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

Area of Science:

• Scientific Computing
• Machine Learning
• Numerical Analysis

Background:

• Sparse approximate inverses (SAIs) are crucial preconditioners for accelerating iterative methods.
• Selecting the optimal SAI for a given iterative solver is a significant challenge in scientific computing.
• Convolutional neural networks (CNNs) offer new approaches to address this challenge.

Purpose of the Study:

• To develop an intelligent framework for the selection of sparse approximate inverses (SAIs) using CNNs.
• To address key issues in applying CNNs to SAI construction: matrix normalization, network architecture, and training data.

Main Methods:

• Proposed a CNN-based framework with three normalization methods for sparse matrices: binary, density, and column-histogram representations.
• Designed corresponding CNN architectures (BinaryNet, DensityNet, HistNet) for each representation.
• Implemented a late fusion strategy to integrate features from independent networks, mitigating interference from heterogeneous data.

Main Results:

• Demonstrated the effectiveness and efficiency of the proposed CNN-based SAI selection framework.
• Successfully constructed a novel training dataset specifically for SAI learning.
• Validated the feasibility and advantages of the intelligent selection approach.

Conclusions:

• The developed CNN framework provides an effective and efficient solution for selecting sparse approximate inverses.
• The proposed normalization techniques, tailored network architectures, and late fusion strategy are key to the framework's success.
• This work advances the application of machine learning in scientific computing for optimizing iterative solvers.