Our

\varphi = 0.46

sits in between, so replacing unique blocks with shared recurrences increases validation loss at matched training compute. For example, at

r{=}4

a 410M looped model performs on par with a 580M non-looped model, but incurs the training cost of a 1B non-looped one. We demonstrate the utility of

\varphi

as a diagnostic tool on two case studies: commonly used truncated backpropagation lowers

\varphi

to

0.38

, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise

\varphi

to

0.65

, a genuine capacity gain. Our method separates true loop improvements from training-side gains, a distinction raw validation loss cannot make.

Kristian Schwethelm, Daniel Rueckert, Georgios Kaissis

Chair for AI in Healthcare and Medicine, Technical University of Munich, Germany · Department of Computing, Imperial College London, UK · Munich Center for Machine Learning (MCML), Germany +1

Truncated Backpropagation Through Time Language Model Scaling Laws Recurrent Neural Networks Recurrent Transformers

Apr 22, 2026

Preconditioned DeltaNet: Curvature-aware Sequence Modeling for Linear Recurrences

To address the increasing long-context compute limitations of softmax attention, several subquadratic recurrent operators have been developed. This work includes models such as Mamba-2, DeltaNet, Gated DeltaNet (GDN), and Kimi Delta Attention (KDA). As the space of recurrences grows, a parallel line of work has arisen to taxonomize them. One compelling view is the test-time regression (TTR) framework, which interprets recurrences as performing online least squares updates that learn a linear map from the keys to values. Existing delta-rule recurrences can be seen as first-order approximations to this objective, but notably ignore the curvature of the least-squares loss during optimization. In this work, we address this by introducing preconditioning to these recurrences. Starting from the theory of online least squares, we derive equivalences between linear attention and the delta rule in the exactly preconditioned case. Next, we realize this theory in practice by proposing a diagonal approximation: this enables us to introduce preconditioned variants of DeltaNet, GDN, and KDA alongside efficient chunkwise parallel algorithms for computing them. Empirically, we find that our preconditioned delta-rule recurrences yield consistent performance improvements across synthetic recall benchmarks and language modeling at the 340M and 1B scale.

Neehal Tumma, Noel Loo, Daniela Rus

1MIT · 2Liquid AI

Long-Context Language Modeling Recurrent Neural Networks Recurrent Transformers Linear Attention Delta-Rule Linear Attention Linear RNNs

Apr 22, 2026

Physics-Enhanced Deep Learning for Proactive Thermal Runaway Forecasting in Li-Ion Batteries

Accurate prediction of thermal runaway in lithium-ion batteries is essential for ensuring the safety, efficiency, and reliability of modern energy storage systems. Conventional data-driven approaches, such as Long Short-Term Memory (LSTM) networks, can capture complex temporal dependencies but often violate thermodynamic principles, resulting in physically inconsistent predictions. Conversely, physics-based thermal models provide interpretability but are computationally expensive and difficult to parameterize for real-time applications. To bridge this gap, this study proposes a Physics-Informed Long Short-Term Memory (PI-LSTM) framework that integrates governing heat transfer equations directly into the deep learning architecture through a physics-based regularization term in the loss function. The model leverages multi-feature input sequences, including state of charge, voltage, current, mechanical stress, and surface temperature, to forecast battery temperature evolution while enforcing thermal diffusion constraints. Extensive experiments conducted on thirteen lithium-ion battery datasets demonstrate that the proposed PI-LSTM achieves an 81.9% reduction in root mean square error (RMSE) and an 81.3% reduction in mean absolute error (MAE) compared to the standard LSTM baseline, while also outperforming CNN-LSTM and multilayer perceptron (MLP) models by wide margins. The inclusion of physical constraints enhances the model's generalization across diverse operating conditions and eliminates non-physical temperature oscillations. These results confirm that physics-informed deep learning offers a viable pathway toward interpretable, accurate, and real-time thermal management in next-generation battery systems.

Salman Khan, Syed Sajid Ullah, Muhammad Zunair Zamir +3

Recurrent Neural Networks Physics-Informed ML Lithium-Ion Batteries

Apr 19, 2026

Machine Learning Hamiltonian Dynamical Systems with Sparse and Noisy Data

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.

Vedanta Thapar, Abhinav Gupta

The Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom · Center for Theoretical Physics, St. Stephen’s College, University of Delhi, Delhi 110007, India

Nonlinear System Identification Recurrent Neural Networks Symplectic RNNs Hamiltonian Neural Networks System Identification Physics-Informed ML+2

Apr 19, 2026

Fully Analog Resonant Recurrent Neural Network via Metacircuit

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.

