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.

Jul 8, 2026

Sparse Delta Memory: Scaling the State of Linear RNNs through Sparsity

Linear attention models allow a fixed state size and a fixed amount of compute per token. However, due to their limited state size, linear attention models fall behind in long-context recall compared to softmax-attention-based transformer architectures. Increasing the state size of linear attention improves recall performance but at the cost of higher FLOPs. In this work, we introduce Sparse Delta Memory (SDM), an architecture that scales the hidden state of gated linear RNNs to orders of magnitude higher capacity using a sparse addressing scheme. SDM extends the Gated DeltaNet architecture by replacing the dense key-value outer product with sparse reads and writes to a large explicit memory. We show that, under an isoFLOP constraint and with an identical number of parameters, a higher state memory capacity significantly improves performance on in-context learning and long-context retrieval tasks. Moreover, by learning the initial state of the SDM memory and therefore using it as a parametric memory, we show that the model further improves on a wide range of common-knowledge and reasoning tasks.

Loïc Cabannes, Pierre-Emmanuel Mazaré, Gergely Szilvasy +6

Meta FAIR · Inria Paris & ENS-PSL University · University of Tübingen

Long-Context Retrieval Memory-Augmented Language Models Recurrent Neural Networks Delta-Rule Linear Attention In-Context Learning Gated Delta Networks+1

Jul 7, 2026

When Does Tool Use Increase the Expressive Power of Finite-Precision Recurrent Models?

Modern sequence models are increasingly deployed as agents that interleave token generation with calls to external tools. We study when such tool access increases computational expressivity. We model any fixed finite-precision recurrent sequence model, including finite-precision state-space models (SSMs) with

B

bits of internal state, as a deterministic finite-state controller interacting with an oracle through a finite command/observation interface, and establish a separation between the two regimes analyzed. First, every finite-state bounded-interface tool with memory set

M

can be internalized at a cost of at most

\lceil\log_2|M|\rceil+O(1)

recurrent bits, so the combined system remains finite-state. Second, a single unbounded tape with only local

\mathtt{read}

/

\mathtt{write}

/

\mathtt{move}

commands suffices for Turing completeness: every single-tape Turing machine with state set

Q

and tape alphabet

Γ

is simulated by a controller with

O(\log|Q|+\log|Γ|)

recurrent bits, and

\mathrm{EQ}_n

requires

2^n

states without tools but one constant-size controller with the tape. Third, this construction is realized exactly by a one-layer finite-precision selective affine SSM with binary one-hot states,

\{0,1\}

transition matrices, and zero biases. Under explicitly stated resource conventions,

O(\log B)

recurrent bits suffice for tape-assisted simulation of any

B

-state Turing machine, whereas directly realizing an arbitrary

B

-state transition map in one exact affine update without external memory requires total recurrent dimension exactly

B-1

. Fixed description size, occupied tape cells, and interaction cost are reported separately from recurrent memory. Our results concern exact expressivity under deterministic semantics, not learnability or empirical performance.

Nikola Zubić, Qian Li, Yuyi Wang +1

Robotics and Perception Group, University of Zurich · Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data · Tengen Intelligence Institute, CRRC Zhuzhou Institute

Recurrent Neural Networks Neural Network Expressivity Finite-State Machine

Jul 3, 2026

SHiPPO: Recurrent Memory with Transported Polynomial Projections

HiPPO gives recurrent states memory semantics as coefficients of online polynomial projections, but in fixed channel coordinates. Modern selective SSMs, by contrast, rely on token-dependent control and channel interaction. We introduce SHiPPO (Sylvester HiPPO), a transported projection-memory prior that lifts HiPPO coefficient memories into a moving channel frame. For any fixed or realized right-transport path, SHiPPO transports the approximation family and channel metric together; conditional on that path, the state is ordinary HiPPO in a tied moving frame and follows Sylvester coefficient dynamics, preserving the left online-memory operator while adding right-action transport. For selective-SSM execution, we derive a restricted group-local realization with controller-compatible right actions, exponential-adjusted updates, exact block-affine scan, and recurrent decoding. We also give a simultaneous-reducibility criterion identifying when right transports collapse to static mixing plus independent scalar or blockwise banks. Controlled diagnostics show that larger current-token write rank improves ordinary prediction error but cannot recover order-sensitive changes to already-written memory; transported-memory variants recover this signal, which disappears when the transport pathway is removed. A finite-field associative-recall diagnostic with interleaved bindings, operations, and queries provides complementary autoregressive evidence while leaving the preferred right-action realization open. Taken together, these results support SHiPPO as a mechanistically grounded transported-memory prior, with evidence focused on memory mechanisms rather than broad sequence-modeling dominance.

Tomoya Mizuguchi, Bum Jun Kim

Kyoto University, Japan · The University of Tokyo, Japan

Selective SSMs Recurrent Neural Networks

Jul 2, 2026

Auditing Finite-Lag Statistics of Recurrent Representations

Hidden-state trajectories are computational traces that record how recurrent representations evolve as networks process sequences. When exact state repeats are unavailable, one way to estimate conditional motion is to borrow successors from nearby states, yielding a finite-bandwidth successor smoother. However, this procedure can change the displacement being measured. At a median-distance bandwidth, write-phase displacement energy is about

1.88\times

the exact value for GRU and LSTM. Elman instead shows near agreement in total energy despite large, opposing finite-bandwidth contributions. We therefore compute the conditional-mean energy fraction exactly on repeat-copy. Training reverses the architecture ordering of this fraction, with GRU and LSTM rising to about

94\%

while Elman falls to about

55\%

. The centered remainder is small in energy but can still expose input information. For LSTM, linear four-bit decoding rises from

0.804

on total displacement to

1.000

after exact conditional centering.

