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Program completion is a translation from the language of logic programs into the language of first-order theories. Its original definition has been extended to programs that include integer arithmetic, accept input, and distinguish between output predicates and auxiliary predicates. For tight programs, that generalization of completion is known to match the stable model semantics, which is the basis of answer set programming. We show that the tightness condition in this theorem can be replaced by a less restrictive "local tightness" requirement. From this fact we conclude that the proof assistant anthem-p2p can be used to verify equivalence between locally tight programs. Under consideration for publication in Theory and Practice of Logic Programming

Multiphysics incompressible fluid dynamics simulations play a crucial role in understanding intricate behaviors of many complex engineering systems that involve interactions between solids, fluids, and various phases like liquid and gas. Numerical modeling of these interactions has generated significant research interest in recent decades and has led to the development of open source simulation tools and commercial software products targeting specific applications or general problem classes in computational fluid dynamics. As the demand increases for these simulations to adapt to platform heterogeneity, ensure composability between different physics models, and effectively utilize inheritance within partial differentiation systems, a fundamental reconsideration of numerical solver design becomes imperative. The discussion presented in this paper emphasizes the importance of these considerations and introduces the Flash-X approach as a potential solution. The software design strategies outlined in the article serve as a guide for Flash-X developers, providing insights into complexities associated with performance portability, composability, and sustainable development. These strategies provide a foundation for improving design of both new and existing simulation tools grappling with these challenges. By incorporating the principles outlined in the Flash-X approach, engineers and researchers can enhance the adaptability, efficiency, and overall effectiveness of their numerical solvers in the ever-evolving field of multiphysics simulations.

Many of the most fundamental laws of nature can be formulated as partial differential equations (PDEs). Understanding these equations is, therefore, of exceptional importance for many branches of modern science and engineering. However, since the general solution of many PDEs is unknown, the efficient approximate solution of these equations is one of humanity's greatest challenges. While multigrid represents one of the most effective methods for solving PDEs numerically, in many cases, the design of an efficient or at least working multigrid solver is an open problem. This thesis demonstrates that grammar-guided genetic programming, an evolutionary program synthesis technique, can discover multigrid methods of unprecedented structure that achieve a high degree of efficiency and generalization. For this purpose, we develop a novel context-free grammar that enables the automated generation of multigrid methods in a symbolically-manipulable formal language, based on which we can apply the same multigrid-based solver to problems of different sizes without having to adapt its internal structure. Treating the automated design of an efficient multigrid method as a program synthesis task allows us to find novel sequences of multigrid operations, including the combination of different smoothing and coarse-grid correction steps on each level of the discretization hierarchy. To prove the feasibility of this approach, we present its implementation in the form of the Python framework EvoStencils, which is freely available as open-source software. This implementation comprises all steps from representing the algorithmic sequence of a multigrid method in the form of a directed acyclic graph of Python objects to its automatic generation and optimization using the capabilities of the code generation framework ExaStencils and the evolutionary computation library DEAP.

Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient propagators in constraint programming (such as the weighted circuit constraint). However, the conventional process of deriving Lagrangian multipliers (e.g., using subgradient methods) is often computationally intensive, limiting its practicality for large-scale or time-sensitive problems. To address this challenge, we propose an innovative unsupervised learning approach that harnesses the capabilities of graph neural networks to exploit the problem structure, aiming to generate accurate Lagrangian multipliers efficiently. We apply this technique to the well-known Held-Karp Lagrangian relaxation for the travelling salesman problem. The core idea is to predict accurate Lagrangian multipliers and to employ them as a warm start for generating Held-Karp relaxation bounds. These bounds are subsequently utilized to enhance the filtering process carried out by branch-and-bound algorithms. In contrast to much of the existing literature, which primarily focuses on finding feasible solutions, our approach operates on the dual side, demonstrating that learning can also accelerate the proof of optimality. We conduct experiments across various distributions of the metric travelling salesman problem, considering instances with up to 200 cities. The results illustrate that our approach can improve the filtering level of the weighted circuit global constraint, reduce the optimality gap by a factor two for unsolved instances up to a timeout, and reduce the execution time for solved instances by 10%.

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by blending seamlessly data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation (HTE). Starting with the second-order high-dimensional PDEs ubiquitous in scientific computing, HTE transforms the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach alleviates the computational bottleneck via Taylor-mode automatic differentiation and significantly reduces memory consumption from the Hessian matrix to HVP. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variance among dimensions. We further extend HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.

Gate-defined quantum dots are a promising candidate system to realize scalable, coupled qubit systems and serve as a fundamental building block for quantum computers. However, present-day quantum dot devices suffer from imperfections that must be accounted for, which hinders the characterization, tuning, and operation process. Moreover, with an increasing number of quantum dot qubits, the relevant parameter space grows sufficiently to make heuristic control infeasible. Thus, it is imperative that reliable and scalable autonomous tuning approaches are developed. In this report, we outline current challenges in automating quantum dot device tuning and operation with a particular focus on datasets, benchmarking, and standardization. We also present ideas put forward by the quantum dot community on how to overcome them.

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than $ 1 $ are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they in fact generalize standard subspace codes (defined in the set of all subspaces of $ \mathbb{F}_q^n $) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed $ n $ and $ q $.

Humans perceive the world by concurrently processing and fusing high-dimensional inputs from multiple modalities such as vision and audio. Machine perception models, in stark contrast, are typically modality-specific and optimised for unimodal benchmarks, and hence late-stage fusion of final representations or predictions from each modality (`late-fusion') is still a dominant paradigm for multimodal video classification. Instead, we introduce a novel transformer based architecture that uses `fusion bottlenecks' for modality fusion at multiple layers. Compared to traditional pairwise self-attention, our model forces information between different modalities to pass through a small number of bottleneck latents, requiring the model to collate and condense the most relevant information in each modality and only share what is necessary. We find that such a strategy improves fusion performance, at the same time reducing computational cost. We conduct thorough ablation studies, and achieve state-of-the-art results on multiple audio-visual classification benchmarks including Audioset, Epic-Kitchens and VGGSound. All code and models will be released.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Behaviors of the synthetic characters in current military simulations are limited since they are generally generated by rule-based and reactive computational models with minimal intelligence. Such computational models cannot adapt to reflect the experience of the characters, resulting in brittle intelligence for even the most effective behavior models devised via costly and labor-intensive processes. Observation-based behavior model adaptation that leverages machine learning and the experience of synthetic entities in combination with appropriate prior knowledge can address the issues in the existing computational behavior models to create a better training experience in military training simulations. In this paper, we introduce a framework that aims to create autonomous synthetic characters that can perform coherent sequences of believable behavior while being aware of human trainees and their needs within a training simulation. This framework brings together three mutually complementary components. The first component is a Unity-based simulation environment - Rapid Integration and Development Environment (RIDE) - supporting One World Terrain (OWT) models and capable of running and supporting machine learning experiments. The second is Shiva, a novel multi-agent reinforcement and imitation learning framework that can interface with a variety of simulation environments, and that can additionally utilize a variety of learning algorithms. The final component is the Sigma Cognitive Architecture that will augment the behavior models with symbolic and probabilistic reasoning capabilities. We have successfully created proof-of-concept behavior models leveraging this framework on realistic terrain as an essential step towards bringing machine learning into military simulations.