A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales

Lei Zhang; Xi-An Li; Zhi-Qin John Xu

A multi-scale DNN algorithm for nonlinear elliptic equations with multiple scales

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arxiv 2009.14597 v2 pith:MV2U5UG2 submitted 2020-09-30 physics.comp-ph math.AP

Xi-An Li , Zhi-Qin John Xu , Lei Zhang This is my paper · ORCID

classification physics.comp-phmath.AP

keywords multi-scalealgorithmellipticmscalednnproblemsalgorithmsnonlinearnumerical

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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.

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Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems
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MR-PIKAN, a multi-resolution training schedule that alternates coarse and fine collocation grids, cuts training time while keeping accuracy on multi-scale forward and inverse PDE problems.

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