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

2026-01-02

arXiv.org

AI Summary

This work addresses the challenge of escalating computational cost and degraded accuracy in FEONet when applied to large-scale problems due to the increasing number of elements. Inspired by the local sparsity inherent in finite element methods, the authors propose a sparse neural network architecture that establishes a direct mapping from parameters to solutions, enabling efficient solution of parametric partial differential equations without requiring training data. By incorporating locally sparse connectivity and a data-free training mechanism, the method substantially reduces both computational and memory overhead while offering theoretical guarantees on approximation accuracy and stability. Numerical experiments demonstrate that the proposed approach achieves computational efficiency significantly higher than that of the original FEONet, while maintaining comparable high accuracy and robustness.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Learning to Search

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• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• parametric PDEs
• finite element method
• operator learning
• computational cost
• scalability

Innovation

• Sparse FEONet
• finite element sparsity
• operator learning
• parametric PDEs
• memory-efficient architecture