Learning Green's Function Efficiently Using Low-Rank Approximations
This work addresses a practical limitation in solving partial differential equations for computational physics and engineering, though it appears incremental as it builds on existing methods like MOD-Net and PINNs.
The paper tackles the computational inefficiency of learning Green's functions with deep learning by proposing a low-rank decomposition method, which reduces computational time compared to MOD-Net while maintaining accuracy similar to PINNs and MOD-Net.
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.