Numerical modeling errors are unavoidable in finite element analysis. The presence of model errors inherently reflects both model accuracy and uncertainty. To date there have been few methods for explicitly quantifying errors at points of interest (e.g. at finite element nodes). The lack of explicit model error approximators has been addressed recently with the emergence of machine learning (ML), which closes the loop between numerical model features/solutions and explicit model error approximations. In this paper, we propose physics-informed neural networks (PINNs) for simultaneous numerical model error approximation and superresolution. To test our approach, numerical data was generated using finite element simulations on a two-dimensional elastic plate with a central opening. Four- and eight-node quadrilateral elements were used in the discretization to represent the reduced-order and higher-order models, respectively. It was found that the developed PINNs effectively predict model errors in both x and y displacement fields with small differences between predictions and ground truth. Our findings demonstrate that the integration of physics-informed loss functions enables neural networks (NNs) to surpass a purely data-driven approach for approximating model errors.

翻译：数值建模误差在有限元分析中不可避免。模型误差的存在本质上反映了模型的精度与不确定性。迄今为止，鲜有方法能够显式量化关注点（例如有限元节点处）的误差。随着机器学习（ML）的兴起，这一缺乏显式模型误差近似器的问题近期得到了解决，机器学习在数值模型特征/解与显式模型误差近似之间建立了闭环。本文提出物理信息神经网络（PINNs），用于同时实现数值模型误差近似与超分辨率。为验证所提方法，我们通过在带中心开口的二维弹性板上进行有限元仿真生成数值数据。离散化过程中分别采用四节点和八节点四边形单元来代表降阶模型和高阶模型。研究发现，所开发的PINNs能有效预测x和y方向位移场的模型误差，其预测值与真实值之间差异较小。我们的结果表明，结合物理信息损失函数使神经网络（NNs）在近似模型误差方面超越了纯数据驱动的方法。