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FinToolBench: Evaluating LLM Agents for Real-World Financial Tool Use

arXiv:2603.08262v1 Announce Type: new Abstract: The integration of Large Language Models (LLMs) into the financial domain is driving a paradigm shift from passive information retrieval to dynamic, agentic interaction. While general-purpose tool learning has witnessed a surge in benchmarks, the financial sector, characterized by high stakes, strict compliance, and rapid data volatility, remains critically underserved. Existing financial evaluations predominantly focus on static textual analysis or document-based QA, ignoring the complex reality of tool execution. Conversely, general tool benchmarks lack the domain-specific rigor required for finance, often relying on toy environments or a negligible number of financial APIs. To bridge this gap, we introduce FinToolBench, the first real-world, runnable benchmark dedicated to evaluating financial tool learning agents. Unlike prior works limited to a handful of mock tools, FinToolBench establishes a realistic ecosystem coupling 760 executable financial tools with 295 rigorous, tool-required queries. We propose a novel evaluation framework that goes beyond binary execution success, assessing agents on finance-critical dimensions: timeliness, intent type, and regulatory domain alignment. Furthermore, we present FATR, a finance-aware tool retrieval and reasoning baseline that enhances stability and compliance. By providing the first testbed for auditable, agentic financial execution, FinToolBench sets a new standard for trustworthy AI in finance. The tool manifest, execution environment, and evaluation code will be open-sourced to facilitate future research.
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Deconstructing Multimodal Mathematical Reasoning: Towards a Unified Perception-Alignment-Reasoning Paradigm

arXiv:2603.08291v1 Announce Type: new Abstract: Multimodal Mathematical Reasoning (MMR) has recently attracted increasing attention for its capability to solve mathematical problems that involve both textual and visual modalities. However, current models still face significant challenges in real-world visual math tasks. They often misinterpret diagrams, fail to align mathematical symbols with visual evidence, and produce inconsistent reasoning steps. Moreover, existing evaluations mainly focus on checking final answers rather than verifying the correctness or executability of each intermediate step. To address these limitations, a growing body of recent research addresses these issues by integrating structured perception, explicit alignment, and verifiable reasoning within unified frameworks. To establish a clear roadmap for understanding and comparing different MMR approaches, we systematically study them around four fundamental questions: (1) What to extract from multimodal inputs, (2) How to represent and align textual and visual information, (3) How to perform the reasoning, and (4) How to evaluate the correctness of the overall reasoning process. Finally, we discuss open challenges and offer perspectives on promising directions for future research.
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CORE-Acu: Structured Reasoning Traces and Knowledge Graph Safety Verification for Acupuncture Clinical Decision Support

arXiv:2603.08321v1 Announce Type: new Abstract: Large language models (LLMs) show significant potential for clinical decision support (CDS), yet their black-box nature -- characterized by untraceable reasoning and probabilistic hallucinations -- poses severe challenges in acupuncture, a field demanding rigorous interpretability and safety. To address this, we propose CORE-Acu, a neuro-symbolic framework for acupuncture clinical decision support that integrates Structured Chain-of-Thought (S-CoT) with knowledge graph (KG) safety verification. First, we construct the first acupuncture Structured Reasoning Trace dataset and a schema-constrained fine-tuning framework. By enforcing an explicit causal chain from pattern identification to treatment principles, treatment plans, and acupoint selection, we transform implicit Traditional Chinese Medicine (TCM) reasoning into interpretable generation constraints, mitigating the opacity of LLM-based CDS. Furthermore, we construct a TCM safety knowledge graph and establish a ``Generate--Verify--Revise'' closed-loop inference system based on a Symbolic Veto Mechanism, employing deterministic rules to intercept hallucinations and enforce hard safety boundaries. Finally, we introduce the Lexicon-Matched Entity-Reweighted Loss (LMERL), which corrects terminology drift caused by the frequency--importance mismatch in general optimization by adaptively amplifying gradient contributions of high-risk entities during fine-tuning. Experiments on 1,000 held-out cases demonstrate CORE-Acu's superior entity fidelity and reasoning quality. Crucially, CORE-Acu achieved 0/1,000 observed safety violations (95\% CI: 0--0.37\%), whereas GPT-4o exhibited an 8.5\% violation rate under identical rules. These results establish CORE-Acu as a robust neuro-symbolic framework for acupuncture clinical decision support, guaranteeing both reasoning auditability and strict safety compliance.
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M$^3$-ACE: Rectifying Visual Perception in Multimodal Math Reasoning via Multi-Agentic Context Engineering

