  The physics-informed neural network (PINN) is capable of recovering partial
differential equation (PDE) coefficients that remain constant throughout the
spatial domain directly from physical measurements. In this work, we propose a
spatially dependent physics-informed neural network (SD-PINN), which enables
the recovery of coefficients in spatially-dependent PDEs using a single neural
network, eliminating the requirement for domain-specific physical expertise. We
apply the SD-PINN to spatially-dependent wave equation coefficients recovery to
reveal the spatial distribution of acoustical properties in the inhomogeneous
medium. The proposed method exhibits robustness to noise owing to the
incorporation of a loss function for the physical constraint that the assumed
PDE must be satisfied. For the coefficients recovery of spatially
two-dimensional PDEs, we store the PDE coefficients at all locations in the 2D
region of interest into a matrix and incorporate the low-rank assumption for
such a matrix to recover the coefficients at locations without available
measurements.
