An intelligent sparse approximate inverse selection method based on convolutional neural networks

As a class of preconditioners, sparse approximate inverses (SAIs) have been proven to be effective in accelerating the convergence of iterative methods. However, given the wide variety of available preconditioners, determining how to select the most suitable one for a specific iterative solver remains a central challenge in scientific computing. In the past decade, the rapid development and widespread adoption of convolutional neural networks (CNNs) have inspired new perspectives for tackling this challenge. When applying CNNs to the parallel construction of SAIs, three key issues must be addressed: normalized representations of sparse matrices, network architectures that accommodate the unique characteristics of sparse data, and appropriate training datasets of sparse matrices tailored for SAI learning. To address these challenges, we propose an innovative CNN-based intelligent selection framework for SAIs. In this framework, three normalization methods for sparse matrices are first introduced, namely binary, density and column-histogram representation. Second, for each representation, a corresponding CNN architecture is specifically designed, and they are named BinaryNet, DensityNet and HistNet, respectively. Third, these three networks operate independently and adopt a late fusion strategy to integrate learned SAI features. This design effectively mitigates the interference that would arise from early fusion, given the heterogeneous nature of the different input representations. Finally, we present, for the first time, a method for constructing a training dataset of sparse matrices specifically designed for SAI learning. Experimental results demonstrate that the proposed framework is both effective and efficient, validating the feasibility and advantages of the approach.