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 demonstrate the effectiveness of our approach on several benchmark problems and show that it outperforms existing state-of-the-art methods 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）是我们理解并预测物理学、工程学和金融学等众多领域自然现象的基础。然而，求解参数化PDEs是一项复杂的任务，需要高效的数值方法。本文提出了一种新颖的方法，利用有限元算子网络（FEONet）求解参数化PDEs。所提方法将深度学习的强大能力与传统数值方法（特别是有限元法）相结合，在无需配对输入-输出训练数据的情况下求解参数化PDEs。我们在多个基准问题上验证了该方法的有效性，并表明其在准确性、泛化能力和计算灵活性方面均优于现有最先进方法。我们的FEONet框架在PDEs对具有不同边界条件和奇异行为复杂领域建模起关键作用的各种应用中展现出潜力。此外，我们利用数值分析中的有限元近似，提供了支持该方法的理论收敛性分析。