We propose a data-driven and machine-learning-based approach to compute
non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods,
addressing the well-known issue of increasing operator complexity. Guided by
the AMG theory on spectrally equivalent coarse-grid operators, we have
developed novel ML algorithms that utilize neural networks (NNs) combined with
smooth test vectors from multigrid eigenvalue problems. The proposed method
demonstrates promise in reducing the complexity of coarse-grid operators while
maintaining overall AMG convergence for solving parametric partial differential
equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian
and linear elasticity problems are provided to showcase the performance and
compare with existing methods for computing non-Galerkin coarse-grid operators.

提出了一种基于数据驱动和机器学习的方法来计算代数多重网格 (AMG) 方法中的非 Galerkin 粗网格算子，同时保持整体的 AMG 收敛性，以解决操作符复杂度增加的问题。

利用机器学习方法简化代数多重网格的算子复杂度

Reducing operator complexity in Algebraic Multigrid with Machine  Learning Approaches

In this paper, we propose physics-informed neural operators (PINO) that uses
available data and/or physics constraints to learn the solution operator of a
family of parametric Partial Differential Equation (PDE). This hybrid approach
allows PINO to overcome the limitations of purely data-driven and physics-based
methods. For instance, data-driven methods fail to learn when data is of
limited quantity and/or quality, and physics-based approaches fail to optimize
on challenging PDE constraints. By combining both data and PDE constraints,
PINO overcomes all these challenges. Additionally, a unique property that PINO
enjoys over other hybrid learning methods is its ability to incorporate data
and PDE constraints at different resolutions. This allows us to combine
coarse-resolution data, which is inexpensive to obtain from numerical solvers,
with higher resolution PDE constraints, and the resulting PINO has no
degradation in accuracy even on high-resolution test instances. This
discretization-invariance property in PINO is due to neural-operator framework
which learns mappings between function spaces and allows evaluation at
different resolutions without the need for re-training. Moreover, PINO succeeds
in the purely physics setting, where no data is available, while other
approaches such as the Physics-Informed Neural Network (PINN) fail due to
optimization challenges, e.g. in multi-scale dynamic systems such as Kolmogorov
flows. This is because PINO learns the solution operator by optimizing PDE
constraints on multiple instances while PINN optimizes PDE constraints of a
single PDE instance. Further, in PINO, we incorporate the Fourier neural
operator (FNO) architecture which achieves orders-of-magnitude speedup over
numerical solvers and also allows us to compute explicit gradients on function
spaces efficiently.

本文提出物理信息神经操作器（PINO），该方法使用现有的数据和物理约束条件来学习参数化偏微分方程（PDE）族的解算器，通过结合数据和 PDE 约束条件，PINO 成功地实现了高分辨率实例的准确性和泛化能力。