Transfer learning-enhanced finite element-integrated neural networks

Physics informed neural networks (PINNs) have attracted increasing attention in computational solid mechanics due to their success in solving complex partial differential equations (PDEs). Nevertheless, the low efficiency and precision always hinder the application of PINNs in boundary value problems. To address this issue, this study proposed a transfer learning enhanced hybrid framework that integrates the finite element method with PINNs to accelerate the training process. The finite element-integrated neural network framework (FEINN) is first introduced, leveraging finite elements for domain discretization and the weak-form governing equation for defining the loss function. A mesh parametric study is subsequently conducted, aiming to identify the optimal discretization configuration by exploring various element sizes, element types, and orders of shape functions. Furthermore, various transfer learning strategies are proposed and fully evaluated to improve the training efficiency and precision of FEINN, including scale transfer learnings (STLs) from coarse mesh to refine mesh and from small domain to large domain, material transfer learnings (MTLs) from elastic material to elastoplastic material and from elastic material to elastic material problems, as well as load transfer learnings (LTLs) form displacement load condition to force load condition. A series of experiments are conducted to showcase the effectiveness of FEINN, identifying the most efficient discretization configuration and validating the efficacy of transfer learning strategies across elastic, elastoplastic, and multi-material scenarios. The results indicate that the element type and size, and shape function order have significant impacts on training efficiency and accuracy. Moreover, the transfer learning techniques can significantly improve the accuracy and training efficiency of FEINN.