Deep Neural Networks for Solving Large Linear Systems Arising from High-Dimensional Problems

Yiqi Gu, Michael K. Ng

Research output: Contribution to journalJournal articlepeer-review

2 Citations (Scopus)

Abstract

This paper studies deep neural networks for solving extremely large linear systems arising from high-dimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such extremely large linear systems. Our idea is to employ a neural network to characterize the solution with many fewer parameters than the size of the solution under a matrix-free setting. We present an error analysis of the proposed method, indicating that the solution error is bounded by the condition number of the matrix and the neural network approximation error. Several numerical examples from partial differential equations, queueing problems, and probabilistic Boolean networks are presented to demonstrate that the solutions of linear systems can be learned quite accurately.

Original languageEnglish
Pages (from-to)A2356-A2381
Number of pages26
JournalSIAM Journal on Scientific Computing
Volume45
Issue number5
Early online date21 Sept 2023
DOIs
Publication statusPublished - Oct 2023

Scopus Subject Areas

  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • very large scale linear systems
  • neural networks
  • partial differential equations
  • Riesz fractional diffusion
  • overflow queuing model
  • probabilistic Boolean networks

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