mlsanityData to 7 Oct 2026

Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs

arXiv 2404.17868 ↗First posted 27 Apr 2024math.NA cs.LG cs.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first establish the convergence of this method for general second-order linear elliptic PDEs with respect to the parameters for neural network approximation. In this regard, we address the role of the condition number of the finite element matrix in the convergence of the method. Secondly, we derive an explicit error estimate for the self-adjoint case. For this, we investigate some regularity properties of the solution in certain function classes for a neural network approximation, verifying the sufficient condition for the solution to have the desired regularity. Finally, we will also conduct some numerical experiments that support the theoretical findings, confirming the role of the condition number of the finite element matrix in the overall convergence.

Key facts. Cited by 3 papers in the map (data through 7 Oct 2026). Builds on 20 of its 52 references that are in the map. First posted on arXiv 27 Apr 2024.

Builds on

20 of 52 references.

References · 20

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  4. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  5. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  6. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  7. 2015Norm-Based Capacity Control in Neural Networkscited by 299
  8. 2019Universal Approximation with Deep Narrow Networkscited by 198
  9. 2018Towards Understanding the Role of Over-Parametrization in Generalization of Neural Networkscited by 140
  10. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  11. 2019A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Casecited by 113
  12. 2023A Mathematical Guide to Operator Learningcited by 74
8 more
  1. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  2. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  3. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  4. 2023HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetworkcited by 25
  5. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  6. 2023Spectral operator learning for parametric PDEs without data reliancecited by 9
  7. 2022opPINN: Physics-Informed Neural Network with operator learning to approximate solutions to the Fokker-Planck-Landau equationcited by 8
  8. 2022Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 3.

3 papers.

  1. 2026Discontinuous Galerkin finite element operator network for solving non-smooth PDEscited by 1
  2. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  3. 2026Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEscited by 0