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<p>Mathematical modelling is fundamental to understanding real-world phenomena. Despite the inherent complexity in designing such models, numerical approaches and, more recently, machine learning techniques, have emerged as powerful tools in this area. This work proposes integrating the finite element method (FEM) into forecasting and introduces parallel techniques for regression problems, with a specific focus on the use of Matérn kernels on local mesh support. This approach generalises the modelling based on radial basis function kernels and offers more flexibility to control the smoothness of the modelled functions. An exhaustive study explores the impact of diverse norms and Matérn kernel variations on the performance of models, and aims to improve the computational efficiency of the model fitting and prediction processes. Furthermore, a heuristic framework is introduced to derive optimal complexity parameters for each Matérn-based FEM kernel. The proposed parallel approaches use dynamic strategies, which significantly reduce the computational time of the algorithms compared to other methods and parallel computing techniques presented in recent years. The proposed methodology is assessed in the context of bias corrections for temperature forecasts made by the Local Data Assimilation and Prediction System (LDAPS) model. A comprehensive comparative analysis which includes machine learning algorithms provides significant insights into the training process, norm selection, and kernel choice, and shows that Matérn-based methods emerge as a choice to be considered for regression problems.</p>

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Design and Comparison of Parallel Dynamic Matérn Kernel-Based Regression Models and Machine Learning Approaches: Application to Bias Correction in Numerical Weather Prediction

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月27日

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  • Violeta Migallón,
  • Héctor Penadés,
  • José Penadés

摘要

Mathematical modelling is fundamental to understanding real-world phenomena. Despite the inherent complexity in designing such models, numerical approaches and, more recently, machine learning techniques, have emerged as powerful tools in this area. This work proposes integrating the finite element method (FEM) into forecasting and introduces parallel techniques for regression problems, with a specific focus on the use of Matérn kernels on local mesh support. This approach generalises the modelling based on radial basis function kernels and offers more flexibility to control the smoothness of the modelled functions. An exhaustive study explores the impact of diverse norms and Matérn kernel variations on the performance of models, and aims to improve the computational efficiency of the model fitting and prediction processes. Furthermore, a heuristic framework is introduced to derive optimal complexity parameters for each Matérn-based FEM kernel. The proposed parallel approaches use dynamic strategies, which significantly reduce the computational time of the algorithms compared to other methods and parallel computing techniques presented in recent years. The proposed methodology is assessed in the context of bias corrections for temperature forecasts made by the Local Data Assimilation and Prediction System (LDAPS) model. A comprehensive comparative analysis which includes machine learning algorithms provides significant insights into the training process, norm selection, and kernel choice, and shows that Matérn-based methods emerge as a choice to be considered for regression problems.