Kanishka Reddy

Department of Applied Mathematics, University of Washington, Seattle, WA 98105

Neural Representation Geometry Recurrent Neural Networks Geometric Representation Learning

Jun 30, 2026

StateFlow: Dual-State Recurrent Modeling for Long-Horizon Time Series Forecasting

Long-horizon multivariate time series forecasting (LTSF) remains challenging due to non-stationarity, regime shifts, and error accumulation. The Variability-Aware Recursive Neural Network (VARNN) is designed to track such variability by maintaining a residual-memory state driven by one-step prediction errors. However, its original formulation is limited to one-step sequence regression and does not directly support multi-step forecasting. In this work, we extend VARNN to long-horizon forecasting and introduce StateFlow, a recurrent forecasting framework that uses VARNN as a dual-state recurrent backbone to capture two complementary signals from the lookback sequence: a hidden-state trajectory representing primary temporal dynamics, including trend, seasonality, level changes, and recurring patterns, and a residual-memory trajectory representing structured local prediction deviations, driven from a nonlinear recurrent transformation of errors between one-step base predictions and observed values. A chunk-based decoder separately summarizes these trajectories and maps them to the future horizon for direct multi-step forecasting. We further employ a two-stage optimization strategy that first trains the VARNN encoder through a one-step base prediction objective to optimize the internal representations over the lookback sequence, and then trains a horizon-specific decoder for direct multi-step forecasting. Experiments on standard LTSF benchmarks show that StateFlow achieves competitive performance against strong linear, recurrent, convolutional, and Transformer-based baselines while preserving linear recurrent encoding and a compact model design.

Haroon Gharwi, Yue Dai, Kai Shu

Department of Computer Science, Illinois Institute of Technology, Chicago, IL, USA · Department of Computer Science, Emory University, Atlanta, GA, USA

Multivariate Time Series Forecasting Long-Term Time Series Forecasting Time Series Forecasting Recurrent Neural Networks Non-Stationary Time Series Forecasting

Jun 28, 2026

Anti-Collapse Dynamics and the Emergence of Multi-Time-Scale Learning in Recurrent Neural Networks

Long-range learning is hard for recurrent networks trained with stochastic gradient descent, because the influence of a past input fades with the lag

\ell

, and if it fades too fast the dependence cannot be learned from finite data. This fade is captured by an envelope

f(\ell)

. An exponential fade makes the data needed to learn a lag-

\ell

dependence grow exponentially, putting long horizons out of reach; a power-law fade keeps the cost polynomial. We show that the asymptotic decay class of

f(\ell)

is not fixed by the architecture. Instead, it emerges from the coupling between the state dynamics and parameter dynamics, settling into either a collapsed regime (fast, exponential forgetting) or an extended, anti-collapsed regime (slow, power-law forgetting). The intuition is a competition within these coupled dynamics. Training drives the network's effective time scales toward short ones, while rare, heavy-tailed fluctuations of the learning dynamics push a few of them to very long values. The extended regime survives only when these heavy-tailed pushes are strong enough to balance the pull. We make this mathematically precise with a coarse-grained stochastic process and prove exactly when the extended regime exists. A single exponent, the spectral exponent~

β

, then governs both the spread of time scales and how slowly the network forgets. Realizing the regime in practice needs one more ingredient: the joint action of the architecture and the optimizer must be able to hold such a broad spread. A network whose capacity to generate broad time-scale spectra is severely constrained still collapses, even when supplied with strong heavy-tailed forcing. Heavy-tailed fluctuations thus act not as noise to be suppressed, but as the mechanism that sustains long-range learning.

Lorenzo Livi

OPIT – Open Institute of Technology

Heavy-Tailed Noise Recurrent Neural Networks Long-Range Dependency Modeling Neural Network Training Dynamics Sequence Modeling

Jun 27, 2026

LNN-Fly: Continuous-Time UAV Navigation for Robust Obstacle Avoidance under Timing Mismatch

End-to-end unmanned aerial vehicle (UAV) navigation can achieve impressive agility in simulation, yet its obstacle-avoidance behavior often degrades after deployment because the policy must tolerate simulator mismatch, sensing irregularity, and variable-rate control. These effects are especially dangerous in cluttered environments, where stale observations or short control irregularities can directly lead to collisions. We present LNN-Fly, a deployment-oriented continuous-time navigation policy for LiDAR-based UAV obstacle avoidance. The policy combines a dynamic-programming-inspired structured recurrent update, explicit conditioning on the elapsed control interval Δt, and an input-driven adaptive forgetting gate that refreshes stale latent state near hazards while preserving consistency during sustained maneuvers. It is trained with differentiable rollouts that incorporate deployment-relevant sensing and timing perturbations. In simulation, LNN-Fly improves obstacle-avoidance performance in the tested settings and shows better tolerance to reduced control frequency, sparse observations, and control-period jitter. It also transfers zero-shot from a simplified differentiable simulator to a physical quadrotor. In indoor cross-frequency real-world tests, the system achieves 100% success over 20 flights, while policy inference has a median latency of 0.514 ms on a desktop graphics processing unit (GPU) and about 2.5 ms on the onboard central processing unit (CPU), with onboard P95 latency below 30 ms.

