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Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy

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arxiv 2412.02586 v1 pith:2UT3VDPQ submitted 2024-12-03 math.NA cs.NA

Zhongshuo Lin , Yifan Wang , Hehu Xie This is my paper · ORCID
classification math.NAcs.NA
keywords subspacenetworkneuralapproximationmethodadaptivedifferentialequations
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Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Adaptive feature capture method for solving partial differential equations with near singular solutions

    math.NA 2025-07 conditional novelty 6.0 of 10

    An adaptive random feature method that repositions feature hyperplanes and collocation points according to the gradient of the current approximation resolves near-singular PDEs to high accuracy.

  2. Domain Decomposition Subspace Neural Network Method for Solving Linear and Nonlinear Partial Differential Equations

    math.NA 2025-05 conditional novelty 4.0 of 10

    A domain-decomposition subspace neural network method solves linear and nonlinear PDEs with errors down to 1e-13 and lower training cost than PINN, DGM, DRM, and LocELM on 1D/2D benchmarks.

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