Today's deep learning architectures are primarily based on perceptron models, which do not capture the oscillatory dynamics characteristic of biological neural activity. Although oscillatory systems have recently gained attention for their closer resemblance to neural behavior, they often lack a structured mechanism to represent rich spatio-temporal dynamics in a controllable and interpretable manner. In this paper, we propose a bio-inspired oscillatory state system (BioOSS), a 2D topographically organized oscillatory state-space model designed to generate diverse oscillation-driven spatio-temporal patterns. BioOSS comprises two coupled state components: punits that represent membrane-potential-like variables inspired by pyramidal-cell activity, and o units that act as velocity-like latent states controlling phase, time scales, and damping. The model incorporates trainable parameters for damping and effective oscillation rates, enabling flexible adaptation to task-specific temporal structures while remaining efficient for long-sequence learning via scanfriendly diagonal dynamics. We evaluate BioOSS on both synthetic and real-world tasks, demonstrating superior performance and enhanced interpretability compared to alternative architectures.

Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying space is transformed with the same group element, as opposed to a local gauge symmetry, where each point is transformed with a different group element, exponentially enlarging the size of the symmetry group. We use gauge equivariant networks to predict topological invariants (Chern numbers) of multiband topological insulators for the first time. The gauge symmetry of the network guarantees that the predicted quantity is a topological invariant. A major technical challenge is that the relevant gauge equivariant networks are plagued by instabilities in their training, severely limiting their usefulness. In particular, for larger gauge groups the instabilities make training impossible. We resolve this problem by introducing a novel gauge equivariant normalization layer which stabilizes the training. Furthermore, we prove a universal approximation theorem for our model. We train on samples with trivial Chern number only but show that our model generalizes to samples with non-trivial Chern number and provide various ablations of our setup.

A.1 Dataset Examples450 In this section of the appendix, we present a detailed overview of several representative tasks from451 each category included in REASONINGGYM. For each task, we describe its structure, complexity452 parameters, and provide examples.453 A.1.1 complex_arithmetic(Algebra)454 Find the solution of an arithmetic operation involving complex numbers.455 The spiral order is clockwise, starting from the top-left corner. Predict the corresponding output grid by applying the rule you found.

The physical sciences are replete with dynamical systems that require the resolution of a wide range of length and time scales. This presents significant computational challenges since direct numerical simulation requires discretization at the finest relevant scales, leading to a high-dimensional state space. In this work, we propose an approach to learn stochastic multiscale models in the form of stochastic differential equations directly from observational data. Drawing inspiration from physics-based multiscale modeling approaches, we resolve the macroscale state on a coarse mesh while introducing a microscale latent state to explicitly model unresolved dynamics. We learn the parameters of the multiscale model using a simulator-free amortized variational inference method with a Product of Experts likelihood that enforces scale separation. We present detailed numerical studies to demonstrate that our learned multiscale models achieve superior predictive accuracy compared to under-resolved direct numerical simulation and closure-type models at equivalent resolution, as well as reduced-order modeling approaches.

Want to get a data center online quickly? As the data-center boom puts pressure on the grid, some companies say the answer isn't just more power plants but software that dials down centers' energy-guzzling ways when demand spikes. At the end of a tense and scoreless first half of a soccer match between the English men's team and rival Germany, millions of Brits let out a collective sigh and did what they so often do in moments of stress: They made tea. That wave of electric kettles clicking on, however, caused a different kind of stress: a huge and sudden increase in demand for electricity. But National Grid, which operates the local transmission network, was ready. Just as those kettles started heating up, an AI program sent instructions to a data center in London to slow down some of the facility's power-hungry chips. This reduction helped make sure there was enough supply to match demand, staving off potential blackouts or damage to electrical hardware.

Large Language Models (LLMs), such as OpenAI's o1 and DeepSeek's R1, excel at advanced reasoning tasks like math and coding via Reinforcement Learning with Verifiable Rewards (RLVR), but still struggle with puzzles solvable by humans without domain knowledge. We introduce ENIGMATA, the first comprehensive suite tailored for improving LLMs with puzzle reasoning skills. It includes 36 tasks across 7 categories, each with: 1) a generator that produces unlimited examples with controllable difficulty, and 2) a rule-based verifier for automatic evaluation. This generator-verifier design supports scalable, multi-task RL training, fine-grained analysis, and seamless RLVR integration. We further propose ENIGMATA-Eval, a rigorous benchmark, and develop optimized multi-task RLVR strategies.

