Neural Algorithmic Reasoning

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Neural algorithmic reasoning asks how neural networks can learn the steps of algorithms and apply them to inputs or problem sizes not seen in training. Common approaches train on algorithm traces, align graph neural networks with message-passing algorithms, or use recurrent and looped models that perform more computation at test time; some methods encode symbolic rules or explicit memory directly. Researchers evaluate these systems on tasks such as CLRS graph algorithms, Sudoku, mazes, arithmetic, and graph optimization, often testing generalization to longer sequences, larger graphs, or deeper reasoning. A central challenge is ensuring that models truly execute and retain useful intermediate computations rather than relying on shortcuts, since accurate answers or decodable representations alone do not guarantee robust reasoning at greater depth.

Latest papers

45 papers14 papers in the last four weeks, up 367% on the four weeks before. 0.1% of all new papers.

Apr 9, 2026

Loom: A Scalable Analytical Neural Computer Architecture

We present Loom, a computer architecture that executes programs compiled from C inside a looped transformer whose weights are derived analytically. The architecture implements a 22-opcode instruction set in 8 transformer layers. Each forward pass executes one instruction; the model is applied iteratively until the program counter reaches zero. The full machine state resides in a single tensor

X \in \mathbb{R}^{d \times n}

of fixed size, and every step has fixed cost for fixed

d

and

n

, independent of program length or execution history. The default configuration uses

d = 155

and

n = 1024

, yielding 4.7 million parameters and 928 instruction slots. A compact configuration at

d = 146

and

n = 512

suffices for a 9

\times

9 Sudoku solver (284 instructions). The weights are program-independent: programs live in the state tensor, and the same fixed-weight model executes any compiled program. We make Loom source code publicly available at https://github.com/mkturkcan/Loom.

Mehmet Kerem Turkcan

Columbia University

Neural Algorithmic Reasoning Recurrent Transformers

Mar 31, 2026

Metriplector: From Field Theory to Neural Architecture

We present Metriplector, a neural architecture primitive in which the input configures an abstract physical system -- fields, sources, and operators -- and the dynamics of that system is the computation. Multiple fields evolve via coupled metriplectic dynamics, and the stress-energy tensor T^{μν}, derived from Noether's theorem, provides the readout. The metriplectic formulation admits a natural spectrum of instantiations: the dissipative branch alone yields a screened Poisson equation solved exactly via conjugate gradient; activating the full structure -- including the antisymmetric Poisson bracket -- gives field dynamics for image recognition, language modeling, and robotic control. We evaluate Metriplector across five domains, each using a task-specific architecture built from this shared primitive with progressively richer physics: 81.03% on CIFAR-100 with 2.26M parameters; 88% CEM success on Reacher robotic control with under 1M parameters; 97.2% exact Sudoku solve rate with zero structural injection; 1.182 bits/byte on language modeling with 3.6x fewer training tokens than a GPT baseline; and F1=1.0 on maze pathfinding, generalizing from 15x15 training grids to unseen 39x39 grids.

Dan Oprisa, Peter Toth

Spheroid Labs∗

Neural Algorithmic Reasoning Language Modeling

Mar 30, 2026

The Long Delay to Arithmetic Generalization: When Learned Representations Outrun Behavior

Grokking in transformers trained on algorithmic tasks is characterized by a long delay between training-set fit and abrupt generalization, but the source of that delay remains poorly understood. In encoder-decoder arithmetic models, we argue that this delay reflects limited access to already learned structure rather than failure to acquire that structure in the first place. We study one-step Collatz prediction and find that the encoder organizes parity and residue structure within the first few thousand training steps, while output accuracy remains near chance for tens of thousands more. Causal interventions support the decoder bottleneck hypothesis. Transplanting a trained encoder into a fresh model accelerates grokking by 2.75 times, while transplanting a trained decoder actively hurts. Freezing a converged encoder and retraining only the decoder eliminates the plateau entirely and yields 97.6% accuracy, compared to 86.1% for joint training. What makes the decoder's job harder or easier depends on numeral representation. Across 15 bases, those whose factorization aligns with the Collatz map's arithmetic (e.g., base 24) reach 99.8% accuracy, while binary fails completely because its representations collapse and never recover. The choice of base acts as an inductive bias that controls how much local digit structure the decoder can exploit, producing large differences in learnability from the same underlying task.

