Learning Singularity-Encoded Green's Functions with Application to Iterative Methods
This work addresses a domain-specific problem in computational mathematics for researchers and practitioners dealing with PDE solvers, offering an incremental improvement through a novel hybrid method.
The paper tackles the challenge of numerically computing Green's functions for elliptic PDEs, which are difficult due to high dimensionality and singularities, by introducing a singularity-encoded learning approach that accelerates iterative solvers, achieving satisfactory resolution and acceleration in experiments with 2D and 4D cases.
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.