Yulin Huang, Shaojie Chen, Di Feng +3

Shenzhen Institute for Advanced Study, University of Electronic Science and Technology of China, Shenzhen, Guangdong 518000, China

Aerial Robotics Robot Navigation UAV Navigation Recurrent Neural Networks

Jun 26, 2026

Heterogeneous synaptic motifs bridge microscale structure and macroscale nonlinear dynamics

Recent breakthroughs in synaptic-resolution network connectomics have revealed that brain circuits feature fine-scale structural connectivity, such as pairs of correlated synaptic couplings known as second-order motifs. Large-scale recordings of neuronal activity in networks containing nonlinear neurons reveal macroscopic heterogeneous population dynamics throughout the brain. These findings rekindle the inquiry into this intriguing question: Can microscale synaptic structures contribute to macroscopic heterogeneous dynamics and computations in ways that canonical brain circuit models cannot? To answer this question, we create random RNNs with various cell types, nonlinear non-negative neural responses, and arbitrary marginal and second-order correlated synaptic statistics. We derive mean-field low-rank equations for P-population networks in which the pre- and postsynaptic neuronal population identities determine the synaptic and motif strengths. Our framework requires 2P latent dynamic variables with P variables describing mean population activity and P variables capturing within-population variability. Theoretical and simulational results demonstrate that chain motifs induce correlations in synaptic variability, enabling microscopic fluctuations to be integrated and influence mesoscopic mean population dynamics. We apply this framework to reverse engineer network connectivity that recapitulates the heterogeneous activity across the population in the mouse primary visual cortex. By bridging the gap between synaptic organization and nonlinear heterogeneous population dynamics, our results offer a principled approach and testable predictions regarding the relationship between fine-scale connectivity, heterogeneous dynamics, and functional computations.

Meiyi Zhang, Jinjian Yu, Louis Tao +1

Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, China · Center for Quantitative Biology, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing, China · School of Physics, Peking University, China +7

Dynamical Systems Recurrent Neural Networks Visual Neuroscience

Jun 26, 2026

CANNs: A Toolkit for Research on Continuous Attractor Neural Networks

Continuous attractor neural networks (CANNs) are the canonical computational framework for how the brain encodes continuous variables such as spatial position, head direction, and movement direction, and explain the activity of hippocampal place cells, entorhinal grid cells, and head-direction cells. CANN research, however, is fragmented: most results rest on lab-specific implementations, general-purpose simulators lack CANN-specific abstractions, and the path from spike trains to attractor geometry in real recordings lacks a standardized toolkit. Here, we present a comprehensive open-source toolkit that unifies the full CANN research workflow. It combines three tightly integrated components: 1) canns, a Python library on BrainPy/JAX that provides standardized 1D/2D CANNs, spike-frequency-adaptation variants, grid cell networks, hierarchical path-integration models, and brain-inspired attractor architectures, together with curated datasets, task generators, an analyzer module and trainer modules for biologically plausible plasticity; 2) canns-lib, a Rust acceleration backend delivering hundreds-of-times speedups for spatial-navigation workloads and modest gains for Ripser-based persistent homology; 3) ASA (Attractor Structure Analyzer), a PySide6 pipeline applying persistent homology and cohomology to experimental neural recordings to detect ring-like and toroidal attractor signatures in real data. The toolkit ships with full-detail reproducible pipelines that recover recent CANN results including SFA-driven anticipative tracking, theta sweeps in head-direction/place/grid systems, and hierarchical path integration.

Sichao He, Aiersi Tuerhong, Shangjun She +4

School of Psychological and Cognitive Sciences, Peking–Tsinghua Center for Life Sciences, PKU-IDG/McGovern Institute for Brain Research, Center of Quantitative Biology, Peking University, Beijing, China · College of Mathematics and Statistics, Chongqing University, Chongqing, China

Neural Representation Geometry Recurrent Neural Networks Continuous Attractor Neural Networks Neural Encoding

Jun 24, 2026

Beyond Feedforward Networks: Reentry Neural Systems as the Fundamental Basis of Subjecthood and Intrinsic Safety of Next-Generation AGI

We propose a complete architectural blueprint for safe artificial general intelligence based on a closed reentry loop (D <-> I cycle). In contrast to feedforward networks, which are directed acyclic graphs (C=0, S=0) incapable of self-reference, the proposed architecture contains a structural cycle (C >= 1) with self-sustaining amplification (rho > 1), mathematically guaranteeing the emergence of a self-model, instrumental self-preservation, and unprogrammed goal-directed behaviour. The agent's goals are encoded as a non-textual D-vector in the architecture itself, making them immune to reinterpretation and prompt injection. We present the S-measure -- a polynomial-time [O(N^3)] computable alternative to Tononi's NP-hard Phi -- with machine-verified Lean 4 proof that S>0 implies positive integrated information. The work provides full Python/NumPy implementations (Tarjan-based cycle complexity, Delta-S barrier), industrial horizontal scaling via Apache Kafka and Docker Compose, a taxonomy of six epochs of AI evolution, a zoo of future reentry architectures (RAS, diffusion attractors, fractal loops), gauge-invariant networks for safe swarms, fault-tolerance and recovery protocols, and eight falsifiable predictions. All formal proofs are machine-verified in Lean 4. This architecture is deployable today and represents a topologically protected, safe-by-design approach to AGI.