In this paper, we propose Tortoise and Hare Guidance (THG), a training-free strategy that accelerates diffusion sampling while maintaining high-fidelity generation. We demonstrate that the noise estimate and the additional guidance term exhibit markedly different sensitivity to numerical error by reformulating the classifier-free guidance (CFG) ODE as a multirate system of ODEs. Our error-bound analysis shows that the additional guidance branch is more robust to approximation, revealing substantial redundancy that conventional solvers fail to exploit. Building on this insight, THG significantly reduces the computation of the additional guidance: the noise estimate is integrated with the tortoise equation on the original, fine-grained timestep grid, while the additional guidance is integrated with the hare equation only on a coarse grid. We also introduce (i) an error-boundaware timestep sampler that adaptively selects step sizes and (ii) a guidance-scale scheduler that stabilizes large extrapolation spans. THG reduces the number of function evaluations (NFE) by up to 30% with virtually no loss in generation fidelity ( ImageReward 0.032) and outperforms state-of-the-art CFG-based training-free accelerators under identical computation budgets. Our findings highlight the potential of multirate formulations for diffusion solvers, paving the way for real-time high-quality image synthesis without any model retraining. The source code is available at https://github.com/yhlee-add/THG.

Large language models (LLMs) often produce reasoning steps that are superficially coherent yet internally inconsistent, leading to unreliable outputs. Since such failures typically arise from implicit or poorly-grounded knowledge, we introduce \emph{Grounded Reasoning in Dependency (GRiD)}, a novel dependency-aware reasoning framework that explicitly grounds reasoning steps in structured knowledge. GRiD represents reasoning as a graph consisting of interconnected knowledge extraction nodes and reasoning nodes, enforcing logical consistency through explicit dependencies. Each reasoning step is validated via a lightweight, step-wise verifier that ensures logical correctness relative to its premises. Extensive experiments across diverse reasoning benchmarks--including StrategyQA, CommonsenseQA, GPQA, and TruthfulQA--demonstrate that GRiD substantially improves reasoning accuracy, consistency, and faithfulness compared to recent state-of-the-art structured reasoning methods. Notably, GRiD enhances performance even when applied purely as a lightweight verification module at inference time, underscoring its generalizability and practical utility. Code is available at: https://github.com/cure-lab/GRiD.

Implicit Neural Representations (INRs) have recently shown impressive results, but their fundamental capacity, implicit biases, and scaling behavior remain poorly understood. We investigate the performance of diverse INRs across a suite of 2D and 3D real and synthetic signals with varying effective bandwidth, as well as both overfitting and generalization tasks including tomography, super-resolution, and denoising. By stratifying performance according to model size as well as signal type and bandwidth, our results shed light on how different INR and grid representations allocate their capacity. We find that, for most tasks and signals, a simple regularized grid with interpolation trains faster and to higher quality than any INR with the same number of parameters. We also find limited settings-namely fitting binary signals such as shape contours-where INRs outperform grids, to guide future development and use of INRs towards the most advantageous applications.

Modern recurrent architectures, such as xLSTM and Mamba, have recently challenged the Transformer in language modeling. However, their structure constrains their applicability to sequences only or requires processing multi-dimensional data structures, such as images or molecular graphs, in a pre-defined sequential order. In contrast, Multi-Dimensional RNNs (MDRNNs) are well suited for data with a higher level structure, like 2D grids, trees, and directed acyclic graphs (DAGs). In this work, we extend the notion of multi-dimensionality to linear RNNs. We introduce parallelizable Linear Source Transition Mark networks (pLSTMs) using Source, Transition, and Mark gates that act on the linegraph of a general DAG. This enables parallelization in analogy to parallel associative scans and the chunkwise-recurrent form of sequential linear RNNs, but for DAGs. For regular grids (1D and 2D), like images, this scheme can be efficiently implemented using einsum operations, concatenations, and padding in logarithmic time.