Learning Green's Function Efficiently Using Low-Rank Approximations

TL;DR

This paper introduces a low-rank approximation method for efficiently learning Green's functions with deep learning, reducing computational costs while maintaining accuracy in solving PDEs.

Contribution

The paper proposes a novel low-rank decomposition architecture that separates domain data learning from Monte-Carlo sampling, improving efficiency over existing methods.

Findings

Reduces computational time compared to MOD-Net

Achieves accuracy comparable to PINNs and MOD-Net

Demonstrates effectiveness through experimental validation

Abstract

Learning the Green's function using deep learning models enables to solve different classes of partial differential equations. A practical limitation of using deep learning for the Green's function is the repeated computationally expensive Monte-Carlo integral approximations. We propose to learn the Green's function by low-rank decomposition, which results in a novel architecture to remove redundant computations by separate learning with domain data for evaluation and Monte-Carlo samples for integral approximation. Using experiments we show that the proposed method improves computational time compared to MOD-Net while achieving comparable accuracy compared to both PINNs and MOD-Net.