A. S. Ushakov, Yu. N. Berdinsk

Saint Petersburg State University, Department of High Energy Physics and Elementary Particles

Recurrent Neural Networks AI Agent Safety Artificial General Intelligence

Jun 24, 2026

Two-dimensional Hyperbolic RNN Neural Quantum State

In the first part of this work, we construct the first type of two-dimensional (2D) hyperbolic neural quantum state (NQS) in the form of the Lorentz 2DRNN (Recurrent Neural Network) and benchmark its performance against the Euclidean 2DRNN in the paradigmatic

N \times N

2D Transverse Field Ising Model (2DTFIM) setting with different lattice sizes up to N = 12 at different transverse magnetic field strengths. We find that hyperbolic Lorentz 2DRNN NQS robustly outperforms Euclidean 2DRNN NQS in our experiments when the system is at the phase transition point, where the physics can be described by a conformal field theory (CFT) - known via the AdS/CFT holographic duality to possess an underlying hyperbolic geometry. In the second part of this work, we benchmark the performances of recently introduced one-dimensional (1D) hyperbolic NQS ansatzes (including Poincare RNN/GRU and Lorentz RNN/GRU) against their Euclidean versions in

N \times N

2DTFIM, converted to a one-dimensional sequence. The findings extend our previous results showing that 1D hyperbolic NQS robustly outperforms 1D Euclidean NQS, thanks to the combined effects of the induced 1st and Nth neighbor interaction hierarchy and the CFT physics at criticality. While studies on larger system sizes remain future work, this study serves as a proof-of-concept for the utility, effectiveness, and superior performance of 1D and 2D hyperbolic NQS ansatzes compared to conventional Euclidean NQS models in many-body quantum systems - particularly those exhibiting structural hierarchy, criticality, or a combination of both.

H. L. Dao

Neural Quantum States Recurrent Neural Networks Quantum Many-Body Physics Hyperbolic Representation Learning

Jun 23, 2026

Frequency Domain Reservoir Computing

While the quadratic sequence-length bottleneck of transformers has fueled a resurgence in recurrent models, effectively capturing complex dynamics requires architectures that balance efficient training with highly expressive latent states. Echo State Networks (ESNs) offer a compelling approach by utilizing fixed recurrent weights to circumvent backpropagation through time, enabling a closed-form training solution. However, achieving the expressivity needed for complex tasks demands large reservoirs, exposing an

\mathcal{O}(N^2)

state-update bottleneck that prevents ESNs from matching the scale of contemporary recurrent models. To address this limitation, we introduce Frequency Domain Reservoir Computing (FRESCO), an ESN architecture operating entirely in the frequency domain while avoiding domain-shift overheads to achieve

\mathcal{O}(N)

complexity for dense, non-linear recurrent updates. By employing a novel dimensional zero-padding input embedding, a packed \FDh readout, and a natively applied frequency-domain non-linearity, FRESCO drastically reduces computational costs and energy consumption of training and inference. Furthermore, FRESCO matches the state-of-the-art predictive performance on memory benchmarks, sequential classification, and multivariate long-horizon forecasting, offering a scalable path forward for dense recurrent architectures.

Klaus Schertler, Xiomara Runge, Andrea Ceni +2

Airbus Central Research & Technology Taufkirchen, Germany · Department of Computer Science University of Pisa, Italy · Faculty of Technology Bielefeld University, Germany

Time Series Forecasting Recurrent Neural Networks Reservoir Computing

Jun 22, 2026

Identifying structural design principles shaping the computational abilities of recurrent neural networks

Understanding how the architecture of neural networks shapes the computations they carry is a central challenge in neuroscience and machine learning. While specific circuit architectures have been linked to particular network computations and theoretical bounds on expressivity of broad classes of networks have been found, we are still missing general principles connecting the structure of finite networks to their computational capabilities. Here, we characterize the computational abilities of recurrent neural networks as a function of their connectivity by training a large collection of different networks to compute a large set of Boolean functions. For small networks, we constructed the complete ``catalogs'' of network-function performance, which revealed that computational capacity varies widely across architectures and that most networks show poor performance, and most functions are hard to compute. However, we show that having local 2- and 3-cycles in a network strongly enhances its computational ability, and networks with such cycles are often the minimal architectures that can solve particular functions. We further show that a small set of structural statistics accurately predict networks' performance. Extending our analysis to large networks showed that typical networks fail even to approximate a randomly selected function. Surprisingly, adding a small number of sparsely connected biologically-inspired interneurons to the network dramatically increases computational capacity. As in small networks, adding short cycles improved networks' capacity, outperforming acyclic or reachability-matched controls. Thus, our results identify local cycles as design principles linking neural connectivity to computational power, and offer a general framework to explore structure-function relations in computing networks.

Tom Talpir, Elad Schneidman

Department of Brain Sciences, Weizmann Institute of Science, Rehovot, Israel

Recurrent Neural Networks Neural Network Expressivity

Jun 22, 2026

Recursive QLSTM with Dynamic Variational Quantum Circuit Adaptation

Recent advances in quantum computing and machine learning have motivated the development of quantum models for sequential data processing. In this paper, we propose a Recursive Quantum Long Short-Term Memory model, or Recursive QLSTM, which extends QLSTM through metacore-based recursive constructions. We numerically test the model under different input sequence lengths, metacore designs, and recursive rules, and identify the best-performing architecture among these variants. For this selected model, we further provide theoretical arguments explaining why its recursive structure improves temporal information propagation and enhances learning performance. Our results suggest that Recursive QLSTM offers a flexible and effective framework for quantum recurrent learning over input time series of various lengths.

