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| Natura: | Preprint |
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2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2601.20978 |
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| _version_ | 1866908795718860800 |
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| author | Khosravi, Omid Tatari, Mehdi |
| author_facet | Khosravi, Omid Tatari, Mehdi |
| contents | In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two-stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solution of Advection Equation with Discontinuous Initial and Boundary Conditions via Physics-Informed Neural Networks Khosravi, Omid Tatari, Mehdi Numerical Analysis In this paper, we investigate several techniques for modeling the one-dimensional advection equation for a specific class of problems with discontinuous initial and boundary conditions using physics-informed neural networks (PINNs). To mitigate the spectral bias phenomenon, we employ a Fourier feature mapping layer as the input representation, adopt a two-stage training strategy in which the Fourier feature parameters and the neural network weights are optimized sequentially, and incorporate adaptive loss weighting. To further enhance the approximation accuracy, a median filter is applied to the spatial data, and the predicted solution is constrained through a bounded linear mapping. Moreover, for certain nonlinear problems, we introduce a modified loss function inspired by the upwind numerical scheme to alleviate the excessive smoothing of discontinuous solutions typically observed in neural network approximations. |
| title | Solution of Advection Equation with Discontinuous Initial and Boundary Conditions via Physics-Informed Neural Networks |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2601.20978 |