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Neural Network Tearing and Interconnecting Methods for PDEs.

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Bibliographic Details
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]
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Database: Engineering Source
Description
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]
ISSN:10648275
DOI:10.1137/25M1752055