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Error estimates of residual minimization using neural networks for linear PDEs

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arxiv 2010.08019 v3 pith:MYUWHYU3 submitted 2020-10-15 math.NA cs.NA

Yeonjong Shin , Zhongqiang Zhang , George Em Karniadakis This is my paper · ORCID
classification math.NAcs.NA
keywords formulationsminimizationnetworksneuralresidualerrorestimatesstrong
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We propose an abstract framework for analyzing the convergence of least-squares methods based on residual minimization when feasible solutions are neural networks. With the norm relations and compactness arguments, we derive error estimates for both continuous and discrete formulations of residual minimization in strong and weak forms. The formulations cover recently developed physics-informed neural networks based on strong and variational formulations.

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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. A discontinuous Galerkin plane wave neural network method for Helmholtz equation and Maxwell's equations

    math.NA 2025-06 conditional novelty 6.0 of 10

    A recursive Galerkin neural network with plane wave activations solves Helmholtz and Maxwell equations with proven convergence, no bounded-parameter assumption, and near-unit condition numbers.

  2. PINN-DG: Residual neural network methods trained with Finite Elements

    math.NA 2025-07 conditional novelty 5.0 of 10

    PINN-DG replaces pointwise derivative losses with finite element interpolation plus discontinuous Galerkin consistency and penalty terms, and proves convergence of the discrete minimizers.

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