A novel normalized reduced-order physics-informed neural network for solving inverse problems

The utilization of Physics-informed Neural Networks (PINNs) in deciphering inverse problems has gained significant attention in recent years. However, the PINN training process for inverse problems is notably restricted due to gradient failures provoked by magnitudes of partial differential equations (PDEs) parameters or source functions. To address these matters, normalized reduced-order physics-informed neural network (nr-PINN) is developed in this study. The goal of the nr-PINN is to reconfigure the original PDE into a system of normalized lower-order PDEs through two sequential steps. To start with, self-homeomorphisms of the PDEs are implemented via scaling factors determined based on measured data. Afterward, each normalized PDE is transformed into a system of lower-order PDEs by primary and secondary variables. Besides, a technique to exactly impose many types of boundary conditions (BCs) by redefining NNs outputs is developed in the context of reduced-order method. The advantages of the nr-PINN model over the original one regarding solution accuracy and training cost are demonstrated through several inverse problems in solid mechanics with different types of PDEs and BCs.