Green Multigrid Network

TL;DR

GreenMGNet addresses the challenge of learning operator solution maps for PDEs by focusing on asymptotically smooth Green's functions. It introduces a piecewise kernel model via AugNN to handle diagonal singularities and applies Multi-Level Multi-Integration (MLMI) to accelerate kernel-vector products during training and inference. The method yields 3.8% to 39.15% improvements in accuracy over GL, while requiring only about 10% of full-grid data, resulting in substantial reductions in training time and GPU memory for 1D and 2D problems. The approach extends beyond Green's functions to other kernels with asymptotic smoothness and offers a scalable pathway for efficient operator learning in elliptic PDE contexts.

Abstract

GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions. Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves $3.8\%$ to $39.15\%$ accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about $10\%$ of the full grid data, resulting in a $55.9\%$ and $92.5\%$ reduction in training time and GPU memory cost for one-dimensional test problems, and a $37.7\%$ and $62.5\%$ reduction for two-dimensional test problems.

Figures (14)

• Figure 1: Schematic of Green Learning
• Figure 2: (a): The slice of Green's function at $x=0.25,x=0.5,x=0.75$ and the cusp appears at $x=y$. (b): The Green's function on $[0,1] \times [0,1]$ and $x=y$(white dash line) divides the domain into two subdomains as an interface.
• Figure 3: (a): the slice of Green's function at $(x_1, x_2, 0, 0)$, where singularity exists at $(0,0,0,0)$. (b): the slice of Green's function at $(x_1, 0, y_1, 0)$ where singularity exists at $x_1=y_1$. The white dash line is the hyperplane $x_1 - y_1 + x_2 - y_2 = 0$ on each slice.
• Figure 4: (a): the architecture of AugNN. (b): Illustration of subdomains introduced in (\ref{['eq:subdomains']}), averaging $G_1$ and $G_2$ is need on $D_4$ for smoothing.
• Figure 5: MLMI only uses the subset of kernel values, which contains values on the coarsest grid points and the neighbors of diagonal points on each finer grid, for matrix-vector product calculation. Neighbors on each finer grid can be further classified as neighbors of $i=2I$, and neighbors of $i \neq 2I$, where $i$ and $I$ are relative indices between two levels. After decomposing the kernel values into subsets, it is possible to recover an approximant of the original full dense kernel by interpolation and corrections.