mlsanityData to 14 Oct 2026

Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEs

arXiv 2211.08900 ↗First posted 16 Nov 2022math.NA cs.LG cs.NAcited by 3

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Abstract (arXiv)

In this paper, we perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based numerical methods for PDEs, the ULGNet expresses the solution as a spectral expansion with respect to the Legendre basis and predicts the coefficients with deep neural networks by solving a variational residual minimization problem. Since the corresponding loss function is equivalent to the residual induced by the linear algebraic system depending on the choice of basis functions, we prove that the minimizer of the discrete loss function converges to the weak solution of the PDEs. Numerical evidence will also be provided to support the theoretical result. Key technical tools include the variant of the universal approximation theorem for bounded neural networks, the analysis of the stiffness and mass matrices, and the uniform law of large numbers in terms of the Rademacher complexity.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 24 of its 67 references that are in the map. First posted on arXiv 16 Nov 2022.

Builds on

24 of 67 references.

References · 24

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12 more
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Citing papers per year: 2023: 1; 2024: 1; 2025: 1.

3 papers.

  1. 2023Spectral operator learning for parametric PDEs without data reliancecited by 11
  2. 2024Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEscited by 3
  3. 2025Reduced-Basis Deep Operator Learning for Parametric PDEs with Independently Varying Boundary and Source Datacited by 1