Laura Gomezjurado Gonzalez

Stanford University

Grokking Neural Algorithmic Reasoning Mathematical Reasoning

Mar 24, 2026

Dynamical Systems Theory Behind a Hierarchical Reasoning Model

Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkably, even when aggressively compressed to an ultra-tiny footprint of just 0.26M parameters, the CMM retains robust predictive power, achieving 85.4 % on Sudoku-Extreme and 82.2 % on the Maze benchmark. These results establish a new frontier for extreme parameter efficiency, proving that mathematically rigorous latent dynamics can effectively replace brute-force scaling in artificial reasoning.

Vasiliy A. Es'kin, Mikhail E. Smorkalov

Department of Radiophysics, University of Nizhny Novgorod, Nizhny Novgorod 603950, Russia · Huawei Nizhny Novgorod Research Center, Nizhny Novgorod 603006, Russia · Skolkovo Institute of Science and Technology, Moscow 121205, Russia

Sudoku Solving Neural Algorithmic Reasoning Algorithmic Reasoning Dynamical Systems Neural ODEs

Mar 5, 2026

Recursive Inference Machines for Neural Reasoning

Neural reasoners such as Tiny Recursive Models (TRMs) solve complex problems by combining neural backbones with specialized inference schemes. Such inference schemes have been a central component of stochastic reasoning systems, where inference rules are applied to a stochastic model to derive answers to complex queries. In this work, we bridge these two paradigms by introducing Recursive Inference Machines (RIMs), a neural reasoning framework that explicitly incorporates recursive inference mechanisms inspired by classical inference engines. We show that TRMs can be expressed as an instance of RIMs, allowing us to extend them through a reweighting component, yielding better performance on challenging reasoning benchmarks, including ARC-AGI-1, ARC-AGI-2, and Sudoku Extreme. Furthermore, we show that RIMs can be used to improve reasoning on other tasks, such as the classification of tabular data, outperforming TabPFNs.

Mieszko Komisarczyk, Saurabh Mathur, Maurice Kraus +2

Department of Computer Science, Technical University of Darmstadt, Germany · Department of Computer Science, The University of Texas at Dallas, USA · Hessian Center for Artificial Intelligence (hessian.ai), Darmstadt, Germany +1

Abstract Reasoning Neural Algorithmic Reasoning

Mar 4, 2026

Modular Neural Computer

This paper introduces the Modular Neural Computer (MNC), a memory-augmented neural architecture for exact algorithmic computation on variable-length inputs. The model combines an external associative memory of scalar cells, explicit read and write heads, a controller multi-layer perceptron (MLP), and a homogeneous set of functional MLP modules. Rather than learning an algorithm end to end from data, it realizes a given algorithm through analytically specified neural components with fixed interfaces and exact behavior. The control flow is represented inside the neural computation through one-hot module gates, where inactive modules are inhibited. Computation unfolds as a sequence of memory transformations generated by a fixed graph. The architecture is illustrated through three case studies: computing the minimum of an array, sorting an array in place, and executing A* search on a fixed problem instance. These examples show that algorithmic procedures can be compiled into modular neural components with external memory while preserving deterministic behavior and explicit intermediate state.

Florin Leon

Department of Computers Faculty of Automatic Control and Computer Engineering “Gheorghe Asachi” Technical University of Iași, Romania

Neural Algorithmic Reasoning Memory-Augmented Neural Networks Modular Neural Networks

Mar 4, 2026

Continuous Modal Logical Neural Networks: Modal Reasoning via Stochastic Accessibility

We propose Fluid Logic, a paradigm in which modal logical reasoning, temporal, epistemic, doxastic, deontic, is lifted from discrete Kripke structures to continuous manifolds via Neural Stochastic Differential Equations (Neural SDEs). Each type of modal operator is backed by a dedicated Neural SDE, and nested formulas compose these SDEs in a single differentiable graph. A key instantiation is Logic-Informed Neural Networks (LINNs): analogous to Physics-Informed Neural Networks (PINNs), LINNs embed modal logical formulas such as (

\Box

bounded) and (

\Diamond

visits_lobe) directly into the training loss, guiding neural networks to produce solutions that are structurally consistent with prescribed logical properties, without requiring knowledge of the governing equations. The resulting framework, Continuous Modal Logical Neural Networks (CMLNNs), yields several key properties: (i) stochastic diffusion prevents quantifier collapse (

\Box

and

\Diamond

differ), unlike deterministic ODEs; (ii) modal operators are entropic risk measures, sound with respect to risk-based semantics with explicit Monte Carlo concentration guarantees; (iii)SDE-induced accessibility provides structural correspondence with classical modal axioms; (iv) parameterizing accessibility through dynamics reduces memory from quadratic in world count to linear in parameters. Three case studies demonstrate that Fluid Logic and LINNs can guide neural networks to produce consistent solutions across diverse domains: epistemic/doxastic logic (multi-robot hallucination detection), temporal logic (recovering the Lorenz attractor geometry from logical constraints alone), and deontic logic (learning safe confinement dynamics from a logical specification).

