In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.

翻译：在本工作中，我们提出了一项结合两种偏微分方程求解方法的研究：连续有限元方法以及基于神经网络的较新技术。近年来，物理信息神经网络在快速求解偏微分方程方面展现出显著优势，尤其在高维问题中。然而，其精度不足可能构成重要缺陷，因此将其与误差估计理论完备的有限元方法相结合具有重要价值。本文提出的完整流程通过引入神经网络预测作为先验信息，对经典有限元逼近空间进行修正。一方面，这种结合能够改进并验证神经网络的预测结果，从而获得快速精确的求解方案；另一方面，理论证明的误差估计表明，此类策略相较于传统方法的提升幅度仅取决于先验信息的质量。我们通过在一维、二维及三维几何参数化问题上的数值实验验证了该方法的有效性。实验表明，在达到相同精度要求时，相较于标准有限元方法，采用增强型有限元方法可使用更粗的网格，从而显著减少计算耗时，这一优势在参数化问题中尤为突出。