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A Deep Conjugate Direction Method for Iteratively Solving Linear Systems

Basic Information

  • Title: A Deep Conjugate Direction Method for Iteratively Solving Linear Systems
  • Authors: Ayano Kaneda, Osman Akar, Jingyu Chen, Victoria Kala, David Hyde, Joseph Teran
  • Affiliations: Waseda University, Tokyo, Japan; University of California, Los Angeles, CA, USA; Vanderbilt University, Nashville, TN, USA; University of California, Davis, CA, USA
  • Published: 2022, at ICLR 2023
  • DOI: arXiv:2205.10763v2 [cs.LG]

Introduction

This paper presents a novel deep learning approach for approximating solutions to large sparse symmetric positive-definite linear systems, which are common in applied sciences, particularly in numerical methods for partial differential equations. It introduces a deep conjugate direction method (DCDM) that leverages deep neural networks to improve search directions for faster convergence in iterative methods like conjugate gradients.

Overview

The work is motivated by the computational intensity of solving linear systems, especially in applications requiring the solution of millions of unknowns. The authors propose a data-driven method that uses a deep neural network to generate efficient search directions, aiming to accelerate convergence without the need for extensive computational resources.

Summary

  • Problem Addressed: Solving large sparse symmetric positive-definite linear systems efficiently.
  • Proposed Solution: DCDM, which employs a convolutional neural network to approximate the inverse of the linear operator, improving search directions for iterative solution methods.
  • Key Results: Demonstrated effectiveness in solving spatially discretized Poisson equations with millions of degrees of freedom, achieving faster convergence with fewer iterations compared to traditional methods.

Contributions

  • A novel approach that integrates deep learning with conjugate gradient methods to optimize search directions for solving linear systems.
  • An unsupervised learning strategy with a specifically designed loss function to train the network.
  • Successful generalization of the proposed method to systems beyond those encountered during training, showcasing its potential for wide applicability in computational fluid dynamics and other areas.

The paper reviews data-driven techniques in solving linear systems, including machine learning estimations of multigrid parameters and preconditioners, and non-iterative machine learning approximations of the inverse of discrete Poisson equations.

Limitations and Future Work

  • The approach may not generalize well to non-sparse or non-symmetric matrices, or to matrices that are computationally expensive to evaluate.
  • Future work could explore the application of DCDM to other classes of PDEs and problems with graph structures, as well as adaptation to non-uniform grids.

Details, Techniques, and Method

DCDM iteratively improves solution approximations using a deep learning-based modification of the conjugate gradients method, ensuring search directions are efficiently chosen for minimizing the matrix norm of the approximation error.

Describe all the Experiments

The method's efficacy was tested on discretized Poisson equations for incompressible flow simulations, demonstrating significant improvement in convergence rates over traditional iterative methods.

Conclusion

The deep conjugate direction method introduces an innovative intersection of deep learning and numerical linear algebra, providing a promising direction for accelerating the solution of large-scale linear systems in computational sciences.