Accurate and compact representation of signed distance functions (SDFs) of implicit surfaces is crucial for efficient storage, computation, and downstream processing of 3D geometry. In this work, we propose a general learning method for approximating precomputed SDF fields of implicit surfaces by a relatively small number of ellipsoidal radial basis functions (ERBFs). The SDF values could be computed from various sources, including point clouds, triangle meshes, analytical expressions, pretrained neural networks, etc. Given SDF values on spatial grid points, our method approximates the SDF using as few ERBFs as possible, achieving a compact representation while preserving the geometric shape of the corresponding implicit surface. To balance sparsity and approximation precision, we introduce a dynamic multi-objective optimization strategy, which adaptively incorporates regularization to enforce sparsity and jointly optimizes the weights, centers, shapes, and orientations of the ERBFs. For computational efficiency, a nearest-neighbor-based data structure restricts computations to points near each kernel center, and CUDA-based parallelism further accelerates the optimization. Furthermore, a hierarchical refinement strategy based on SDF spatial grid points progressively incorporates coarse-to-fine samples for parameter initialization and optimization, improving convergence and training efficiency. Extensive experiments on multiple benchmark datasets demonstrate that our method can represent SDF fields with significantly fewer parameters than existing sparse implicit representation approaches, achieving better accuracy, robustness, and computational efficiency. The corresponding executable program is publicly available at https://github.com/lianbobo/SE-RBFNet.git

ABSTRACT Graphons, as limits of graph sequences, provide a framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons yields operator-level convergence rates, enabling transferability analyses of GNNs. This note summarizes known bounds under no assumptions, global Lipschitz continuity, and piecewise-Lipschitz continuity, highlighting tradeoffs between assumptions and rates, and illustrating their empirical tightness on synthetic and real data. Index T erms-- graph neural operator, graphon, convergence rates, graph neural networks, transferability 1. INTRODUCTION Graph neural networks (GNNs) are widely used in drug discovery [1, 2], social networks [3, 4], recommendation systems [5], and NLP [6, 7, 8]. GNNs operate on graph-structured data via message passing and aggregation [9], but training on large graphs is computationally expensive.

Reinforcement learning (RL) has become a predominant technique to align language models (LMs) with human preferences or promote outputs which are deemed to be desirable by a given reward function. Standard RL approaches optimize average reward, while methods explicitly focused on reducing the probability of undesired outputs typically come at a cost to average-case performance. To improve this tradeoff, we introduce RePULSe, a new training method that augments the standard RL loss with an additional loss that uses learned proposals to guide sampling low-reward outputs, and then reduces those outputs' probability. We run experiments demonstrating that RePULSe produces a better tradeoff of expected reward versus the probability of undesired outputs and is more adversarially robust, compared to standard RL alignment approaches and alternatives.

In recent years, large language models (LLMs) have shown remarkable capabilities in various artificial intelligence problems. However, they fail to plan reliably, even when prompted with a detailed definition of the planning task. Attempts to improve their planning capabilities, such as chain-of-thought prompting, fine-tuning, and explicit "reasoning" still yield incorrect plans and usually fail to generalize to larger tasks. In this paper, we show how to use LLMs to generate correct plans, even for out-of-distribution tasks of increasing size. For a given planning domain, we ask an LLM to generate several domain-dependent heuristic functions in the form of Python code, evaluate them on a set of training tasks within a greedy best-first search, and choose the strongest one. The resulting LLM-generated heuristics solve many more unseen test tasks than state-of-the-art domain-independent heuristics for classical planning. They are even competitive with the strongest learning algorithm for domain-dependent planning. These findings are especially remarkable given that our proof-of-concept implementation is based on an unoptimized Python planner and the baselines all build upon highly optimized C++ code. In some domains, the LLM-generated heuristics expand fewer states than the baselines, revealing that they are not only efficiently computable, but sometimes even more informative than the state-of-the-art heuristics. Overall, our results show that sampling a set of planning heuristic function programs can significantly improve the planning capabilities of LLMs.

