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Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs

arXiv 2601.00672 ↗First posted 2 Jan 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs, SIAM J. Sci. Comput., 47(2), C501-C528, 2025. FEONet realizes the parameter-to-solution map on a finite element space and admits a training procedure that does not require training data, while exhibiting high accuracy and robustness across a broad class of problems. However, its computational cost increases and accuracy may deteriorate as the number of elements grows, posing notable challenges for large-scale problems. In this paper, we propose a new sparse network architecture motivated by the structure of the finite elements to address this issue. Throughout extensive numerical experiments, we show that the proposed sparse network achieves substantial improvements in computational cost and efficiency while maintaining comparable accuracy. We also establish theoretical results demonstrating that the sparse architecture can approximate the target operator effectively and provide a stability analysis ensuring reliable training and prediction.

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References · 16

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  8. 2020Minimum Width for Universal Approximationcited by 92
  9. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  10. 2023HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetworkcited by 25
  11. 2021Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Controlcited by 24
  12. 2023Minimum width for universal approximation using ReLU networks on compact domaincited by 17
4 more
  1. 2023Finite Element Operator Network for Solving Elliptic-type parametric PDEscited by 14
  2. 2023Spectral operator learning for parametric PDEs without data reliancecited by 9
  3. 2024Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEscited by 3
  4. 2025Minimum width for universal approximation using squashable activation functionscited by 2

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