arXiv:2603.08369v1 Announce Type: new Abstract: Multimodal large language models have recently shown promising progress in visual mathematical reasoning. However, their performance is often limited by a critical yet underexplored bottleneck: inaccurate visual perception. Through systematic analysis, we find that the most failures originate from incorrect or incomplete visual evidence extraction rather than deficiencies in reasoning capability. Moreover, models tend to remain overly confident in their initial perceptions, making standard strategies such as prompt engineering, multi-round self-reflection, or posterior guidance insufficient to reliably correct errors. To address this limitation, we propose M3-ACE, a multi-agentic context engineering framework designed to rectify visual perception in multimodal math reasoning. Instead of directly aggregating final answers, our approach decouples perception and reasoning by dynamically maintaining a shared context centered on visual evidence lists. Multiple agents collaboratively contribute complementary observations, enabling the system to expose inconsistencies and recover missing perceptual information. To support stable multi-turn collaboration, we further introduce two lightweight tools: a Summary Tool that organizes evidence from different agents into consistent, complementary, and conflicting components, and a Refine Tool that filters unreliable samples and guides iterative correction. Extensive experiments demonstrate that M3-ACE substantially improves visual mathematical reasoning performance across multiple benchmarks. Our method establishes new state-of-the-art results 89.1 on the MathVision benchmark and achieves consistent improvements on other related datasets, including MathVista and MathVerse. These results highlight the importance of perception-centric multi-agent collaboration for advancing multimodal reasoning systems.
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RetroAgent: From Solving to Evolving via Retrospective Dual Intrinsic Feedback

arXiv:2603.08561v1 Announce Type: new Abstract: Large language model (LLM)-based agents trained with reinforcement learning (RL) have shown strong potential on complex interactive tasks. However, standard RL paradigms favor static problem-solving over continuous adaptation: agents often converge to suboptimal strategies due to insufficient exploration, while learned knowledge remains implicit within parameters rather than explicitly retrievable, limiting effective experiential learning. To address these limitations, we introduce RetroAgent, an online RL framework that empowers agents to master complex interactive environments not just by solving, but by evolving. Concretely, RetroAgent features a hindsight self-reflection mechanism that produces dual intrinsic feedback: (1) intrinsic numerical feedback that that tracks incremental subtask completion relative to prior attempts, rewarding promising explorations, and (2) intrinsic language feedback that distills reusable lessons into a memory buffer, retrieved via our proposed Similarity & Utility-Aware Upper Confidence Bound (SimUtil-UCB) strategy balancing relevance, utility, and exploration to effectively leverage past experiences. Extensive experiments on two model families across four challenging agentic tasks demonstrate that RetroAgent significantly outperforms existing methods, achieving state-of-the-art results -- e.g., surpassing Group Relative Policy Optimization (GRPO)-trained agents by +18.3% on ALFWorld, +15.4% on WebShop, +27.1% on Sokoban, and +8.9% on MineSweeper -- while exhibiting strong test-time adaptation and generalization to out-of-distribution scenarios.
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CoCo: Code as CoT for Text-to-Image Preview and Rare Concept Generation

arXiv:2603.08652v1 Announce Type: new Abstract: Recent advancements in Unified Multimodal Models (UMMs) have significantly advanced text-to-image (T2I) generation, particularly through the integration of Chain-of-Thought (CoT) reasoning. However, existing CoT-based T2I methods largely rely on abstract natural-language planning, which lacks the precision required for complex spatial layouts, structured visual elements, and dense textual content. In this work, we propose CoCo (Code-as-CoT), a code-driven reasoning framework that represents the reasoning process as executable code, enabling explicit and verifiable intermediate planning for image generation. Given a text prompt, CoCo first generates executable code that specifies the structural layout of the scene, which is then executed in a sandboxed environment to render a deterministic draft image. The model subsequently refines this draft through fine-grained image editing to produce the final high-fidelity result. To support this training paradigm, we construct CoCo-10K, a curated dataset containing structured draft-final image pairs designed to teach both structured draft construction and corrective visual refinement. Empirical evaluations on StructT2IBench, OneIG-Bench, and LongText-Bench show that CoCo achieves improvements of +68.83%, +54.8%, and +41.23% over direct generation, while also outperforming other generation methods empowered by CoT. These results demonstrate that executable code is an effective and reliable reasoning paradigm for precise, controllable, and structured text-to-image generation. The code is available at: https://github.com/micky-li-hd/CoCo
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OfficeQA Pro: An Enterprise Benchmark for End-to-End Grounded Reasoning