Zixin Zhou, Tianxi Jiang, Menglong Yang +3

Institute of Humanoid Robots, CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei 230026, People’s Republic of China · State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China

Efficient Neural Network Inference Time Series Classification Recurrent Neural Networks Neuromorphic Computing Edge ML Physical Neural Networks

Apr 19, 2026

LASER: Low-Rank Activation SVD for Efficient Recursion

Recursive architectures such as Tiny Recursive Models (TRMs) perform implicit reasoning through iterative latent computation, yet the geometric structure of these reasoning trajectories remains poorly understood. We investigate the activation manifold of TRMs during recursive unrolling and find that activations occupy an effectively linear, low-dimensional subspace whose principal directions can be tracked dynamically with cheap power iterations. This suggests that weight-sharing concentrates iterative computation along a small number of dominant eigendirections, and we find that this concentration varies sharply across computational sites. We exploit this structure through LASER (Low-Rank Activation SVD for Efficient Recursion), a dynamic compression framework that maintains an evolving low-rank basis via matrix-free subspace tracking with a fidelity-triggered reset mechanism, achieving

{\sim}60\%

activation memory savings with no statistically significant accuracy degradation. Our analysis raises questions about how recursive architectures allocate representational capacity during implicit reasoning, and whether this concentration can be exploited to improve the efficiency and stability of latent computation.

Ege Çakar, Ketan Ali Raghu, Lia Zheng

Harvard University Cambridge, MA, USA

Low-Rank Matrix Decomposition Recurrent Neural Networks Low-Rank Compression Low-Rank Approximation Neural Network Compression

Apr 18, 2026

The Topological Trouble With Transformers

Transformers encode structure in sequences via an expanding contextual history. However, their purely feedforward architecture fundamentally limits dynamic state tracking. State tracking -- the iterative updating of latent variables reflecting an evolving environment -- involves inherently sequential dependencies that feedforward networks struggle to maintain. Consequently, feedforward models push evolving state representations deeper into their layer stack with each new input step, rendering information inaccessible in shallow layers and ultimately exhausting the model's depth. While this depth limit can be bypassed by dynamic depth models and by explicit or latent thinking that externalizes state representations, these solutions are computationally and memory inefficient. In this article, we argue that temporally extended cognition requires refocusing from explicit thought traces to implicit activation dynamics via recurrent architectures. We introduce a taxonomy of recurrent and continuous-thought transformer architectures, categorizing them by their recurrence axis (depth versus step) and their ratio of input tokens to recurrence steps. Finally, we outline promising research directions, including enhanced state-space models and coarse-grained recurrence, to better integrate state tracking into modern foundation models.

Michael C. Mozer, Shoaib Ahmed Siddiqui, Rosanne Liu

Google DeepMind

Transformer FFNs Recurrent Neural Networks Recurrent Transformers State Tracking

Apr 17, 2026

Multilevel neural networks with dual-stage feature fusion for human activity recognition

Human activity recognition (HAR) refers to the process of identifying human actions and activities using data collected from sensors. Neural networks, such as convolutional neural networks (CNNs), long short-term memory (LSTM) networks, convolutional LSTM, and their hybrid combinations, have demonstrated exceptional performance in various research domains. Developing a multilevel individual or hybrid model for HAR involves strategically integrating multiple networks to capitalize on their complementary strengths. The structural arrangement of these components is a critical factor influencing the overall performance. This study explores a novel framework of a two-level network architecture with dual-stage feature fusion: late fusion, which combines the outputs from the first network level, and intermediate fusion, which integrates the features from both the first and second levels. We evaluated

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different network architectures of CNNs, LSTMs, and convolutional LSTMs, incorporating late fusion with and without intermediate fusion, to identify the optimal configuration. Experimental evaluation on two public benchmark datasets demonstrates that architectures incorporating both late and intermediate fusion achieve higher accuracy than those relying on late fusion alone. Moreover, the optimal configuration outperforms baseline models, thereby validating its effectiveness for HAR.

Abeer FathAllah Brery, Ascensión Gallardo-Antolín, Israel Gonzalez-Carrasco +1

Human Activity Recognition Recurrent Neural Networks Convolutional LSTM Convolutional Neural Networks Feature Fusion

Apr 17, 2026

Combining Convolution and Delay Learning in Recurrent Spiking Neural Networks

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

Lúcio Folly Sanches Zebendo, Eleonora Cicciarella, Michele Rossi

University of Padova, Department of Information Engineering · Department of Mathematics “Tullio Levi Civita” at the University of Padova, Italy

Convolutional RNNs Efficient Neural Network Inference Recurrent Neural Networks Spiking Neural Networks

Apr 17, 2026

Impact of leaky dynamics on predictive path integration accuracy in recurrent neural networks

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.

Yanlin Zhang, Yan Zhang, Muhua Zheng +1

School of physics and Electronic Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, China · Teaching Department, Jiangsu University Jingjiang College, Zhenjiang 212028, Jiangsu, China

Recurrent Neural Networks Continuous Attractor Neural Networks

Apr 16, 2026

A Nonlinear Separation Principle via Contraction Theory: Applications to Neural Networks, Control, and Learning

This paper establishes a nonlinear separation principle based on contraction theory and derives sharp stability conditions for recurrent neural networks (RNNs). First, we introduce a nonlinear separation principle that guarantees global exponential stability for the interconnection of a contracting state-feedback controller and a contracting observer, alongside parametric extensions for robustness and equilibrium tracking. Second, we derive sharp linear matrix inequality (LMI) conditions that guarantee the contractivity of both firing rate and Hopfield neural network architectures. We establish structural relationships among these certificates-demonstrating that continuous-time models with monotone non-decreasing activations maximize the admissible weight space-and extend these stability guarantees to interconnected systems and Graph RNNs. Third, we combine our separation principle and LMI framework to solve the output reference tracking problem for RNN-modeled plants. We provide LMI synthesis methods for feedback controllers and observers, and rigorously design a low-gain integral controller to eliminate steady-state error. Finally, we derive an exact, unconstrained algebraic parameterization of our contraction LMIs to design highly expressive implicit neural networks, achieving competitive accuracy and parameter efficiency on standard image classification benchmarks.

