SD-PINN: Physics informed neural networks for spatially dependent PDEs

The physics-informed neural network (PINN) is able to identify partial differential equation (PDE) coefficients which are constant across the space directly from physical measurements. In this paper, we propose a modification of PINN, named as SD-PINN, which can recover the coefficients in spatially-dependent PDEs using only one neural network without the requirement of domain-specific physical knowledge. The network structure is a simple fully connected neural network, and multiple physical information like the time-invariance and spatial-smoothness of the PDE coefficients is incorporated as loss functions. The method is robust to noise due to introduced physical constraints, which is verified by experiments.