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

Abstract: This study investigates the Finite Element Operator Network (FEONet), a method designed for learning operators in parametric contexts. Originally presented by J. Y. Lee, S. Ko, and Y. Hong in their 2025 paper, Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs (SIAM J. Sci. Comput., 47(2), C501-C528), FEONet establishes a parameter-to-solution mapping within a finite element framework. It is distinguished by its ability to train without the need for dataset samples, while delivering robust performance and high precision across a wide variety of problems. Nevertheless, the method faces significant hurdles in large-scale applications, as computational demands rise and accuracy can degrade as the element count increases. To mitigate these limitations, we introduce a novel sparse network architecture inspired by the inherent structure of finite elements. Our extensive numerical experiments demonstrate that this sparse design significantly reduces computational expenses and enhances efficiency without compromising accuracy relative to the original approach. Furthermore, we provide theoretical proofs confirming the sparse architecture’s capacity to effectively approximate the target operator, alongside a stability analysis that guarantees dependable training and prediction outcomes.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC