用有限元局部稀疏性设计低开销算子网络，高效求解参数化偏微分方程。
Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs
- 基于有限元结构设计稀疏算子网络，降低计算开销。
- 在大规模问题中保持高精度，计算效率提升显著。
- 理论证明有效逼近与训练稳定性，适合大规模参数化PDE求解。
本文研究了有限元算子网络（FEONet），一种用于参数化问题的算子学习方法，最初由J. Y. Lee、S. Ko和Y. Hong提出（SIAM J. Sci. Comput., 47(2), C501-C528, 2025）。FEONet在有限元空间上实现参数到解的映射，无需训练数据即可训练，且对广泛问题具有高精度和鲁棒性。然而，随着单元数量增加，其计算成本上升，精度可能下降，给大规模问题带来挑战。本文提出一种受有限元结构启发的新型稀疏网络架构以解决该问题。通过大量数值实验，我们表明所提稀疏网络在显著降低计算成本和提升效率的同时，仍保持相近的精度。此外，我们建立了理论结果，证明该稀疏架构能有效逼近目标算子，并提供稳定性分析以确保可靠训练与预测。
原文摘要 · Abstract (English)
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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