Antonin Sulc

Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Logical Reasoning Neuro-Symbolic Reasoning Neural Algorithmic Reasoning Epistemic Logic

Mar 3, 2026

Form Follows Function: Recursive Stem Model

Recursive reasoning models such as Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM) show that small, weight-shared networks can solve compute-heavy and NP puzzles by iteratively refining latent states, but their training typically relies on deep supervision and/or long unrolls that increase wall-clock cost and can bias the model toward greedy intermediate behavior. We introduce Recursive Stem Model (RSM), a recursive reasoning approach that keeps the TRM-style backbone while changing the training contract so the network learns a stable, depth-agnostic transition operator. RSM fully detaches the hidden-state history during training, treats early iterations as detached "warm-up" steps, and applies loss only at the final step. We further grow the outer recursion depth

H

and inner compute depth

L

independently and use a stochastic outer-transition scheme (stochastic depth over

H

) to mitigate instability when increasing depth. This yields two key capabilities: (i)

>20\times

faster training than TRM while improving accuracy (

\approx 5\times

reduction in error rate), and (ii) test-time scaling where inference can run for arbitrarily many refinement steps (

\sim 20,000 H_{\text{test}} \gg 20 H_{\text{train}}

), enabling additional "thinking" without retraining. On Sudoku-Extreme, RSM reaches 97.5% exact accuracy with test-time compute (within ~1 hour of training on a single A100), and on Maze-Hard (

30 \times 30

) it reaches ~80% exact accuracy in ~40 minutes using attention-based instantiation. Finally, because RSM implements an iterative settling process, convergence behavior provides a simple, architecture-native reliability signal: non-settling trajectories warn that the model has not reached a viable solution and can be a guard against hallucination, while stable fixed points can be paired with domain verifiers for practical correctness checks.

Navid Hakimi

Recursive Reasoning Models Neural Algorithmic Reasoning Algorithmic Reasoning Test-Time Scaling

Mar 2, 2026

Symbol-Equivariant Recurrent Reasoning Models

Reasoning problems such as Sudoku and ARC-AGI remain challenging for neural networks. The structured problem solving architecture family of Recurrent Reasoning Models (RRMs), including Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), offer a compact alternative to large language models, but currently handle symbol symmetries only implicitly via costly data augmentation. We introduce Symbol-Equivariant Recurrent Reasoning Models (SE-RRMs), which enforce permutation equivariance at the architectural level through symbol-equivariant layers, guaranteeing identical solutions under symbol or color permutations. SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize from just training on 9x9 to smaller 4x4 and larger 16x16 and 25x25 instances, to which existing RRMs cannot extrapolate. On ARC-AGI-1 and ARC-AGI-2, SE-RRMs achieve competitive performance with substantially less data augmentation and only 2 million parameters, demonstrating that explicitly encoding symmetry improves the robustness and scalability of neural reasoning. Code is available at https://github.com/ml-jku/SE-RRM.

Richard Freinschlag, Timo Bertram, Erich Kobler +2

ELLIS Unit Linz, LIT AI Lab & Institute for Machine Learning, Johannes Kepler University Linz, Austria · Clinical Research Institute for Medical AI, Johannes Kepler University Linz, Austria

Permutation-Equivariant Neural Networks Sudoku Solving Neural Algorithmic Reasoning Algorithmic Reasoning Recurrent Neural Networks Equivariant Neural Networks+1

Feb 24, 2026

MINAR: Mechanistic Interpretability for Neural Algorithmic Reasoning

The recent field of neural algorithmic reasoning (NAR) studies the ability of graph neural networks (GNNs) to emulate classical algorithms like Bellman-Ford, a phenomenon known as algorithmic alignment. At the same time, recent advances in large language models (LLMs) have spawned the study of mechanistic interpretability, which aims to identify granular model components like circuits that perform specific computations. In this work, we introduce Mechanistic Interpretability for Neural Algorithmic Reasoning (MINAR), an efficient circuit discovery toolbox that adapts attribution patching methods from mechanistic interpretability to the GNN setting. We show through two case studies that MINAR recovers faithful neuron-level circuits from GNNs trained on algorithmic tasks. Our study sheds new light on the process of circuit formation and pruning during training, as well as giving new insight into how GNNs trained to perform multiple tasks in parallel reuse circuit components for related tasks. Our code is available at https://github.com/pnnl/MINAR.

