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（FEINN）框架从弱解属于 $H^1$ 的偏微分方程（PDE）推广至弱解属于 $H(\textbf{curl})$ 或 $H(\textbf{div})$ 的偏微分方程。为此，我们考虑满足 de Rham Hilbert 子复形的插值试验空间，从而为多种偏微分方程提供稳定且保结构的神经网络离散化方案。这一方法被称为兼容 FEINN，已成功用于精确逼近 $H(\textbf{curl})$ 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。此外，为展示该方法的普适性，我们证明兼容 FEINN 在求解表面偏微分方程（如球面上的 Darcy 方程）时亦能达到高精度。该框架还可集成自适应网格细化技术，以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网络。最后，我们将兼容 FEINN 与伴随神经网络方法在求解反问题方面进行比较。我们提出一种单循环算法，通过包含 PDE 残差与数据失配项的损失函数，训练神经网络以求解未知量与缺失参数。该算法应用于从部分或含噪声观测数据中识别 $H(\textbf{curl})$ 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，甚至更优。