Anand Gokhale, Anton V. Proskurnikov, Yu Kawano +1

Center for Control, Dynamical Systems, and Computation, UC Santa Barbara, Santa Barbara, CA 93106 USA · Department of Electronics and Telecommunications, Politecnico di Torino, Italy, 10129 · Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

Dynamical Systems Contraction Theory Recurrent Neural Networks Neural Network Stability Neural Network Control

Apr 16, 2026

NEAT-NC: NEAT guided Navigation Cells for Robot Path Planning

To navigate a space, the brain makes an internal representation of the environment using different cells such as place cells, grid cells, head direction cells, border cells, and speed cells. All these cells, along with sensory inputs, enable an organism to explore the space around it. Inspired by these biological principles, we developed NEATNC, a Neuro-Evolution of Augmenting Topology guided Navigation Cells. The goal of the paper is to improve NEAT algorithm performance in path planning in dynamic environments using spatial cognitive cells. This approach uses navigation cells as inputs and evolves recurrent neural networks, representing the hippocampus part of the brain. The performance of the proposed algorithm is evaluated in different static and dynamic scenarios. This study highlights NEAT's adaptability to complex and different environments, showcasing the utility of biological theories. This suggests that our approach is well-suited for real-time dynamic path planning for robotics and games.

Hibatallah Meliani, Khadija Slimani, Samira Khoulji

ISI Laboratory, National School of Applied Sciences (ENSA), Abdelmalek Essaadi University Tetouan, Morocco · EsieaLab LDR, Higher School of Computer Science, Electronics and Automation (ESIEA) Paris, France

Evolutionary Optimization Robot Navigation Recurrent Neural Networks Robot Motion Planning Neuroevolution

Apr 15, 2026

Neuromorphic Spiking Ring Attractor for Proprioceptive Joint-State Estimation

Maintaining stable internal representations of continuous variables is fundamental for effective robotic control. Continuous attractor networks provide a biologically inspired mechanism for encoding such variables, yet neuromorphic realizations have rarely addressed proprioceptive estimation under resource constraints. This work introduces a spiking ring-attractor network representing a robot joint angle through self-sustaining population activity. Local excitation and broad inhibition support a stable activity bump, while velocity-modulated asymmetries drive its translation and boundary conditions confine motion within mechanical limits. The network reproduces smooth trajectory tracking and remains stable near joint limits, showing reduced drift and improved accuracy compared to unbounded models. Such compact hardware-compatible implementation preserves multi-second stability demonstrating a near-linear relationship between bump velocity and synaptic modulation.

Federica Ferrari, Flavia Davidhi, Bernard Maacaron +6

Istituto Italiano di Tecnologia, Genoa, Italy · Department of Neurosurgery, University Hospital Zürich, Switzerland · Institute of Neuroinformatics, Univ. of Zurich and ETH Zurich, Switzerland +3

Continuous Attractor Neural Networks Robot State Estimation Spiking Neural Networks Neuromorphic Computing Recurrent Neural Networks

Apr 15, 2026

Modeling of Self-sustained Neuron Population without External Stimulus

Self-sustained neural activity in the absence of ongoing external input is a fundamental feature of nervous system dynamics, yet the conditions under which it can emerge in biophysically grounded network models remain incompletely understood. We studied whether a recurrent network of Hodgkin-Huxley neurons with spike-timing-dependent plasticity and intrinsic stochasticity can maintain autonomous activity after brief transient stimulation. The simulated network comprised 200 neurons (160 excitatory, 40 inhibitory) with 80% connection probability, incorporating excitatory and inhibitory STDP, probabilistic vesicle release, probabilistic synapse formation, receptor variability, and voltage-dependent inhibition. After a brief 200 ms initialization stimulus to 30 excitatory neurons, the network received no further external input. In one 1800 s simulation and two additional 500 s simulations, the network maintained sparse, irregular activity without ongoing drive. In the 1800 s run, 67% of neurons exhibited mean firing rates below 1 Hz, the population mean firing rate was 1.13 +/- 1.34 Hz, participation increased across longer observation windows, and population-mean Fano factors remained near 1-2, consistent with irregular spike timing. Raster activity also showed spontaneous qualitative reorganizations in collective firing patterns over time. These findings suggest that recurrent Hodgkin-Huxley networks with plastic and stochastic synapses can sustain long-duration autonomous activity in a sparse firing regime after brief initialization.

İhsan Ertuğrul Karakaş, Özden Özel, İlkay Ulusoy +1

Middle East Technical University · Başkent University

Neural Processes Spiking Neural Networks Recurrent Neural Networks