Jesse He, Helen Jenne, Max Vargas +4

Pacific Northwest National Laboratory, Richland, WA · Halıcı˘oğlu Data Science Institute, University of California, San Diego, San Diego, CA · Department of Computer and Information Science, University of Pennsylvania, Pennsylvaina, PA +1

Neural Algorithmic Reasoning Mechanistic Interpretability GNN Explainability Graph Neural Networks Activation Patching

Feb 18, 2026

Parallelizable Neural Turing Machines

We introduce a parallelizable simplification of Neural Turing Machine (NTM), referred to as P-NTM, which redesigns the core operations of the original architecture to enable efficient scan-based parallel execution. We evaluate the proposed architecture on a synthetic benchmark of algorithmic problems involving state tracking, memorization, and basic arithmetic, solved via autoregressive decoding. We compare it against a revisited stable implementation of the standard NTM, as well as conventional recurrent and attention-based architectures. Results show that, despite its simplifications, the proposed model attains length generalization performance comparable to the original, learning to solve all problems, including unseen sequence lengths, with perfect accuracy. It also improves training efficiency, with parallel execution of P-NTM being up to an order of magnitude faster than the standard NTM. Ultimately, this work contributes toward the development of efficient neural architectures capable of expressing a broad class of algorithms.

Gabriel Faria, Arnaldo Candido Junior

University of São Paulo∗ · ∗Instituto de Ciências Matemáticas e de Computação (ICMC), University of São Paulo. · São Paulo State University

Neural Algorithmic Reasoning Recurrent Neural Networks

Nov 30, 2025

Efficiently Learning Branching Networks for Multitask Algorithmic Reasoning

Algorithmic reasoning -- the ability to perform step-by-step logical inference -- is a synthetic benchmark for evaluating multi-step reasoning abilities, designed for graph neural networks and also for transformer models. Prior work has evaluated reasoning for executing a single algorithmic task, whereas a more desirable objective is to perform multiple algorithmic reasoning tasks simultaneously. We start by noting that this is inherently difficult due to differences arising from the execution traces of the algorithms (such as depth- vs. breadth-first search), which cause interference when they are trained together. In this paper, we introduce {branching neural networks}, a new architecture for multitask algorithmic reasoning. The main idea is to search for a recursive tree-structured partition of

n

algorithmic tasks into a

k

-ary tree (divided into

L

layers). Naive search requires

O(k^{nL})

complexity; we develop an algorithm that reduces this to

O(nL)

by solving a convex relaxation at each layer to approximate an optimal partition. Our approach clusters these tasks using gradient-based affinity and can be used on top of any base model. We validate our approach on algorithmic reasoning benchmarks and their extensions with text descriptions. We show that gradient-based affinity scores help estimate true performance with less than 5% error, measured across eight different architectures with up to 34 billion parameters. On the CLRS benchmark, our approach outperforms existing graph neural networks by 3.7% and baselines by 1.2%, while reducing runtime by 48% and memory usage by 26%. The learned branching structure shows a hierarchical clustering of related algorithms. On three text-based graph reasoning benchmarks, our approach improves over baseline methods by 3.2%. Finally, we validate our approach for overlapping community detection.

Dongyue Li, Zhenshuo Zhang, Minxuan Duan +2

†Northeastern University, Boston, Massachusetts · ‡University of Pennsylvania, Philadelphia, Pennsylvania

Neural Algorithmic Reasoning Multi-Task Learning Algorithmic Reasoning

Sep 23, 2025

Tackling GNARLy Problems: Graph Neural Algorithmic Reasoning Reimagined through Reinforcement Learning

Neural algorithmic reasoning (NAR) is a paradigm that trains neural networks to execute classic algorithms by supervised learning. Despite its successes, important limitations remain: inability to construct valid solutions without post-processing and to reason about multiple correct ones, poor performance on combinatorial NP-hard problems, and inapplicability to problems for which strong algorithms are not yet known. To address these limitations, we reframe the problem of learning algorithm trajectories as a Markov decision process, which imposes structure on the solution construction procedure and unlocks the powerful tools of imitation and reinforcement learning (RL). We propose the GNARL framework, encompassing the methodology to translate problem formulations from NAR to RL and a learning architecture suitable for a wide range of graph-based problems. We achieve high rates of reaching correct states on several CLRS-30 problems and performance matching or exceeding much narrower NAR approaches for NP-hard problems. Remarkably, GNARL remains applicable when no expert algorithm is available, though reward-driven learning can exhibit greater variability than direct supervision.

