Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915454963941376 |
|---|---|
| author | Cen, Siyu Jin, Bangti Quan, Qimeng Zhou, Zhi |
| author_facet | Cen, Siyu Jin, Bangti Quan, Qimeng Zhou, Zhi |
| contents | Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve PDE parameter identifications. These approaches employ neural networks as ansatz functions to approximate the parameters and / or the states, and have demonstrated impressive empirical performance. In this paper, we provide a comprehensive survey on these unsupervised learning techniques on one model problem, diffusion coefficient identification, from the classical numerical analysis perspective, and outline a general framework for deriving rigorous error bounds on the discrete approximations obtained using the Galerkin finite element method, hybrid method and deep neural networks. Throughout we highlight the crucial role of conditional stability estimates in the error analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs Cen, Siyu Jin, Bangti Quan, Qimeng Zhou, Zhi Numerical Analysis Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve PDE parameter identifications. These approaches employ neural networks as ansatz functions to approximate the parameters and / or the states, and have demonstrated impressive empirical performance. In this paper, we provide a comprehensive survey on these unsupervised learning techniques on one model problem, diffusion coefficient identification, from the classical numerical analysis perspective, and outline a general framework for deriving rigorous error bounds on the discrete approximations obtained using the Galerkin finite element method, hybrid method and deep neural networks. Throughout we highlight the crucial role of conditional stability estimates in the error analysis. |
| title | Numerical Analysis of Unsupervised Learning Approaches for Parameter Identification in PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2508.15381 |