A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales. (English) Zbl 1474.65445
Summary: 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 Z. Liu et al. [Commun. Comput. Phys. 28, No. 5, 1970–2001 (2020; Zbl 1473.65348)] 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.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J66 | Nonlinear boundary value problems for nonlinear elliptic equations |
| 41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |
| 68T07 | Artificial neural networks and deep learning |
Keywords:
multi-scale elliptic problem; \(p\)-Laplacian equation; deep neural network (DNN); variational formulation; activation functionCitations:
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