We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we consider interpolation trial spaces that satisfy the de Rham Hilbert subcomplex, providing stable and structure-preserving neural network discretisations for a wide variety of PDEs. This approach, coined compatible FEINNs, has been used to accurately approximate the $H(\textbf{curl})$ inner product. We numerically observe that the trained network outperforms finite element solutions by several orders of magnitude for smooth analytical solutions. Furthermore, to showcase the versatility of the method, we demonstrate that compatible FEINNs achieve high accuracy in solving surface PDEs such as the Darcy equation on a sphere. Additionally, the framework can integrate adaptive mesh refinements to effectively solve problems with localised features. We use an adaptive training strategy to train the network on a sequence of progressively adapted meshes. Finally, we compare compatible FEINNs with the adjoint neural network method for solving inverse problems. We consider a one-loop algorithm that trains the neural networks for unknowns and missing parameters using a loss function that includes PDE residual and data misfit terms. The algorithm is applied to identify space-varying physical parameters for the $H(\textbf{curl})$ model problem from partial or noisy observations. We find that compatible FEINNs achieve accuracy and robustness comparable to, if not exceeding, the adjoint method in these scenarios.

翻译：我们将有限元插值神经网络框架从弱解属于$H^1$空间的偏微分方程推广至弱解属于$H(\textbf{curl})$或$H(\textbf{div})$空间的偏微分方程。为此，我们构建满足德拉姆希尔伯特子复形结构的插值试探空间，从而为各类偏微分方程提供稳定且保结构的神经网络离散格式。这种被称为兼容FEINN的方法已被用于精确逼近$H(\textbf{curl})$内积。数值实验表明，对于光滑解析解，训练后的网络在精度上超越有限元解数个数量级。为展示该方法的普适性，我们证明兼容FEINN在求解球面达西方程等曲面偏微分方程时同样能获得高精度。此外，该框架可结合自适应网格细化技术有效求解具有局部特征的问题。我们采用自适应训练策略，在逐步优化的网格序列上训练神经网络。最后，我们将兼容FEINN与伴随神经网络方法在反问题求解中进行比较。通过设计单循环算法，采用包含PDE残差项与数据失配项的损失函数，同步训练未知场与缺失参数的神经网络。该算法应用于从局部或含噪观测数据中识别$H(\textbf{curl})$模型问题的空间变化物理参数。研究发现，在此类场景下兼容FEINN达到的精度与鲁棒性至少与伴随方法相当，甚至更具优势。