Alex Schutz, Victor-Alexandru Darvariu, Efimia Panagiotaki +2

Oxford Robotics Institute, University of Oxford · Stateful Robotics

Graph Neural Networks Reinforcement Learning Neural Algorithmic Reasoning RL for Combinatorial Optimization Algorithmic Reasoning Neural Combinatorial Optimization

Feb 11, 2025

Scaling Online Complex Event Detection with Synthetic Supervision and Mamba-Based Neural Algorithmic Reasoning

Modern machine learning models excel at detecting individual actions, sounds, or scene attributes from short, localized observations. However, many real-world tasks, such as in smart cities and healthcare, require reasoning over high-level complex events (CEs): spatiotemporal, rule-governed patterns of short-term atomic events (AEs). Complex event detection (CED) is challenging due to long temporal dependencies, generalization beyond the training horizon, sparse CE-level supervision without temporally aligned fine-grained AE labels, and cognitively demanding annotation, as CE labels often depend on ordering, duration, negation, and completion-time semantics. These challenges are further amplified in an online setting that requires causal, streaming inference with limited computation. We identify the primary bottleneck in online CED as learning robust CE rules, and propose a Neural Algorithmic Reasoning framework that decouples rule learning from low-level sensor semantics by (i) generating large-scale synthetic AE-level concept traces to pretrain a Mamba-based CE-rule reasoner, and (ii) introducing an adapter that learns to map raw sensor inputs into the reasoner's latent space using limited, labeled sensor data. We introduce a controlled simulator-generated online multilabel CED testbed built from real-world multimodal sensor clips and rule-generated CE labels, with stress-test settings that vary sensor noise, distribution shift, and the window size used to segment streaming sensor sequences. Experiments on this controlled benchmark show that NAROCE is competitive with the strongest baselines and often outperforms them under these stress tests and longer-horizon generalization, while using 5x fewer labeled sensor sequences and 10-20x fewer FLOPs than all non-Mamba baselines. Code and dataset available at https://github.com/nesl/naroce_dailyoce.

Liying Han, Gaofeng Dong, Xiaomin Ouyang +4

Department of Electrical and Computer Engineering University of California, Los Angeles · Department of Computer Science and Engineering Hong Kong University of Science and Technology · U.S. Army DEVCOM Army Research Laboratory +1

Neural Algorithmic Reasoning Algorithmic Reasoning Data-Efficient Learning

Oct 4, 2023

Inferring Inference

Patterns of microcircuitry in the cerebral cortex suggest that the brain has an array of repeated canonical computational units. However, neural representations are distributed, so the relevant computations may only be related indirectly to single-neuron transformations. It thus remains an open challenge how to define canonical distributed computations. Here we integrate normative and algorithmic theories of neural computation to present a mathematical framework for inferring canonical distributed computations from large-scale neural activity patterns. At the normative level, we hypothesize that the brain learns a structured internal model of its environment and its latent causes. At the algorithmic level, we propose that this inference process is a nonlinear message-passing algorithm on a graph-structured model of the world. Given a time series of neural activity during a perceptual inference task, our analysis framework finds (i) the neural representation of the relevant latent variables, (ii) interactions between these latent variables that define the brain's internal model of the world, and (iii) message functions that specify the inference algorithm. With enough data, key computational properties are identifiable due to constraints from the symmetries inherent in any canonical computation, up to a joint global transformation of all interactions and the message-passing functions. As a demonstration of this framework, we analyze artificial neural recordings generated by a simulated brain that implicitly implements an approximate inference algorithm. Given simulated inputs and neural activity we successfully recover the latent variables, their neural representation and dynamics, and canonical message functions that govern the dynamics. Analysis of these models reveals features of experimental design required to extract canonical computations from neural data.

Rajkumar Vasudeva Raju, Zhe Li, Scott Linderman +1

Department of ECE, Rice University · Department of Neuroscience, Baylor College of Medicine · Department of Statistics, Stanford University +3

Neural Algorithmic Reasoning System Identification Neural Encoding Neural Decoding