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：我们提出了一种求解偏微分方程约束的正问题和逆问题的通用框架，其中将神经网络插值到有限元空间以表示（部分）未知量。该框架克服了边界条件施加、物理信息神经网络中配置点选择以及变分物理信息神经网络集成等方面的挑战。数值实验证实了该框架处理各类正问题和逆问题的能力。特别地，训练后的神经网络对光滑问题具有良好的泛化能力，其精度比有限元解高数个数量级。最后，我们提出了一种有效的一环求解器，通过初始数据拟合步骤（以获得廉价初始值）来求解逆问题。