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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2502.19392 |
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| _version_ | 1866913994627874816 |
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| author | Akram, Wasim Gautam, Sagar Verma, Deepanshu Mohan, Manil T. |
| author_facet | Akram, Wasim Gautam, Sagar Verma, Deepanshu Mohan, Manil T. |
| contents | The article focuses on error estimates as well as stability analysis of deep learning methods for stationary and non-stationary viscous Burgers equation in two and three dimensions. The local well-posedness of homogeneous boundary value problem for non-stationary viscous Burgers equation is established by using semigroup techniques and fixed point arguments. By considering a suitable approximate problem and deriving appropriate energy estimates, we prove the existence of a unique strong solution. Additionally, we extend our analysis to the global well-posedness of the non-homogeneous problem. For both the stationary and non-stationary cases, we derive explicit error estimates in suitable Lebesgue and Sobolev norms by optimizing a loss function in a Deep Neural Network approximation of the solution with fixed complexity. Finally, numerical results on prototype systems are presented to illustrate the derived error estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19392 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error estimates for viscous Burgers' equation using deep learning method Akram, Wasim Gautam, Sagar Verma, Deepanshu Mohan, Manil T. Numerical Analysis The article focuses on error estimates as well as stability analysis of deep learning methods for stationary and non-stationary viscous Burgers equation in two and three dimensions. The local well-posedness of homogeneous boundary value problem for non-stationary viscous Burgers equation is established by using semigroup techniques and fixed point arguments. By considering a suitable approximate problem and deriving appropriate energy estimates, we prove the existence of a unique strong solution. Additionally, we extend our analysis to the global well-posedness of the non-homogeneous problem. For both the stationary and non-stationary cases, we derive explicit error estimates in suitable Lebesgue and Sobolev norms by optimizing a loss function in a Deep Neural Network approximation of the solution with fixed complexity. Finally, numerical results on prototype systems are presented to illustrate the derived error estimates. |
| title | Error estimates for viscous Burgers' equation using deep learning method |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.19392 |