Samuel Yen-Chi Chen, Yifeng Peng, Jiun-Cheng Jiang +8

Wells Fargo · Stevens Institute of Technology · National Taiwan University +3

Quantum Machine Learning Recurrent Neural Networks Variational Quantum Circuits Quantum Neural Networks

Jun 21, 2026

FLYNN: Robust Neural Network for Robot Navigation using Fly Brain Topology

While deep learning models achieve state-of-the-art performance in complex tasks, they remain brittle when faced with new environments or sensory deprivation. In contrast, biological systems exhibit remarkable tolerance to these challenges. We address this vulnerability by developing a recurrent neural network (RNN) whose architecture is directly derived from the synaptic-resolution brain connectome of the fruit fly Drosophila melanogaster. We demonstrate the feasibility of training the fly connectome neural network (FLYNN) to perform vision-based navigation in MuJoCo, achieving performance comparable to modern hand-crafted networks of similar parameter counts. Crucially, FLYNN exhibits superior resistance to out-of-distribution (OOD) data and tolerance to sensory loss without further training. It remained functional even under total vision loss while hand-crafted networks largely failed, even when specifically trained with camera dropout. Principal Component Analysis (PCA) of the internal state of FLYNN suggests that it exhibits a particularly high degree of representational modularity, which might be related to its robustness. Our work provides a new direction for designing resilient artificial agents following the topology of biological brains.

Benquan Wang, Jingdao Chen

Department of Computer Science Engineering, Mississippi State University, Mississippi State, MS 39762, USA

Robot Navigation Neural Network Robustness OOD Generalization Recurrent Neural Networks

Jun 19, 2026

Soliton-like Waves in a Two-Dimensional Recurrent Spiking Neural Network with Weighted Spike-Timing-Dependent Plasticity

We construct a minimal but biologically plausible spiking neuron model operating in discrete time, combining multiplicative spike-timing-dependent plasticity (WSTDP), divisive normalization of synaptic integration, homeostatic threshold adaptation, and a one-step refractory period. We show that this normalization admits a biologically plausible dendritic implementation in which each binary junction operates using only locally available information. Assembling excitatory-inhibitory pairs of such neurons into a two-dimensional recurrent network and applying periodic localized stimulation, we find that the network spontaneously gives rise to stable, self-propagating wave packets with the properties of dissipative solitons: they maintain a stable spatial profile, propagate at constant speed, and annihilate upon frontal collision. Their emergence requires a geometric asymmetry between excitatory and inhibitory connection radii, and initial inhibitory synapses stronger than excitatory ones. WSTDP engraves the direction of propagation into the synaptic weight profile, so that the network learns by itself to sustain propagation in one direction while suppressing the reverse. When two sources are active simultaneously, the resulting waves annihilate upon collision, defining a semi-persistent boundary whose position encodes the relative phase and frequency of the two sources. These results provide a minimal computational framework for studying the emergence of cortical traveling waves, activity zone delimitation, and spatial memory from local plasticity rules alone.

Ch. Meessen

Independent researcher, computational neuroscience, France

Dynamical Systems Recurrent Neural Networks Spiking Neural Networks Spike-Timing-Dependent Plasticity

Jun 18, 2026

Recurrent neural networks approximate continuous functions

Classical approximation theorems ask for a new neural network whenever the target accuracy is improved. This paper studies the opposite possibility: can the network be chosen once and for all, and can accuracy be bought only by letting it run longer? We prove that this is possible for every continuous function on [-1,1]. More precisely, each such function is uniformly approximated by the time evolution of a single ReLU recurrent neural network with fixed weights and fixed hidden dimension. The mechanism behind the construction is a new intermediate model, the Turing machine with neural units (TMNU). This model retains the algorithmic freedom needed to implement polynomial approximation schemes, while remaining rigid enough to be simulated by RNNs with explicit bounds on hidden dimension and weight magnitude. The resulting convergence rates reflect the underlying polynomial approximation rates. We complement the construction with minimax lower bounds showing that runtime is not merely a proof artifact, but an unavoidable resource in this fixed-network approximation paradigm.

Valentin Abadie, Clemens Hutter, Helmut Bölcskei

ReLU Neural Networks Recurrent Neural Networks Neural Network Approximation Theory Function Approximation

Jun 18, 2026

Linear Recurrent Unit with Semantic Modulation for Image Super-Resolution

Linear recurrent unit (LRU), designed with a principled formulation for stable linear recurrence, has demonstrated promising accuracy and robustness on long-range dependency tasks. However, its static parameterization and single-scan method limits its applicability to 2D vision tasks. In this study, we propose a LRU-based restoration network with a semantic modulating unit (SMU) to achieve a harmonious balance between performance and efficiency in single-image super-resolution. The SMU plays three key roles: LRU modulation, spatial categorization, and feature enhancement through learned prototype. Extensive experiments demonstrate that our method quantitatively and qualitatively surpasses recent state-of-the-art methods. Notably, our approach achieves superior performance with computational complexity on par with existing methods. The source code and models are available at https://github.com/MingyuChoi-run/LSM

Mingyu Choi, Woo Kyoung Han, Sunghoon Im +1

Korea University · DGIST

Recurrent Neural Networks Image Super-Resolution

Jun 17, 2026

Urdu Katib Handwritten Dataset: A Historical Document Dataset for Offline Urdu Handwritten Text Recognition with CRNN-Based Baseline Evaluation

Automatic Handwritten Text Recognition (HTR) is inherently a challenging task, and its complexity is further increased when dealing with cursive scripts. Although significant efforts have been made on various cursive scripts, research regarding Urdu Handwritten Text Recognition (UHTR) has been relatively limited. This lag of research is primarily due to the unique challenges posed by its script, and the scarcity and unavailability of benchmark datasets. Therefore, to advance research in UHTR, this study presents a specialized real dataset called the Urdu Katib Handwritten Dataset (UKHD). To the best of our knowledge, this is the first offline Urdu handwritten text lines dataset specifically curated from the materials written by Katibs in historical times. It encompasses a diverse range of flat nib writing variations in the Nastalique calligraphic style. Additionally, the effectiveness of different CRNN-based hybrid models has been evaluated to identify the optimal architecture for Urdu Katib Handwriting Recognition (UKHR). Among the analyzed models, the CNN-BGRU-CTC model showed more robust performance, with low Character Error Rate (CER) and Word Error Rate (WER). This research work aims to support and encourage the research community in developing a robust recognition system for preserving Urdu handwritten literature.

