Learning Green's Function Efficiently Using Low-Rank Approximations

Abstract

Learning the Green’s function using deep learning models enables efficient parametrization of partial differential equations. However, a practical limitation of this approach is the repeated computation of Monte Carlo integral approximations, which makes the learning process computationally expensive. To address this, we propose DecGreenNet, a novel algorithm that uses low-rank decomposition to learn the Green’s function efficiently. This novel architecture predicts the solution of PDE at a grid element using the product of two networks; one taking each grid element as input and the other taking the Monte Carlo samples as input. Experimental results show that the proposed method achieves faster training times compared to MOD-Net while maintaining comparable or lower prediction error relative to both PINNs and MOD-Net. We also provide a theoretical analysis for Green’s function based PINNs, including both DecGreenNet and MOD-Net, using a clipped Green’s function. Our analysis shows that both MOD-Net and DecGreenNet obtains similar convergence rates.