论文标题
具有多个尺度的非线性椭圆方程的多尺度DNN算法
A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales
论文作者
论文摘要
基于深神经网络（DNN）的算法引起了科学计算社区的越来越多的关注。基于DNN的算法易于实现，对于非线性问题而自然，并且显示出很大的潜力来克服维度的诅咒。在这项工作中，我们利用Liu，Cai和Xu（2020）提出的基于多尺度DNN的算法（MSCALEDNN）来解决具有可能非线性的多规模椭圆问题，例如P-Laplacian问题。我们通过平滑而局部的激活函数来改善MSCALEDNN算法。在低维和高维欧几里得空间中具有可分离或不可分割量表的多尺度椭圆问题的数值示例用于证明MSCALEDNN数值方案的有效性和准确性。
Algorithms based on deep neural networks (DNNs) have attracted increasing attention from the scientific computing community. DNN based algorithms are easy to implement, natural for nonlinear problems, and have shown great potential to overcome the curse of dimensionality. In this work, we utilize the multi-scale DNN-based algorithm (MscaleDNN) proposed by Liu, Cai and Xu (2020) to solve multi-scale elliptic problems with possible nonlinearity, for example, the p-Laplacian problem. We improve the MscaleDNN algorithm by a smooth and localized activation function. Several numerical examples of multi-scale elliptic problems with separable or non-separable scales in low-dimensional and high-dimensional Euclidean spaces are used to demonstrate the effectiveness and accuracy of the MscaleDNN numerical scheme.
