A novel unsupervised PINN framework with dynamically self-adaptive strategy for solid mechanics

Unbalance in multi-task training has always posed significant challenges in deep learning, particularly for Physics-Informed Neural Networks (PINN). A novel unsupervised PINN framework configured with two neural network architectures has been developed for solid mechanics problems with pure boundary data. This framework features an innovatively formulated loss function, where loss weights are dynamically updated through a self-adaptive strategy using either the gradient normalization algorithm or the augmented Lagrangian algorithm, effectively tackling training imbalances across different types of boundary data. The PINN model consistently approximates solutions for solid mechanics problems with improved convergence and accuracy, as validated through a uniaxial tensile test case. The novel framework is robust and adaptable to two neural network architectures with different numbers of layers and neurons. About 30000 training epochs can reduce the prediction error of deformation to below 10^-6, meeting the requirements of computational solid mechanics. An information entropy of 1.3 bits, calculated from 500 well-trained PINN models, indicates that the novel unsupervised PINN framework exhibits low uncertainty. The findings uncover the so-called “squared rule” for selecting a suitable threshold in the convergence criterion. This study successfully addresses the critical scientific problem inherent in multitask learning, positioning PINN as a potentially universal and user-friendly method, offering an alternative for numerical computations.