arXiv:2603.08655v1 Announce Type: new Abstract: We introduce OfficeQA Pro, a benchmark for evaluating AI agents on grounded, multi-document reasoning over a large and heterogeneous document corpus. The corpus consists of U.S. Treasury Bulletins spanning nearly 100 years, comprising 89,000 pages and over 26 million numerical values. OfficeQA Pro consists of 133 questions that require precise document parsing, retrieval, and analytical reasoning across both unstructured text and tabular data. Frontier LLMs including Claude Opus 4.6, GPT-5.4, and Gemini 3.1 Pro Preview achieve less than 5% accuracy on OfficeQA Pro when relying on parametric knowledge, and less than 12% with additional access to the web. When provided directly with the document corpus, frontier agents still struggle on over half of questions, scoring 34.1% on average. We find that providing agents with a structured document representation produced by Databricks' ai_parse_document yields a 16.1% average relative performance gain across agents. We conduct additional ablations to study the effects of model selection, table representation, retrieval strategy, and test-time scaling on performance. Despite these improvements, significant headroom remains before agents can be considered reliable at enterprise-grade grounded reasoning.
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Agentic Critical Training

arXiv:2603.08706v1 Announce Type: new Abstract: Training large language models (LLMs) as autonomous agents often begins with imitation learning, but it only teaches agents what to do without understanding why: agents never contrast successful actions against suboptimal alternatives and thus lack awareness of action quality. Recent approaches attempt to address this by introducing self-reflection supervision derived from contrasts between expert and alternative actions. However, the training paradigm fundamentally remains imitation learning: the model imitates pre-constructed reflection text rather than learning to reason autonomously. We propose Agentic Critical Training (ACT), a reinforcement learning paradigm that trains agents to identify the better action among alternatives. By rewarding whether the model's judgment is correct, ACT drives the model to autonomously develop reasoning about action quality, producing genuine self-reflection rather than imitating it. Across three challenging agent benchmarks, ACT consistently improves agent performance when combined with different post-training methods. It achieves an average improvement of 5.07 points over imitation learning and 4.62 points over reinforcement learning. Compared to approaches that inject reflection capability through knowledge distillation, ACT also demonstrates clear advantages, yielding an average improvement of 2.42 points. Moreover, ACT enables strong out-of-distribution generalization on agentic benchmarks and improves performance on general reasoning benchmarks without any reasoning-specific training data, highlighting the value of our method. These results suggest that ACT is a promising path toward developing more reflective and capable LLM agents.
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A Robust Incomplete Multimodal Low-Rank Adaptation Approach for Emotion Recognition

arXiv:2507.11202v1 Announce Type: cross Abstract: Multimodal Emotion Recognition (MER) often encounters incomplete multimodality in practical applications due to sensor failures or privacy protection requirements. While existing methods attempt to address various incomplete multimodal scenarios by balancing the training of each modality combination through additional gradients, these approaches face a critical limitation: training gradients from different modality combinations conflict with each other, ultimately degrading the performance of the final prediction model. In this paper, we propose a unimodal decoupled dynamic low-rank adaptation method based on modality combinations, named MCULoRA, which is a novel framework for the parameter-efficient training of incomplete multimodal learning models. MCULoRA consists of two key modules, modality combination aware low-rank adaptation (MCLA) and dynamic parameter fine-tuning (DPFT). The MCLA module effectively decouples the shared information from the distinct characteristics of individual modality combinations. The DPFT module adjusts the training ratio of modality combinations based on the separability of each modality's representation space, optimizing the learning efficiency across different modality combinations. Our extensive experimental evaluation in multiple benchmark datasets demonstrates that MCULoRA substantially outperforms previous incomplete multimodal learning approaches in downstream task accuracy.
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Isotonic Layer: A Universal Framework for Generic Recommendation Debiasing

