标题： 有限元神经网络插值 第一部分：求解偏微分方程的可解释和自适应离散化 显示英文标题

标题： Finite Element Neural Network Interpolation. Part I: Interpretable and Adaptive Discretization for Solving PDEs

摘要： 我们提出了有限元神经网络插值（FENNI）框架，这是一种稀疏神经网络架构，扩展了之前在分层深度学习神经网络（HiDeNN）中引入的嵌入式有限元神经网络（EFENN）的工作。 由于其基于网格的结构，EFENN所需的可训练参数显著少于全连接神经网络，且每个权重和偏置都有明确的解释。 我们的FENNI框架在EFENN框架内对HiDeNN方法进行了改进。 首先，我们提出了一种基于参考单元的架构，其中形状函数定义在参考单元上，从而实现了插值函数的可变性，并能够直接使用高斯求积规则来评估损失函数。 其次，我们提出了一种基于框架可解释性的实用多网格训练策略。 第三，HiDeNN的组合rh自适应从1D扩展到2D，并引入了一种基于雅可比矩阵的节点添加准则，结合了h自适应和r自适应。 从深度学习的角度来看，通过rh自适应和多网格方法实现的自适应网格行为对应于迁移学习，使FENNI能够在训练过程中动态优化网络架构。 该框架的能力在1D和2D测试案例中得到验证，其准确性与计算成本与解析解和经典有限元法求解器进行比较。 在这些案例中，多网格训练策略显著提高了训练阶段的效率和鲁棒性。 最后，我们在EFENN框架内引入了一种变分损失，表明其性能与基于能量的损失相当，并优于基于残差的损失。 在第二部分中，该框架被扩展到参数空间上的代理建模。 显示英文摘要

摘要： We present the Finite Element Neural Network Interpolation (FENNI) framework, a sparse neural network architecture extending previous work on Embedded Finite Element Neural Networks (EFENN) introduced with the Hierarchical Deep-learning Neural Networks (HiDeNN). Due to their mesh-based structure, EFENN requires significantly fewer trainable parameters than fully connected neural networks, with individual weights and biases having a clear interpretation. Our FENNI framework, within the EFENN framework, brings improvements to the HiDeNN approach. First, we propose a reference element-based architecture where shape functions are defined on a reference element, enabling variability in interpolation functions and straightforward use of Gaussian quadrature rules for evaluating the loss function. Second, we propose a pragmatic multigrid training strategy based on the framework's interpretability. Third, HiDeNN's combined rh-adaptivity is extended from 1D to 2D, with a new Jacobian-based criterion for adding nodes combining h- and r-adaptivity. From a deep learning perspective, adaptive mesh behavior through rh-adaptivity and the multigrid approach correspond to transfer learning, enabling FENNI to optimize the network's architecture dynamically during training. The framework's capabilities are demonstrated on 1D and 2D test cases, where its accuracy and computational cost are compared against an analytical solution and a classical FEM solver. On these cases, the multigrid training strategy drastically improves the training stage's efficiency and robustness. Finally, we introduce a variational loss within the EFENN framework, showing that it performs as well as energy-based losses and outperforms residual-based losses. This framework is extended to surrogate modeling over the parametric space in Part II.