Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910671525904384 |
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| author | Cai, Zhiqiang Choi, Junpyo Liu, Min |
| author_facet | Cai, Zhiqiang Choi, Junpyo Liu, Min |
| contents | The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent least-squares formulation and \cite{cai2023least} employs ReLU neural network (NN) functions with $\lceil \log_2(d+1)\rceil+1$-layer representations for approximating solutions. In this paper, we show theoretically that the method is also capable of accurately approximating non-constant jumps along discontinuous interfaces that are not necessarily straight lines. Theoretical results are confirmed through multiple numerical examples with $d=2,3$ and various non-constant jumps and interface shapes, showing that the LSNN method with $\lceil \log_2(d+1)\rceil+1$ layers approximates solutions accurately with degrees of freedom less than that of mesh-based methods and without the common Gibbs phenomena along discontinuous interfaces having non-constant jumps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_07445 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps Cai, Zhiqiang Choi, Junpyo Liu, Min Numerical Analysis 65N15, 65N99 The least-squares ReLU neural network (LSNN) method was introduced and studied for solving linear advection-reaction equation with discontinuous solution in \cite{Cai2021linear,cai2023least}. The method is based on an equivalent least-squares formulation and \cite{cai2023least} employs ReLU neural network (NN) functions with $\lceil \log_2(d+1)\rceil+1$-layer representations for approximating solutions. In this paper, we show theoretically that the method is also capable of accurately approximating non-constant jumps along discontinuous interfaces that are not necessarily straight lines. Theoretical results are confirmed through multiple numerical examples with $d=2,3$ and various non-constant jumps and interface shapes, showing that the LSNN method with $\lceil \log_2(d+1)\rceil+1$ layers approximates solutions accurately with degrees of freedom less than that of mesh-based methods and without the common Gibbs phenomena along discontinuous interfaces having non-constant jumps. |
| title | Least-Squares Neural Network (LSNN) Method For Linear Advection-Reaction Equation: Non-constant Jumps |
| topic | Numerical Analysis 65N15, 65N99 |
| url | https://arxiv.org/abs/2306.07445 |