Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918135941038080 |
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| author | Zhu, Qiao Chaikovskii, Dmitrii Jin, Bangti Zhang, Ye |
| author_facet | Zhu, Qiao Chaikovskii, Dmitrii Jin, Bangti Zhang, Ye |
| contents | Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations Zhu, Qiao Chaikovskii, Dmitrii Jin, Bangti Zhang, Ye Numerical Analysis 35B25, 65D17, 65N99, 68T07 Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed time-dependent reaction-advection-diffusion equations exhibit internal moving transition layers with sharp gradients, and thus the standard PINN becomes ineffective. In this work, we propose a deep asymptotic expansion (DAE) method, which is inspired by asymptotic analysis and leverages deep learning to approximate the smooth part of the expansion. We first derive the governing equations for transition layers, which are then solved using PINN. Numerical experiments show that the DAE outperforms the standard PINN, gPINN, and PINN with adaptive sampling. We also show its robustness with respect to training point distributions, network architectures, and random seeds. |
| title | Deep asymptotic expansion method for solving singularly perturbed time-dependent reaction-advection-diffusion equations |
| topic | Numerical Analysis 35B25, 65D17, 65N99, 68T07 |
| url | https://arxiv.org/abs/2505.23002 |