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Volume 21, Issue 5
Deep Surrogate Model for Learning Green’s Function Associated with Linear Reaction-Diffusion Operator

Junqing Jia, Lili Ju & Xiaoping Zhang

DOI: 10.4208/ijnam2024-1028

Int. J. Numer. Anal. Mod., 21 (2024), pp. 697-715.

Published online: 2024-10

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  • Abstract

In this paper, we present a deep surrogate model for learning the Green’s function associated with the reaction-diffusion operator in rectangular domain. The U-Net architecture is utilized to effectively capture the mapping from source to solution of the target partial differential equations (PDEs). To enable efficient training of the model without relying on labeled data, we propose a novel loss function that draws inspiration from traditional numerical methods used for solving PDEs. Furthermore, a hard encoding mechanism is employed to ensure that the predicted Green’s function is perfectly matched with the boundary conditions. Based on the learned Green’s function from the trained deep surrogate model, a fast solver is developed to solve the corresponding PDEs with different sources and boundary conditions. Various numerical examples are also provided to demonstrate the effectiveness of the proposed model.

  • Keywords

Reaction-diffusion operator, Green’s function, surrogate model, deep learning, fast solver.

  • AMS Subject Headings

34B27, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-21-697, author = {Jia , JunqingJu , Lili and Zhang , Xiaoping}, title = {Deep Surrogate Model for Learning Green’s Function Associated with Linear Reaction-Diffusion Operator }, journal = {International Journal of Numerical Analysis and Modeling}, year = {2024}, volume = {21}, number = {5}, pages = {697--715}, abstract = {

In this paper, we present a deep surrogate model for learning the Green’s function associated with the reaction-diffusion operator in rectangular domain. The U-Net architecture is utilized to effectively capture the mapping from source to solution of the target partial differential equations (PDEs). To enable efficient training of the model without relying on labeled data, we propose a novel loss function that draws inspiration from traditional numerical methods used for solving PDEs. Furthermore, a hard encoding mechanism is employed to ensure that the predicted Green’s function is perfectly matched with the boundary conditions. Based on the learned Green’s function from the trained deep surrogate model, a fast solver is developed to solve the corresponding PDEs with different sources and boundary conditions. Various numerical examples are also provided to demonstrate the effectiveness of the proposed model.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2024-1028}, url = {http://global-sci.org/intro/article_detail/ijnam/23449.html} }
TY - JOUR T1 - Deep Surrogate Model for Learning Green’s Function Associated with Linear Reaction-Diffusion Operator AU - Jia , Junqing AU - Ju , Lili AU - Zhang , Xiaoping JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 697 EP - 715 PY - 2024 DA - 2024/10 SN - 21 DO - http://doi.org/10.4208/ijnam2024-1028 UR - https://global-sci.org/intro/article_detail/ijnam/23449.html KW - Reaction-diffusion operator, Green’s function, surrogate model, deep learning, fast solver. AB -

In this paper, we present a deep surrogate model for learning the Green’s function associated with the reaction-diffusion operator in rectangular domain. The U-Net architecture is utilized to effectively capture the mapping from source to solution of the target partial differential equations (PDEs). To enable efficient training of the model without relying on labeled data, we propose a novel loss function that draws inspiration from traditional numerical methods used for solving PDEs. Furthermore, a hard encoding mechanism is employed to ensure that the predicted Green’s function is perfectly matched with the boundary conditions. Based on the learned Green’s function from the trained deep surrogate model, a fast solver is developed to solve the corresponding PDEs with different sources and boundary conditions. Various numerical examples are also provided to demonstrate the effectiveness of the proposed model.

Jia , JunqingJu , Lili and Zhang , Xiaoping. (2024). Deep Surrogate Model for Learning Green’s Function Associated with Linear Reaction-Diffusion Operator . International Journal of Numerical Analysis and Modeling. 21 (5). 697-715. doi:10.4208/ijnam2024-1028
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