arXiv:2603.06589v1 Announce Type: cross Abstract: Model calibration and debiasing are fundamental to the reliability and fairness of large scale recommendation systems. We introduce the Isotonic Layer, a novel, differentiable framework that integrates piecewise linear fitting directly into neural architectures. By partitioning the feature space into discrete segments and optimizing non negative slopes via a constrained dot product mechanism, we enforce a global monotonic inductive bias. This ensures model outputs remain logically consistent with critical features such as latent relevance, recency, or quality scores. We further generalize this architecture by parameterizing segment wise slopes as learnable embeddings. This enables the model to adaptively capture context specific distortions, such as position based CTR bias through specialized isotonic profiles. Our approach utilizes a dual task formulation that decouples the recommendation objective into latent relevance estimation and bias aware calibration. A major contribution of this work is the ability to perform highly granular, customized calibration for arbitrary combinations of context features, a level of control difficult to achieve with traditional non parametric methods. We also extend this to Multi Task Learning environments with dedicated embeddings for distinct objectives. Extensive empirical evaluations on real world datasets and production AB tests demonstrate that the Isotonic Layer effectively mitigates systematic bias and enhances calibration fidelity, significantly outperforming production baselines in both predictive accuracy and ranking consistency.
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ARC-AGI-2 Technical Report

arXiv:2603.06590v1 Announce Type: cross Abstract: The Abstraction and Reasoning Corpus (ARC) is designed to assess generalization beyond pattern matching, requiring models to infer symbolic rules from very few examples. In this work, we present a transformer-based system that advances ARC performance by combining neural inference with structure-aware priors and online task adaptation. Our approach is built on four key ideas. First, we reformulate ARC reasoning as a sequence modeling problem using a compact task encoding with only 125 tokens, enabling efficient long-context processing with a modified LongT5 architecture. Second, we introduce a principled augmentation framework based on group symmetries, grid traversals, and automata perturbations, enforcing invariance to representation changes. Third, we apply test-time training (TTT) with lightweight LoRA adaptation, allowing the model to specialize to each unseen task by learning its transformation logic from demonstrations. Fourth, we design a symmetry-aware decoding and scoring pipeline that aggregates likelihoods across augmented task views, effectively performing ``multi-perspective reasoning'' over candidate solutions. We demonstrate that these components work synergistically: augmentations expand hypothesis space, TTT sharpens local reasoning, and symmetry-based scoring improves solution consistency. Our final system achieves a significant improvement over transformer baselines and surpasses prior neural ARC solvers, closing the gap toward human-level generalization.
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Distributionally Robust Geometric Joint Chance-Constrained Optimization: Neurodynamic Approaches

arXiv:2603.06597v1 Announce Type: cross Abstract: This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and belong to a certain distributional uncertainty set. In our paper, we study three uncertainty sets for the unknown distributions. The neurodynamic duplex is designed based on three projection equations. The main contribution of our work is to propose a neural network-based method to solve distributionally robust joint chance-constrained optimization problems that converges in probability to the global optimum without the use of standard state-of-the-art solving methods. We show that neural networks can be used to solve multiple instances of a problem. In the numerical experiments, we apply the proposed approach to solve a problem of shape optimisation and a telecommunication problem.
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FuzzingRL: Reinforcement Fuzz-Testing for Revealing VLM Failures

arXiv:2603.06600v1 Announce Type: cross Abstract: Vision Language Models (VLMs) are prone to errors, and identifying where these errors occur is critical for ensuring the reliability and safety of AI systems. In this paper, we propose an approach that automatically generates questions designed to deliberately induce incorrect responses from VLMs, thereby revealing their vulnerabilities. The core of this approach lies in fuzz testing and reinforcement finetuning: we transform a single input query into a large set of diverse variants through vision and language fuzzing. Based on the fuzzing outcomes, the question generator is further instructed by adversarial reinforcement fine-tuning to produce increasingly challenging queries that trigger model failures. With this approach, we can consistently drive down a target VLM's answer accuracy -- for example, the accuracy of Qwen2.5-VL-32B on our generated questions drops from 86.58\% to 65.53\% in four RL iterations. Moreover, a fuzzing policy trained against a single target VLM transfers to multiple other VLMs, producing challenging queries that degrade their performance as well.
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Evo: Autoregressive-Diffusion Large Language Models with Evolving Balance

