Coupled Physics-informed Neural Networks for Inferring Solutions of Partial Differential Equations with Unknown Source Terms

Physics-informed neural networks (PINNs) provide a transformative development for approximating the solutions to partial differential equations (PDEs). This work proposes a coupled physics-informed neural network (C-PINN) for the nonhomogeneous PDEs with unknown dynamical source terms, which is used to describe the systems with external forces and cannot be well approximated by the existing PINNs. In our method, two neural networks, NetU and NetG, are proposed. NetU is constructed to generate a quasi-solution satisfying PDEs under study. NetG is used to regularize the training of NetU. Then, the two networks are integrated into a data-physics-hybrid cost function. Finally, we propose a hierarchical training strategy to optimize and couple the two networks. The performance of C-PINN is proved by approximating several classical PDEs.