Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.

Publication:

JOURNAL OF COMPUTATIONAL PHYSICS

http://dx.doi.org/10.1016/j.jcp.2026.114894

Author:

Qi Sun

School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092, China

Corresponding author.

E-mail addresses: qsun_irl@tongji.edu.cn

Shengyan Li

School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

mail addresses: 2410285@tongji.edu.cn

Bowen Zheng

LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

E-mail addresses:zhengbowen@lsec.cc.ac.cn

Lili Ju

Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

E-mail addresses:ju@math.sc.edu

Xuejun Xu

School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University, Shanghai, 200092, China

E-mail addresses:xuxj@tongji.edu.cn