arXiv:2603.06617v1 Announce Type: cross Abstract: We introduce \textbf{Evo}, a duality latent trajectory model that bridges autoregressive (AR) and diffusion-based language generation within a continuous evolutionary generative framework. Rather than treating AR decoding and diffusion generation as separate paradigms, Evo reconceptualizes text generation as a latent flow: each token is associated with a vector-valued embedding that evolves over a progression variable $t_i \in [0, 1]$, indicating its semantic maturity. Low $t_i$ values correspond to confident AR-like refinement, while high values invoke diffusion-style planning, allowing the model to adaptively balance AR and diffusion based on uncertainty. Theoretically, we show that both AR and diffusion models emerge as discretizations of a shared probability flow, and we derive Evo's training objective from a unified variational ELBO. The model is implemented as a time-conditioned Transformer governed by a shared vector field, trained end-to-end to jointly infer latent codes and their progression times. During decoding, Evo performs efficient, semantics-aware refinement, achieving high-quality outputs without sacrificing speed. Empirically, Evo 8B achieves state-of-the-art or highly competitive results on 15 diverse benchmarks, including reasoning (GSM8K, ARC-C), code generation (HumanEval, MBPP), and general language understanding, while maintaining fast inference speed. Our results demonstrate that Evo delivers a new paradigm for LLM design with strong generation quality, robust symbolic reasoning, and decoding efficiency.
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GraphSkill: Documentation-Guided Hierarchical Retrieval-Augmented Coding for Complex Graph Reasoning

arXiv:2603.06620v1 Announce Type: cross Abstract: The growing demand for automated graph algorithm reasoning has attracted increasing attention in the large language model (LLM) community. Recent LLM-based graph reasoning methods typically decouple task descriptions from graph data, generate executable code augmented by retrieval from technical documentation, and refine the code through debugging. However, we identify two key limitations in existing approaches: (i) they treat technical documentation as flat text collections and ignore its hierarchical structure, leading to noisy retrieval that degrades code generation quality; and (ii) their debugging mechanisms focus primarily on runtime errors, yet ignore more critical logical errors. To address them, we propose {\method}, an \textit{agentic hierarchical retrieval-augmented coding framework} that exploits the document hierarchy through top-down traversal and early pruning, together with a \textit{self-debugging coding agent} that iteratively refines code using automatically generated small-scale test cases. To enable comprehensive evaluation of complex graph reasoning, we introduce a new dataset, {\dataset}, covering small-scale, large-scale, and composite graph reasoning tasks. Extensive experiments demonstrate that our method achieves higher task accuracy and lower inference cost compared to baselines\footnote{The code is available at \href{https://github.com/FairyFali/GraphSkill}{\textcolor{blue}{https://github.com/FairyFali/GraphSkill}}.}.
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Grouter: Decoupling Routing from Representation for Accelerated MoE Training

arXiv:2603.06626v1 Announce Type: cross Abstract: Traditional Mixture-of-Experts (MoE) training typically proceeds without any structural priors, effectively requiring the model to simultaneously train expert weights while searching for an optimal routing policy within a vast combinatorial space. This entanglement often leads to sluggish convergence and training instabilities. This paper introduces Grouter, a preemptive routing method that by distilling high-quality structures from fully-trained MoE models and serving as a fixed router for target models. By decoupling structural optimization from weight updates, Grouter significantly accelerates both the speed and quality of model convergence. To ensure the framework's versatility, we also introduce expert folding to adapt Grouter across varying model configurations and expert tuning to rebalance workloads across different data distributions. Furthermore, by leveraging the structural priors provided by preemptive routing, we can implement targeted optimizations to further enhance training throughput. Experiments demonstrate that Grouter achieves superior performance and efficiency which boosts pre-training data utilization by 4.28x and achieves up to 33.5% throughput acceleration, establishing preemptive routing as a fundamental paradigm for scalable MoE training.
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Photons = Tokens: The Physics of AI and the Economics of Knowledge

arXiv:2603.06630v1 Announce Type: cross Abstract: Debates about artificial intelligence capabilities and risks are often conducted without quantitative grounding. This paper applies the methodology of MacKay (2009) -- who reframed energy policy as arithmetic -- to the economy of AI computation. We define the token, the elementary unit of large language model input and output, as a physical quantity with measurable thermodynamic cost. Using Landauer's principle, Shannon's channel capacity, and current infrastructure data, we construct a supply-and-demand balance sheet for global token production. We then derive a finite question budget: the number of meaningful queries humanity can direct at AI systems under physical, information-theoretic, and economic constraints. We apply Coase's theory of the firm and the durable-goods monopoly problem to the AI value chain -- from photon to atom to chip to power to token to question -- to identify where economic value concentrates and where regulatory intervention is warranted. We argue that the expansion of the token budget does not resolve a deeper constraint: under structural uncertainty, the decisive variable is not how many questions can be answered but which questions are worth asking -- a problem of agency and direction that computation alone cannot solve. We connect limits of measurement in the token economy to a structural parallel between Goodhart's law and the Heisenberg uncertainty principle, and to Arrow's impossibility result for efficient information pricing. The framework yields order-of-magnitude estimates that discipline policy discussion: at current efficiency, the projected 2028 US AI energy allocation of 326~TWh could support roughly $6.5 \times 10^{17}$ tokens per year, or 225,000 tokens per person per day -- more than three orders of magnitude above estimated mid-2024 utilization.
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HEARTS: Benchmarking LLM Reasoning on Health Time Series

