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Machine learning-based reliability analysis methods have shown great advancements for their computational efficiency and accuracy. Recently, many efficient learning strategies have been proposed to enhance the computational performance. However, few of them explores the theoretical optimal learning strategy. In this article, we propose several theorems that facilitates such exploration. Specifically, cases that considering and neglecting the correlations among the candidate design samples are well elaborated. Moreover, we prove that the well-known U learning function can be reformulated to the optimal learning function for the case neglecting the Kriging correlation. In addition, the theoretical optimal learning strategy for sequential multiple training samples enrichment is also mathematically explored through the Bayesian estimate with the corresponding lost functions. Simulation results show that the optimal learning strategy considering the Kriging correlation works better than that neglecting the Kriging correlation and other state-of-the art learning functions from the literatures in terms of the reduction of number of evaluations of performance function. However, the implementation needs to investigate very large computational resource.

This work presents the cubature scheme for the fitting of spatio-temporal Poisson point processes. The methodology is implemented in the R Core Team (2024) package stopp (D'Angelo and Adelfio, 2023), published on the Comprehensive R Archive Network (CRAN) and available from //CRAN.R-project.org/package=stopp. Since the number of dummy points should be sufficient for an accurate estimate of the likelihood, numerical experiments are currently under development to give guidelines on this aspect.

Bayesian inference for complex models with an intractable likelihood can be tackled using algorithms performing many calls to computer simulators. These approaches are collectively known as "simulation-based inference" (SBI). Recent SBI methods have made use of neural networks (NN) to provide approximate, yet expressive constructs for the unavailable likelihood function and the posterior distribution. However, they do not generally achieve an optimal trade-off between accuracy and computational demand. In this work, we propose an alternative that provides both approximations to the likelihood and the posterior distribution, using structured mixtures of probability distributions. Our approach produces accurate posterior inference when compared to state-of-the-art NN-based SBI methods, while exhibiting a much smaller computational footprint. We illustrate our results on several benchmark models from the SBI literature.

This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

As language models (LMs) become more capable, it is increasingly important to align them with human preferences. However, the dominant paradigm for training Preference Models (PMs) for that purpose suffers from fundamental limitations, such as lack of transparency and scalability, along with susceptibility to overfitting the preference dataset. We propose Compositional Preference Models (CPMs), a novel PM framework that decomposes one global preference assessment into several interpretable features, obtains scalar scores for these features from a prompted LM, and aggregates these scores using a logistic regression classifier. Through these simple steps, CPMs allow to control which properties of the preference data are used to train the preference model and to build it based on features that are believed to underlie the human preference judgment. Our experiments show that CPMs not only improve generalization and are more robust to overoptimization than standard PMs, but also that best-of-n samples obtained using CPMs tend to be preferred over samples obtained using conventional PMs. Overall, our approach demonstrates the benefits of endowing PMs with priors about which features determine human preferences while relying on LM capabilities to extract those features in a scalable and robust way.

Regression models that incorporate smooth functions of predictor variables to explain the relationships with a response variable have gained widespread usage and proved successful in various applications. By incorporating smooth functions of predictor variables, these models can capture complex relationships between the response and predictors while still allowing for interpretation of the results. In situations where the relationships between a response variable and predictors are explored, it is not uncommon to assume that these relationships adhere to certain shape constraints. Examples of such constraints include monotonicity and convexity. The scam package for R has become a popular package to carry out the full fitting of exponential family generalized additive modelling with shape restrictions on smooths. The paper aims to extend the existing framework of shape-constrained generalized additive models (SCAM) to accommodate smooth interactions of covariates, linear functionals of shape-constrained smooths and incorporation of residual autocorrelation. The methods described in this paper are implemented in the recent version of the package scam, available on the Comprehensive R Archive Network (CRAN).

Benchmarking models via classical simulations is one of the main ways to judge ideas in quantum machine learning before noise-free hardware is available. However, the huge impact of the experimental design on the results, the small scales within reach today, as well as narratives influenced by the commercialisation of quantum technologies make it difficult to gain robust insights. To facilitate better decision-making we develop an open-source package based on the PennyLane software framework and use it to conduct a large-scale study that systematically tests 12 popular quantum machine learning models on 6 binary classification tasks used to create 160 individual datasets. We find that overall, out-of-the-box classical machine learning models outperform the quantum classifiers. Moreover, removing entanglement from a quantum model often results in as good or better performance, suggesting that "quantumness" may not be the crucial ingredient for the small learning tasks considered here. Our benchmarks also unlock investigations beyond simplistic leaderboard comparisons, and we identify five important questions for quantum model design that follow from our results.

Machine learning (ML) methods, which fit to data the parameters of a given parameterized model class, have garnered significant interest as potential methods for learning surrogate models for complex engineering systems for which traditional simulation is expensive. However, in many scientific and engineering settings, generating high-fidelity data on which to train ML models is expensive, and the available budget for generating training data is limited. ML models trained on the resulting scarce high-fidelity data have high variance and are sensitive to vagaries of the training data set. We propose a new multifidelity training approach for scientific machine learning that exploits the scientific context where data of varying fidelities and costs are available; for example high-fidelity data may be generated by an expensive fully resolved physics simulation whereas lower-fidelity data may arise from a cheaper model based on simplifying assumptions. We use the multifidelity data to define new multifidelity Monte Carlo estimators for the unknown parameters of linear regression models, and provide theoretical analyses that guarantee the approach's accuracy and improved robustness to small training budgets. Numerical results verify the theoretical analysis and demonstrate that multifidelity learned models trained on scarce high-fidelity data and additional low-fidelity data achieve order-of-magnitude lower model variance than standard models trained on only high-fidelity data of comparable cost. This illustrates that in the scarce data regime, our multifidelity training strategy yields models with lower expected error than standard training approaches.

Machine-learning models have demonstrated great success in learning complex patterns that enable them to make predictions about unobserved data. In addition to using models for prediction, the ability to interpret what a model has learned is receiving an increasing amount of attention. However, this increased focus has led to considerable confusion about the notion of interpretability. In particular, it is unclear how the wide array of proposed interpretation methods are related, and what common concepts can be used to evaluate them. We aim to address these concerns by defining interpretability in the context of machine learning and introducing the Predictive, Descriptive, Relevant (PDR) framework for discussing interpretations. The PDR framework provides three overarching desiderata for evaluation: predictive accuracy, descriptive accuracy and relevancy, with relevancy judged relative to a human audience. Moreover, to help manage the deluge of interpretation methods, we introduce a categorization of existing techniques into model-based and post-hoc categories, with sub-groups including sparsity, modularity and simulatability. To demonstrate how practitioners can use the PDR framework to evaluate and understand interpretations, we provide numerous real-world examples. These examples highlight the often under-appreciated role played by human audiences in discussions of interpretability. Finally, based on our framework, we discuss limitations of existing methods and directions for future work. We hope that this work will provide a common vocabulary that will make it easier for both practitioners and researchers to discuss and choose from the full range of interpretation methods.