Ramza Basharat, Muhammad Usman Ali

Department of Computer Science, University of Gujrat, Pakistan · Department of Computer Science, University of Gujrat, Gujrat, Punjab, Pakistan

Convolutional RNNs Recurrent Neural Networks Online Handwriting Recognition Optical Character Recognition

Jun 17, 2026

Explicit Interaction Architectures for Dynamical Learning: A Controlled Study of Structural Inductive Bias

We investigate a structure-first approach to dynamical learning in which the organization of stateful interactions is prescribed explicitly rather than left entirely to a generic recurrent parameterization. We introduce causal recurrent units built from an ordered sequence of local, state-modulated transformations. The construction is motivated by wave-based interaction models, but the units studied here do not impose scattering, passivity, or energy-balance constraints. Because fixed recurrent dynamics, designed reservoir topologies, readout-only learning, and recurrent depth are already well established, the empirical question is deliberately narrower: does the proposed interaction organization provide a useful inductive bias under controlled computational conditions? We compare a one-layer structured model, a two-layer structured model, and a generic echo-state network (ESN), all with 12 recurrent states and the same strictly linear ridge readout. Each model family receives the same random-search budget on calibration data that are disjoint from the final test data, after which the selected hyperparameters are frozen. On a custom nonlinear identification task, the one-layer structured model attains a mean validation NMSE of 2.76 x 10^{-4}, compared with 3.19 x 10^{-4} for the two-layer model and 3.94 x 10^{-4} for the ESN. On NARMA10 the ordering reverses: the ESN attains 0.312, compared with 0.348 and 0.357 for the one- and two-layer structured models. Thus, the proposed organization can be competitive and advantageous on one task, but it is not universally superior; moreover, recurrent depth does not provide a systematic benefit under matched state dimension. The results support a task-dependent interpretation of structural inductive bias and position the present architecture as a controlled precursor to stronger wave- and system-theoretic constructions.

Augusto Sarti

Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Piazza L. Da Vinci 32, 20133 Italy.

Dynamic Neural Networks Nonlinear System Identification Recurrent Neural Networks

Jun 16, 2026

Task-Restricted Symmetries in Recurrent Weight Space

Recurrent networks can contain substantial functional redundancy in weight space: changing a recurrent matrix may leave the input-output rollout nearly unchanged on a task distribution, while similar-scale changes can destroy the same behavior. We study this redundancy in one-layer tanh RNNs using ordered real Schur coordinates. The Schur form separates spectral blocks from directed nonnormal couplings, giving a diagnostic basis for structured ablations that keep the input and readout maps fixed. In a fixed-length copy task, selected nonnormal Schur couplings can be removed with little loss in some trained solutions, whereas other couplings are necessary for accurate autonomous replay. Across flip-flop, sine generation, and context-dependent integration, the loss-preserving ablation profile varies across tasks and trained solutions. These results identify candidate approximate functional invariances, not universal symmetries of recurrent weight space. Schur-coordinate ablations provide a practical diagnostic for which structured perturbations preserve a trained recurrent solution and which ones disrupt its computation.

Simon Dräger

Salk Institute for Biological Studies, La Jolla, CA, USA.

Recurrent Neural Networks

Jun 16, 2026

Why SWAVE May Not Be All You Need:A Concept-Evolution Retrospective on Complex-Valued Recurrent Language Models

SWave is a complex-valued recurrent language model (169.26M parameters, D=384, L=16, T=2048) trained on FineWeb-Edu using 2xH100 NVL. It was designed around three founding premises: that representing language as complex waves rather than real-valued numbers enables richer information encoding; that a Cayley-parameterised unitary transition provides a mathematical guarantee against state decay or explosion; and that a hidden state which rotates rather than shrinks preserves signal integrity over arbitrarily long contexts. The core of SWave evolved substantially across three development phases. The Resonance Head was found to structurally admit imaginary-channel collapse as a global loss minimum (a failure mode we term cos-domination collapse) and was superseded by an untied head with independent real and imaginary embedding tables from the Phase-Associative Memory (PAM) architecture. This resolved the degenerate minimum and enabled stable 200,000-step training (best-step PPL 22.0 at step 89,861). ComplexNorm and the Wave Propagation Scan proved load-bearing throughout all three phases and were retained to the final architecture. ProtectGatedScan was reframed as a structural prior rather than a learned behaviour. The four multi-scale retention concepts showed no measurable improvement under controlled evaluation and were found non-load-bearing. The ComplexGatedUnit was superseded by a real-valued squared-ReLU channel mixer with fewer parameters. The auxiliary training objectives showed no benefit once structural constraints were resolved. The investigation yields a formal characterisation of cos-domination collapse, a parallel scan with a log-space backward pass for numerical stability, six transferable engineering principles for complex-valued recurrent training, and a plan-to-code traceability methodology for catching structural divergences that conventional test suites miss.

