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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2410.00053 |
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| _version_ | 1866916417080655872 |
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| author | Huang, Jizu You, Rukang Zhou, Tao |
| author_facet | Huang, Jizu You, Rukang Zhou, Tao |
| contents | Multi-scale deep neural networks (MscaleDNNs) with downing-scaling mapping have demonstrated superiority over traditional DNNs in approximating target functions characterized by high frequency features. However, the performance of MscaleDNNs heavily depends on the parameters in the downing-scaling mapping, which limits their broader application. In this work, we establish a fitting error bound to explain why MscaleDNNs are advantageous for approximating high frequency functions. Building on this insight, we construct a hybrid feature embedding to enhance the accuracy and robustness of the downing-scaling mapping. To reduce the dependency of MscaleDNNs on parameters in the downing-scaling mapping, we propose frequency-adaptive MscaleDNNs, which adaptively adjust these parameters based on a posterior error estimate that captures the frequency information of the fitted functions. Numerical examples, including wave propagation and the propagation of a localized solution of the schr$\ddot{\text{o}}$dinger equation with a smooth potential near the semi-classical limit, are presented. These examples demonstrate that the frequency-adaptive MscaleDNNs improve accuracy by two to three orders of magnitude compared to standard MscaleDNNs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00053 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frequency-adaptive Multi-scale Deep Neural Networks Huang, Jizu You, Rukang Zhou, Tao Machine Learning Multi-scale deep neural networks (MscaleDNNs) with downing-scaling mapping have demonstrated superiority over traditional DNNs in approximating target functions characterized by high frequency features. However, the performance of MscaleDNNs heavily depends on the parameters in the downing-scaling mapping, which limits their broader application. In this work, we establish a fitting error bound to explain why MscaleDNNs are advantageous for approximating high frequency functions. Building on this insight, we construct a hybrid feature embedding to enhance the accuracy and robustness of the downing-scaling mapping. To reduce the dependency of MscaleDNNs on parameters in the downing-scaling mapping, we propose frequency-adaptive MscaleDNNs, which adaptively adjust these parameters based on a posterior error estimate that captures the frequency information of the fitted functions. Numerical examples, including wave propagation and the propagation of a localized solution of the schr$\ddot{\text{o}}$dinger equation with a smooth potential near the semi-classical limit, are presented. These examples demonstrate that the frequency-adaptive MscaleDNNs improve accuracy by two to three orders of magnitude compared to standard MscaleDNNs. |
| title | Frequency-adaptive Multi-scale Deep Neural Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.00053 |