Machine learning has been successfully applied to various fields of scientific computing in recent years. In this work, we propose a sparse radial basis function neural network method to solve elliptic partial differential equations (PDEs) with multiscale coefficients. Inspired by the deep mixed residual method, we rewrite the second-order problem into a first-order system and employ multiple radial basis function neural networks (RBFNNs) to approximate unknown functions in the system. To aviod the overfitting due to the simplicity of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the $L_2$ loss for the residual of the first-order system and boundary conditions, and the $\ell_1$ regularization term for the weights of radial basis functions (RBFs). An algorithm for optimizing the specific loss function is introduced to accelerate the training process. The accuracy and effectiveness of the proposed method are demonstrated through a collection of multiscale problems with scale separation, discontinuity and multiple scales from one to three dimensions. Notably, the $\ell_1$ regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like $\mathcal{O}(\varepsilon^{-n\tau})$, where $\varepsilon$ is the smallest scale, $n$ is the dimensionality, and $\tau$ is typically smaller than $1$. It is worth mentioning that the proposed method not only has the numerical convergence and thus provides a reliable numerical solution in three dimensions when a classical method is typically not affordable, but also outperforms most other available machine learning methods in terms of accuracy and robustness.

翻译：近年来，机器学习已成功应用于科学计算的各个领域。本文提出了一种稀疏径向基函数神经网络方法，用于求解具有多尺度系数的椭圆型偏微分方程。受深度混合残差方法的启发，我们将二阶问题改写为一阶系统，并采用多个径向基函数神经网络逼近系统中的未知函数。为避免径向基函数神经网络过拟合，我们在损失函数中引入了额外的正则化项。因此，损失函数包含两部分：一阶系统残差与边界条件的$L_2$损失，以及径向基函数权重的$\ell_1$正则化项。我们引入了一种针对该损失函数的优化算法以加速训练过程。通过一系列包含尺度分离、间断性和多尺度特征的一维至三维多尺度问题，验证了所提方法的精度与有效性。值得注意的是，$\ell_1$正则化能够实现用更少的径向基函数表征解的目标。因此，径向基函数的总数尺度为$\mathcal{O}(\varepsilon^{-n\tau})$，其中$\varepsilon$为最小尺度，$n$为维度，$\tau$通常小于1。值得强调的是，该方法不仅具有数值收敛性，从而在经典方法难以承受的三维计算中提供可靠的数值解，而且在精度和鲁棒性方面优于大多数现有机器学习方法。