Solving multiscale elliptic problems by sparse radial basis function neural networks

Abstract

Machine learning has been successfully applied to various fields of scientific computing in recent years. For problems with multiscale features such as flows in porous media and mechanical properties of composite materials, however, it remains difficult for machine learning methods to resolve such multiscale features, especially when the small scale information is present. 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 avoid the overfitting and improve the performance of RBFNN, an additional regularization is introduced in the loss function. Thus the loss function contains two parts: the L2 loss for the residual of the first-order system and boundary conditions, and the ℓ1 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 ℓ1 regularization can achieve the goal of representing the solution by fewer RBFs. As a consequence, the total number of RBFs scales like O(ε−nτ), where ε is the smallest scale, n is the dimensionality, and τ 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. Executable files are available at https://github.com/wangzhiwensuda/SRBFNN-for-multiscale-elliptic-problems. The program was implemented in PyTorch and was run on Linux.