Learning Green’s Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver

Yuankai Teng, Xiaoping Zhang, Zhu Wang, Lili Ju
Proceedings of Mathematical and Scientific Machine Learning, PMLR 190:1-16, 2022.

Abstract

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and so on. Due to their important applications in scientific research and engineering, many numerical methods have been developed in the past decades for efficient and accurate solutions of these equations. Inspired by the rapidly growing impact of deep learning techniques, we propose in this paper a novel neural network method, “GF-Net”, for learning the Green’s functions of the classic linear reaction-diffusion equations in the unsupervised fashion. The proposed method overcomes the challenges for finding the Green’s functions of the equations on arbitrary domains by utilizing the physics-informed neural network approach and domain decomposition. As a consequence, it particularly leads to a fast algorithm for solving the target equations subject to various sources and Dirichlet boundary conditions without network retraining. We also numerically demonstrate the effectiveness of the proposed method by extensive experiments in the square, annular and L-shape domains.

Cite this Paper

BibTeX
@InProceedings{pmlr-v190-teng22a, title = {Learning Green’s Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver}, author = {Teng, Yuankai and Zhang, Xiaoping and Wang, Zhu and Ju, Lili}, booktitle = {Proceedings of Mathematical and Scientific Machine Learning}, pages = {1--16}, year = {2022}, editor = {Dong, Bin and Li, Qianxiao and Wang, Lei and Xu, Zhi-Qin John}, volume = {190}, series = {Proceedings of Machine Learning Research}, month = {15--17 Aug}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v190/teng22a/teng22a.pdf}, url = {https://proceedings.mlr.press/v190/teng22a.html}, abstract = {Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and so on. Due to their important applications in scientific research and engineering, many numerical methods have been developed in the past decades for efficient and accurate solutions of these equations. Inspired by the rapidly growing impact of deep learning techniques, we propose in this paper a novel neural network method, “GF-Net”, for learning the Green’s functions of the classic linear reaction-diffusion equations in the unsupervised fashion. The proposed method overcomes the challenges for finding the Green’s functions of the equations on arbitrary domains by utilizing the physics-informed neural network approach and domain decomposition. As a consequence, it particularly leads to a fast algorithm for solving the target equations subject to various sources and Dirichlet boundary conditions without network retraining. We also numerically demonstrate the effectiveness of the proposed method by extensive experiments in the square, annular and L-shape domains.} }
Endnote
%0 Conference Paper %T Learning Green’s Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver %A Yuankai Teng %A Xiaoping Zhang %A Zhu Wang %A Lili Ju %B Proceedings of Mathematical and Scientific Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Bin Dong %E Qianxiao Li %E Lei Wang %E Zhi-Qin John Xu %F pmlr-v190-teng22a %I PMLR %P 1--16 %U https://proceedings.mlr.press/v190/teng22a.html %V 190 %X Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and so on. Due to their important applications in scientific research and engineering, many numerical methods have been developed in the past decades for efficient and accurate solutions of these equations. Inspired by the rapidly growing impact of deep learning techniques, we propose in this paper a novel neural network method, “GF-Net”, for learning the Green’s functions of the classic linear reaction-diffusion equations in the unsupervised fashion. The proposed method overcomes the challenges for finding the Green’s functions of the equations on arbitrary domains by utilizing the physics-informed neural network approach and domain decomposition. As a consequence, it particularly leads to a fast algorithm for solving the target equations subject to various sources and Dirichlet boundary conditions without network retraining. We also numerically demonstrate the effectiveness of the proposed method by extensive experiments in the square, annular and L-shape domains.
APA
Teng, Y., Zhang, X., Wang, Z. & Ju, L.. (2022). Learning Green’s Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver. Proceedings of Mathematical and Scientific Machine Learning, in Proceedings of Machine Learning Research 190:1-16 Available from https://proceedings.mlr.press/v190/teng22a.html.

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