Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. Our FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis.

翻译：偏微分方程（PDEs）是我们理解和预测物理学、工程学及金融学等多个领域自然现象的基础。然而，求解参数化偏微分方程是一项复杂任务，需要高效的数值方法。本文提出了一种利用有限元算子网络（FEONet）求解参数化偏微分方程的新方法。该方法将深度学习与传统数值方法（特别是有限元法）相结合，在没有配对输入-输出训练数据的情况下求解参数化偏微分方程。我们在多个基准问题上进行了各类实验，证实了该方法在不同设置和环境下均展现出优异性能，验证了其在精度、泛化能力和计算灵活性方面的多用途性。我们的FEONet框架在PDEs对模拟具有复杂边界条件和奇异行为的领域起关键作用的各类应用中展现出潜力。此外，我们利用数值分析中的有限元逼近方法，提供了理论收敛性分析以支持所提方法。