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

TL;DR

This paper introduces a sparse FEONet architecture that reduces computational costs for parametric PDEs by leveraging finite element sparsity, maintaining accuracy and stability for large-scale problems.

Contribution

The paper proposes a novel sparse network architecture for FEONet, significantly improving efficiency while preserving accuracy and providing theoretical guarantees.

Findings

Substantial reduction in computational cost and memory usage.

Maintains comparable accuracy to dense FEONet.

Theoretical approximation and stability guarantees.

Abstract

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.