Exact Dirichlet boundary Physics-informed Neural Network EPINN for solid mechanics

Physics-informed neural networks (PINNs) have been rapidly developed for solving partial differential equations. The Exact Dirichlet boundary condition Physics-informed Neural Network (EPINN) is proposed to achieve efficient simulation of solid mechanics problems based on the principle of least work with notably reduced training time. There are five major building features in the EPINN framework. First, for the 1D solid mechanics problem, the neural networks are formulated to exactly replicate the shape function of linear or quadratic truss elements. Second, for 2D and 3D problems, the tensor decomposition was adopted to build the solution field without the need of generating the finite element mesh of complicated structures to reduce the number of trainable weights in the PINN framework. Third, the principle of least work was adopted to formulate the loss function. Fourth, the exact Dirichlet boundary condition (i.e., displacement boundary condition) was implemented. Finally, the meshless finite difference (MFD) was adopted to calculate gradient information efficiently. By minimizing the total energy of the system, the loss function is selected to be the same as the total work of the system, which is the total strain energy minus the external work done on the Neumann boundary conditions (i.e., force boundary conditions). The exact Dirichlet boundary condition was implemented as a hard constraint compared to the soft constraint (i.e., added as additional terms in the loss function), which exactly meets the requirement of the principle of least work. The EPINN framework is implemented in the Nvidia Modulus platform and GPU-based supercomputer and has achieved notably reduced training time compared to the conventional PINN framework for solid mechanics problems. Typical numerical examples are presented. The convergence of EPINN is reported and the training time of EPINN is compared to conventional PINN architecture and finite element solvers. Compared to conventional PINN architecture, EPINN achieved a speedup of more than 13 times for 1D problems and more than 126 times for 3D problems. The simulation results show that EPINN can even reach the convergence speed of finite element software. In addition, the prospective implementations of the proposed EPINN framework in solid mechanics are proposed, including nonlinear time-dependent simulation and super-resolution network.