Solving multiscale elliptic problems by sparse radial basis function neural networks

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