The Kernel-Free Boundary Integral (KFBI) method presents an iterative
solution to boundary integral equations arising from elliptic partial
differential equations (PDEs). This method effectively addresses elliptic PDEs
on irregular domains, including the modified Helmholtz, Stokes, and elasticity
equations. The rapid evolution of neural networks and deep learning has
invigorated the exploration of numerical PDEs. An increasing interest is
observed in deep learning approaches that seamlessly integrate mathematical
principles for investigating numerical PDEs. We propose a hybrid KFBI method,
integrating the foundational principles of the KFBI method with the
capabilities of deep learning. This approach, within the framework of the
boundary integral method, designs a network to approximate the solution
operator for the corresponding integral equations by mapping the parameters,
inhomogeneous terms and boundary information of PDEs to the boundary density
functions, which can be regarded as the solution of the integral equations. The
models are trained using data generated by the Cartesian grid-based KFBI
algorithm, exhibiting robust generalization capabilities. It accurately
predicts density functions across diverse boundary conditions and parameters
within the same class of equations. Experimental results demonstrate that the
trained model can directly infer the boundary density function with
satisfactory precision, obviating the need for iterative steps in solving
boundary integral equations. Furthermore, applying the inference results of the
model as initial values for iterations is also reasonable; this approach can
retain the inherent second-order accuracy of the KFBI method while accelerating
the traditional KFBI approach by reducing about 50% iterations.

在边界积分方程的框架下，基于深度学习的核自由边界积分法 (KFBI) 综合了 KFBI 方法的基本原理和深度学习的能力，通过将偏微分方程的参数、非齐次项和边界信息映射到边界密度函数，设计了一种网络近似解算符的方法。在使用 Cartesian 网格 KFBI 算法生成的数据进行训练后，该方法展示了强大的泛化能力，准确预测了同类方程中不同边界条件和参数的密度函数。实验证明，该训练模型可以直接推断边界密度函数，并且其预测精度令人满意，无需迭代解边界积分方程。此外，使用模型的推断结果作为迭代的初始值也是合理的，这种方法可以保持 KFBI 方法的内在二阶精度，同时通过减少约 50％的迭代步骤来加速传统的 KFBI 方法。

在复杂领域解决参数化偏微分方程的混合无核边界积分方法和算子学习

A Hybrid Kernel-Free Boundary Integral Method with Operator Learning for  Solving Parametric Partial Differential Equations In Complex Domains

In this paper, a novel method to perform model-based clustering of time
series is proposed. The procedure relies on two iterative steps: (i) K global
forecasting models are fitted via pooling by considering the series pertaining
to each cluster and (ii) each series is assigned to the group associated with
the model producing the best forecasts according to a particular criterion.
Unlike most techniques proposed in the literature, the method considers the
predictive accuracy as the main element for constructing the clustering
partition, which contains groups jointly minimizing the overall forecasting
error. Thus, the approach leads to a new clustering paradigm where the quality
of the clustering solution is measured in terms of its predictive capability.
In addition, the procedure gives rise to an effective mechanism for selecting
the number of clusters in a time series database and can be used in combination
with any class of regression model. An extensive simulation study shows that
our method outperforms several alternative techniques concerning both
clustering effectiveness and predictive accuracy. The approach is also applied
to perform clustering in several datasets used as standard benchmarks in the
time series literature, obtaining great results.

本文提出了一种新的基于模型的时间序列聚类方法，该方法通过考虑每个聚类所属的系列来拟合 K 个全局预测模型，然后将每个系列分配给产生最佳预测的模型关联的组。该方法以预测准确性作为构建聚类分区的主要元素，产生了一种新的聚类范例，其中聚类解决方案的质量是通过其预测能力来衡量的。此外，该过程还产生了一种选择时间序列数据库中聚类数量的有效机制，并且可以与任何类回归模型结合使用。大量模拟研究表明，该方法优于其他几种替代技术，包括聚类效果和预测准确性。该方法还应用于多个时间序列文献中使用的标准基准数据集的聚类中，产生了良好的结果。