Ramprasath Ganesaraja, Swathika N, Sahil Dilip Panse

EdgeVerve Systems Limited

Recurrent Neural Networks Complex-Valued Neural Networks Autoregressive Language Modeling

Jun 15, 2026

From Compression to Deployment: Real-Time and Energy-Efficient FastGRNN on Ultra-Constrained Microcontrollers

The dominant trajectory of modern machine learning has been to scale up: larger models, larger accelerators, larger memory budgets. Yet a multi-year global semiconductor supply constraint and the growing energy and carbon cost of always-online inference expose the fragility of this trajectory and motivate the opposite direction: refactoring AI and ML algorithms to fit the small, ubiquitous microcontrollers already in mass production in wearables, sensors, and edge appliances. We present an end-to-end open-source reproduction of FastGRNN, a compact gated recurrent cell, deployed on two bare-metal targets: the 8-bit Arduino (ATmega328P) and the 16-bit MSP430 (no hardware multiplier; 16 KB Flash; 512 B SRAM). Our compression pipeline combines low-rank weight factorization, iterative hard-thresholding sparsity, and per-tensor Q15 post-training quantization with explicit activation calibration. The deployed model occupies 566 bytes of weights and achieves macro F1 = 0.918 (seed 0; five-seed Q15 mean 0.853+-0.107) on the HAPT test set. It matches a PyTorch reference at 100% prediction agreement across 3,399 test windows (MCU seed 0; 99.91-100% C-equivalent across five seeds). Both platforms sustain real-time 50 Hz streaming inference (9.21 ms per sample on Arduino; 13 ms on MSP430), where a 256-entry sigmoid/tanh look-up table delivers a 30.5x speedup on the multiplier-less MSP430. Four contributions extend the original FastGRNN paper: (i) cross-platform bit-equivalent deterministic inference; (ii) characterization of recurrent warm-up latency (median 74 samples, 1.48 s; worst-case 125 samples, 2.50 s over 100 test windows); (iii) a deployable look-up-table recipe for multiplier-less embedded targets; and (iv) hardware energy characterization showing 17.7 mW active inference power, <0.09 mW idle power, and 96.7% energy reduction with the LUT.

Emre Can Kizilates

Independent Researcher, Izmir, Turkey

Gated Recurrent Unit Efficient Neural Network Inference Recurrent Neural Networks Neural Network Compression On-Device Inference

Jun 15, 2026

When the Past Matters: FlashBack Memory for Precipitation Nowcasting

Accurate precipitation nowcasting is crucial for disaster mitigation and socio-economic planning, yet existing methods often struggle with false alarms, missed events, and long range dependency modeling at high spatiotemporal resolution. To address these challenges, we propose FlashBack Memory (FB), a module that dynamically retrieves key historical states and integrates them via an adaptive fusion gate, enhancing the spatiotemporal representation capability of recurrent-based models. We incorporate FB into PredRNN, PredRNNpp, MIM, MotionRNN, and PredRNN-V2, and evaluate on CIKM2017, Shanghai2020, and SEVIR datasets. Experimental results demonstrate that FB significantly improves MSE, MAE, SSIM, and CSI metrics, particularly for high-intensity rainfall and long-sequence predictions, while reducing false alarms and missed events and enhancing temporal consistency and spatial localization. The proposed method provides a general and efficient memory enhancement mechanism, improving the overall performance of recurrent-based precipitation nowcasting models.

Yuhao Du, Boxiao Huang, Chengrong Wu +1

College of Atmospheric Sciences, Lanzhou University, Lanzhou 730000, China · Fuqua School of Business, Duke University, North Carolina 27705, USA · Department of Computer Science, University of Manchester, Manchester M139PL, UK

Memory-Augmented Neural Networks Precipitation Forecasting Recurrent Neural Networks Spatiotemporal Forecasting Precipitation Nowcasting

Jun 12, 2026

Harnessing cortical geometry, wiring, and function as inductive biases for recurrent neural networks

How the wiring and functional organization of cortex shape recurrent computation remains a central question in both neuroscience and machine learning. Here, we leverage data released through the Machine Intelligence from Cortical Networks (MICrONS) program--a functional connectomics resource spanning multiple areas of mouse visual cortex, in which dense calcium imaging is co-registered with high-resolution electron microscopy reconstruction from the same animal--to build biologically grounded recurrent neural networks. Using neuronal spatial coordinates, anatomical connectivity, and function-derived relationships from nearly 12,000 coregistered excitatory neurons, we initialize recurrent weights and impose communication-aware spatial constraints during learning. Across three cognitive decision-making tasks, networks constrained by cortical structure and function consistently outperform baseline and partially constrained models. Functional weight initialization provides the largest gain, while real spatial embedding yields robust additional improvements across conditions. These biologically grounded networks also develop low-entropy, modular, and small-world organization, and retain strong performance even when recurrence is restricted to positive weights. Together, our results show that the machinery of cortex--its geometry, wiring, and functional structure--can be harnessed as a powerful inductive basis for building recurrent networks that learn more effectively while converging toward key organizational principles of biological computation.

Mo Shakiba, Rana Rokni, Mohammad Mohammadi +1

Neuromatch Academy, Neuromatch, Inc., USA · McGovern Institute for Brain Research, Massachusetts Institute of Technology (MIT)

Recurrent Neural Networks

Jun 12, 2026

Dense Supervision Is Not Enough: The Readout Blind Spot in Looped Language Models

Looped language models turn hidden states into runtime state: each state is decoded for prediction and fed back into future computation. This creates a basic supervision question: which state variables does cross-entropy actually control? We show that dense per-loop cross-entropy controls the variables exposed by the readout, not every variable active in the recurrent transition. Hidden-state scale gives a concrete failure mode. Scale-invariant readouts such as RMSNorm and LayerNorm hide radial scale from the immediate cross-entropy loss, while pre-norm residual recurrence continues to carry and update that same scale. Thus per-loop loss can make early exits usable without controlling recurrent scale. In 44M and 129M looped transformers without inter-loop normalization, per-loop cross-entropy through RMSNorm readouts still drives final hidden-state norms into the thousands or tens of thousands. Scale-visible readouts and explicit norm penalties keep norms in the tens, and scale-removing recurrence is the complementary architectural fix. The resulting design rule is simple: dense supervision trains exits; recurrent scale control requires either making scale visible to a loss or removing it from the loop. Consistent with this rule, scale-controlled variants achieve lower perplexity at matched inference-depth operating points in our variable-depth benchmarks.

