XI-DeepONet: An operator learning method for elliptic interface problems
•Proposing an operator learning architecture to solve the elliptic interface problems.•Effectively handling the low regularity of solutions and complex interface geometries.•The network can infer solutions with various interface geometries once trained.•The network can be trained without any paired...
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| Vydáno v: | Journal of computational physics Ročník 538; s. 114164 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier Inc
01.10.2025
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| Témata: | |
| ISSN: | 0021-9991 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | •Proposing an operator learning architecture to solve the elliptic interface problems.•Effectively handling the low regularity of solutions and complex interface geometries.•The network can infer solutions with various interface geometries once trained.•The network can be trained without any paired input-output data.•The numerical results demonstrate the robustness and effectiveness.
Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges when dealing with problems involving complex geometries. Machine learning-based methods, on the other hand, are mesh-free, thus providing a promising alternative. In particular, operator learning methods have been proposed to learn the mapping from the input space to the solution space, enabling rapid inference of solutions to partial differential equations (PDEs) once trained. In this work, we address the parametric elliptic interface problem. Building upon the deep operator network (DeepONet), we propose an extended interface deep operator network (XI–DeepONet) that embeds the level set function used to represent the interface into the branch and trunk networks. XI–DeepONet exhibits three unique features: (1) The interface geometry is incorporated into the neural network as an additional input, enabling the network to infer solutions for new interface geometries once trained; (2) By augmenting the level set function within its trunk network, it is capable of efficiently capturing the cusp (where the derivatives are discontinuous) of the solutions at the interface. (3) The network can be trained without requiring any input-output data pairs, thus completely avoiding the need for meshes of any kind, directly or indirectly. We conduct a comprehensive series of numerical experiments to demonstrate the accuracy and robustness of the proposed method. |
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| ISSN: | 0021-9991 |
| DOI: | 10.1016/j.jcp.2025.114164 |