arXiv:2603.06638v1 Announce Type: cross Abstract: The rise of large language models (LLMs) has shifted time series analysis from narrow analytics to general-purpose reasoning. Yet, existing benchmarks cover only a small set of health time series modalities and tasks, failing to reflect the diverse domains and extensive temporal dependencies inherent in real-world physiological modeling. To bridge these gaps, we introduce HEARTS (Health Reasoning over Time Series), a unified benchmark for evaluating hierarchical reasoning capabilities of LLMs over general health time series. HEARTS integrates 16 real-world datasets across 12 health domains and 20 signal modalities, and defines a comprehensive taxonomy of 110 tasks grouped into four core capabilities: Perception, Inference, Generation, and Deduction. Evaluating 14 state-of-the-art LLMs on more than 20K test samples reveals intriguing findings. First, LLMs substantially underperform specialized models, and their performance is only weakly related to general reasoning scores. Moreover, LLMs often rely on simple heuristics and struggle with multi-step temporal reasoning. Finally, performance declines with increasing temporal complexity, with similar failure modes within model families, indicating that scaling alone is insufficient. By making these gaps measurable, HEARTS provides a standardized testbed and living benchmark for developing next-generation LLM agents capable of reasoning over diverse health signals.
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HURRI-GAN: A Novel Approach for Hurricane Bias-Correction Beyond Gauge Stations using Generative Adversarial Networks

arXiv:2603.06649v1 Announce Type: cross Abstract: The coastal regions of the eastern and southern United States are impacted by severe storm events, leading to significant loss of life and properties. Accurately forecasting storm surge and wind impacts from hurricanes is essential for mitigating some of the impacts, e.g., timely preparation of evacuations and other countermeasures. Physical simulation models like the ADCIRC hydrodynamics model, which run on high-performance computing resources, are sophisticated tools that produce increasingly accurate forecasts as the resolution of the computational meshes improves. However, a major drawback of these models is the significant time required to generate results at very high resolutions, which may not meet the near real-time demands of emergency responders. The presented work introduces HURRI-GAN, a novel AI-driven approach that augments the results produced by physical simulation models using time series generative adversarial networks (TimeGAN) to compensate for systemic errors of the physical model, thus reducing the necessary mesh size and runtime without loss in forecasting accuracy. We present first results in extrapolating model bias corrections for the spatial regions beyond the positions of the water level gauge stations. The presented results show that our methodology can accurately generate bias corrections at target locations spatially beyond gauge stations locations. The model's performance, as indicated by low root mean squared error (RMSE) values, highlights its capability to generate accurate extrapolated data. Applying the corrections generated by HURRI-GAN on the ADCIRC modeled water levels resulted in improving the overall prediction on the majority of the testing gauge stations.
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Geodesic Gradient Descent: A Generic and Learning-rate-free Optimizer on Objective Function-induced Manifolds

arXiv:2603.06651v1 Announce Type: cross Abstract: Euclidean gradient descent algorithms barely capture the geometry of objective function-induced hypersurfaces and risk driving update trajectories off the hypersurfaces. Riemannian gradient descent algorithms address these issues but fail to represent complex hypersurfaces via a single classic manifold. We propose geodesic gradient descent (GGD), a generic and learning-rate-free Riemannian gradient descent algorithm. At each iteration, GGD uses an n-dimensional sphere to approximate a local neighborhood on the objective function-induced hypersurface, adapting to arbitrarily complex geometries. A tangent vector derived from the Euclidean gradient is projected onto the sphere to form a geodesic, ensuring the update trajectory stays on the hypersurface. Parameter updates are performed using the endpoint of the geodesic. The maximum step size of the gradient in GGD is equal to a quarter of the arc length on the n-dimensional sphere, thus eliminating the need for a learning rate. Experimental results show that compared with the classic Adam algorithm, GGD achieves test MSE reductions ranging from 35.79% to 48.76% for fully connected networks on the Burgers' dataset, and cross-entropy loss reductions ranging from 3.14% to 11.59% for convolutional neural networks on the MNIST dataset.
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