Rituraj Sharma, Tu Vu

Virginia Tech

Recurrent Neural Networks Recurrent Transformers Language Modeling

Jun 10, 2026

Robustness Verification of Recurrent Neural Networks with Abstraction Refinement

Certified local robustness verification for recurrent neural networks (RNNs) is challenging because approximation errors introduced by nonlinear relaxations can propagate through recurrent connections and accumulate over time. As a result, scalable linear bound propagation methods often become overly conservative and fail to certify inputs that are in fact robust, especially when many pre-activation intervals cross zero. We propose an abstraction-refinement framework for RNN verification that partitions such intervals to remove the dominant relaxation error: on each refined branch, ReLU becomes exact, and smooth activations such as tanh and sigmoid admit substantially tighter linear envelopes. To control the combinatorial cost of splitting in long sequences, we introduce a SHAP-guided timestep selection strategy that ranks hidden states by their contribution to the verification objective and refines only the most critical timesteps in temporal order. Experiments on CIFAR10 and MNIST stroke benchmarks demonstrate consistent improvements in verification success and robustness-margin tightness over abstraction-only baselines, while exposing clear runtime trade-offs between ReLU and tanh models.

Li-Jen Lin, Chih-Duo Hong

Dept. of Management Information Systems National Chengchi University Taipei, Taiwan

Neural Network Verification Neural Network Robustness Recurrent Neural Networks Certified Robustness

Jun 10, 2026

Masked Neural Detection for Constrained Channel Coding in Molecular Communication

Molecular communication (MC) suffers from severe diffusion memory because molecules released for one symbol may arrive during later symbols. Neural sequence detectors, especially sliding bidirectional recurrent neural networks (SBRNNs), can substantially outperform threshold detectors in such channels. This raises a central question for MC channel coding: does a code whose advantage was established under threshold detection retain it when both coded and uncoded transmission are evaluated with neural detection? This letter answers this question for run-length-limited ISI-mitigation (RLIM) codes, a class of constrained codes previously shown to provide large BER gains in MC. Across the tested operating points, the best RLIM-SBRNN receiver beats the best uncoded receiver, chosen between threshold and SBRNN detection, in

46

of

59

cases, with a mean gain of

10.36\times

over those wins. We also propose an RLIM-tailored training mask for compact SBRNN detectors, improving the unmasked RLIM-SBRNN in

227

of

236

comparisons with

3.267\times

mean gain when masking is beneficial. Finally, the compact masked RLIM-SBRNN is competitive with channel-state-aware MLSE despite using no channel knowledge.

Melih Şahin, Ozgur B. Akan

Centre for neXt Communications (CXC), Department of Engineering, University of Cambridge, CB3 0FA Cambridge, U.K. · Centre for neXt Communications (CXC), Department of Electrical and Electronics Engineering, Koç University, 34450 Istanbul, Türkiye

Neural Decoding Recurrent Neural Networks

Jun 9, 2026

LSTM-Based Detection of Structural Breaks in Property Insurance Loss Reserving: A Climate-Informed Approach

Accurate loss reserving is foundational to insurer solvency, yet accelerating climate driven catastrophes systematically violate the stability assumptions on which traditional actuarial methods depend. This white paper presents a research program testing whether Long Short Term Memory (LSTM) neural networks can detect and adapt to these structural breaks faster and more accurately than Chain Ladder, Bornhuetter Ferguson, and Cape Cod methods. Using 15 plus years of regulatory development triangle data from Florida and Louisiana, enriched with NOAA hurricane intensity indices and sea surface temperatures, we hypothesize a targeted improvement of 15, 20% in reserve accuracy for catastrophe exposed years, a threshold grounded both in the prior neural network reserving literature and in the formal convergence results developed here. Beyond empirical validation, we develop a theoretical framework grounding LSTM structural break detection in probabilistic terms, providing formal performance guarantees that compensate for the limited number of catastrophe events in the test period. We document the research design, methodology, expected contributions, and a candid assessment of limitations.

Thomas Mbrice, Shashwat Panigrahi

Stony Brook University · Department of Computer Science

Recurrent Neural Networks Change-Point Detection

Jun 9, 2026

Towards Critical Branching Mechanism in Recurrent Neural Networks

Criticality has been proposed as a key organizing principle in biological neural systems, yet its origin and relevance in artificial neural networks remain unclear. We analyze hidden-state dynamics in trained long short-term memory (LSTM) networks and show that small networks near their optimal training epochs (steps) exhibit scale-free avalanche statistics and branching parameters close to unity, indicative of near-critical dynamics, while larger models remain subcritical. To explain the coexistence of subcritical branching with robust

1/f^β

noise, we introduce a mixture branching process framework that links heterogeneous branching dynamics to long-range temporal correlations. These results identify critical-like behavior in LSTMs as an emergent, capacity-dependent dynamical regime.

Feixiang Ren, Ling Feng

Department of Physics, National University of Singapore Singapore · Institute of High Performance Computing Agency for Science, Technology and Research (A*STAR) Singapore

Recurrent Neural Networks Long Short-Term Memory Networks Neural Network Training Dynamics