Applying complex legal rules characterized by multiple, heterogeneously weighted criteria presents a fundamental challenge in judicial decision-making, often hindering the consistent realization of legislative intent. This challenge is particularly evident in the quantification of non-pecuniary damages in personal injury cases. This paper introduces Soppia, a structured prompting framework designed to assist legal professionals in navigating this complexity. By leveraging advanced AI, the system ensures a comprehensive and balanced analysis of all stipulated criteria, fulfilling the legislator's intent that compensation be determined through a holistic assessment of each case. Using the twelve criteria for non-pecuniary damages established in the Brazilian CLT (Art. 223-G) as a case study, we demonstrate how Soppia (System for Ordered Proportional and Pondered Intelligent Assessment) operationalizes nuanced legal commands into a practical, replicable, and transparent methodology. The framework enhances consistency and predictability while providing a versatile and explainable tool adaptable across multi-criteria legal contexts, bridging normative interpretation and computational reasoning toward auditable legal AI.

Reasoning models represent a significant advance in LLM capabilities, particularly for complex reasoning tasks such as mathematics and coding. Previous studies confirm that parallel test-time compute-sampling multiple solutions and selecting the best one-can further enhance the predictive performance of LLMs. However, strategies in this area often require complex scoring, thus increasing computational cost and complexity. In this work, we demonstrate that the simple and counterintuitive heuristic of selecting the shortest solution is highly effective. We posit that the observed effectiveness stems from models operating in two distinct regimes: a concise, confident conventional regime and a verbose overthinking regime characterized by uncertainty, and we show evidence of a critical point where the overthinking regime begins to be significant. By selecting the shortest answer, the heuristic preferentially samples from the conventional regime. We confirm that this approach is competitive with more complex methods such as self-consistency across two challenging benchmarks while significantly reducing computational overhead. The shortest-answer heuristic provides a Pareto improvement over self-consistency and applies even to tasks where output equality is not well defined.

Three Chinese nationals have been arrested in Georgia on suspicion of attempting to illegally purchase 2kg of uranium. Lasha Maghradze, deputy head of the nation's State Security Service (SSG), told a news briefing the group planned to pay $400,000 (£300,570) for the nuclear material in the capital, Tblisi, before transporting it to China via Russia. The alleged plot was unearthed by intelligence agents while one member of the group was attempting to buy the radioactive substance on the black market, he said. The three pleaded not guilty at a court in Tblisi and have been placed in custody to prevent them fleeing the country, according to public broadcaster Georgia Today. They face up to five years in prison under a provision of Georgia's criminal code banning the purchasing of nuclear material.

Things to Do in L.A. Tap to enable a layout that focuses on the article. California will be the first state to ban most law enforcement, including federal immigration agents, from covering their faces while conducting official business under a bill signed by Gov. Gavin Newsom on Saturday. This is read by an automated voice. Please report any issues or inconsistencies here . SACRAMENTO -- Though hailed by some for signing new laws to combat antisemitism in California schools, Gov. Gavin Newsom expressed enough reservations about the bills to urge state lawmakers to make some changes.

Game Theory What's behind bans on away fans? Maccabi Tel Aviv fans have been banned from Villa Park for their Europa League game against Aston Villa. But who decides when away fans are banned and what constitutes a high-risk game? Samantha Johnson looks at the reasons and the politics behind banning orders in European football. Afghan Women's Team: The Fight to Play U-20 World Cup Who would you play for?

Is Trump losing patience with Putin? Will sanctions against Russian oil giants hurt Putin? How much of Europe's oil still comes from Russia? Russian overnight attack on Ukraine's Kyiv kills at least 3, wounds dozens At least three people have been killed and dozens wounded in an overnight Russian air attack on Kyiv, according to the mayor of the Ukrainian capital, as Russia's war on Ukraine approaches its four-year mark. Mayor Vitali Klitschko said on Sunday that "several" Russian drones were operating over the city, and warned people to "remain in shelters".