We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework overcomes the challenges related to the imposition of boundary conditions, the choice of collocation points in physics-informed neural networks, and the integration of variational physics-informed neural networks. A numerical experiment set confirms the framework's capability of handling various forward and inverse problems. In particular, the trained neural network generalises well for smooth problems, beating finite element solutions by some orders of magnitude. We finally propose an effective one-loop solver with an initial data fitting step (to obtain a cheap initialisation) to solve inverse problems.

翻译：我们提出了一种通用框架，用于求解偏微分方程约束的正问题和反问题。该框架将神经网络插值到有限元空间，以表示（部分）未知量。这一框架克服了与边界条件施加、物理信息神经网络中配点选择以及变分物理信息神经网络集成相关的挑战。数值实验证实了该框架处理各种正问题和反问题的能力。特别地，训练后的神经网络在光滑问题上具有良好的泛化性能，其精度比有限元解高出数个数量级。最后，我们提出了一种有效的一循环求解器，结合初始数据拟合步骤（以获得廉价的初始化），用于求解反问题。