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Neural Network Tearing and Interconnecting Methods for PDEs.

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Title: Neural Network Tearing and Interconnecting Methods for PDEs.
Authors: Jeon, Young Jae1 (AUTHOR) yjjeon@hku.ac.kr, Kim, Hyea Hyun2 (AUTHOR) hhkim@hku.ac.kr
Source: SIAM Journal on Scientific Computing. 2026, Vol. 48 Issue 3, pC553-C578. 26p.
Subjects: Partial differential equations, Domain decomposition methods, Parallel programming, Artificial neural networks, Optimization algorithms, Finite element method, Elliptic equations
Abstract: Neural network tearing and interconnecting methods are proposed and tested for second-order elliptic problems to address the longer training time in neural network solutions for PDEs. The methods are developed for nonoverlapping subdomain partitions of the problem domain, and such nonoveralapping partitions are useful for modeling coefficient discontinuity, multiphysics, and interface discontinuity inside the problem domain. A gradient-based iterative algorithm is first derived for the solution on the subdomain interface, where, at each iteration, local neural network solutions are trained for the provided interface solution values independently, allowing parallel computation, and the interface solution values are updated by using the trained local neural network solutions. The convergence of the iterative algorithm is shown by using well-established finite element tearing and interconnecting methods. The gradient-based iterative algorithm is then further enhanced by proposing preconditioning schemes on the gradient value. The performance of the proposed preconditioning schemes is demonstrated for various test examples. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 195222010
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
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  Data: <searchLink fieldCode="AR" term="%22Jeon%2C+Young+Jae%22">Jeon, Young Jae</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> yjjeon@hku.ac.kr</i><br /><searchLink fieldCode="AR" term="%22Kim%2C+Hyea+Hyun%22">Kim, Hyea Hyun</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> hhkim@hku.ac.kr</i>
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  Data: <searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Domain+decomposition+methods%22">Domain decomposition methods</searchLink><br /><searchLink fieldCode="DE" term="%22Parallel+programming%22">Parallel programming</searchLink><br /><searchLink fieldCode="DE" term="%22Artificial+neural+networks%22">Artificial neural networks</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Elliptic+equations%22">Elliptic equations</searchLink>
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  Data: Neural network tearing and interconnecting methods are proposed and tested for second-order elliptic problems to address the longer training time in neural network solutions for PDEs. The methods are developed for nonoverlapping subdomain partitions of the problem domain, and such nonoveralapping partitions are useful for modeling coefficient discontinuity, multiphysics, and interface discontinuity inside the problem domain. A gradient-based iterative algorithm is first derived for the solution on the subdomain interface, where, at each iteration, local neural network solutions are trained for the provided interface solution values independently, allowing parallel computation, and the interface solution values are updated by using the trained local neural network solutions. The convergence of the iterative algorithm is shown by using well-established finite element tearing and interconnecting methods. The gradient-based iterative algorithm is then further enhanced by proposing preconditioning schemes on the gradient value. The performance of the proposed preconditioning schemes is demonstrated for various test examples. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of SIAM Journal on Scientific Computing is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1137/25M1752055
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      – Code: eng
        Text: English
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        PageCount: 26
        StartPage: C553
    Subjects:
      – SubjectFull: Partial differential equations
        Type: general
      – SubjectFull: Domain decomposition methods
        Type: general
      – SubjectFull: Parallel programming
        Type: general
      – SubjectFull: Artificial neural networks
        Type: general
      – SubjectFull: Optimization algorithms
        Type: general
      – SubjectFull: Finite element method
        Type: general
      – SubjectFull: Elliptic equations
        Type: general
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      – TitleFull: Neural Network Tearing and Interconnecting Methods for PDEs.
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            NameFull: Jeon, Young Jae
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            NameFull: Kim, Hyea Hyun
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            – D: 01
              M: 05
              Text: 2026
              Type: published
              Y: 2026
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