  Partial differential equations (PDEs) are a model candidate for soft sensors
in industrial processes with spatiotemporal dependence. Although
physics-informed neural networks (PINNs) are a promising machine learning
method for solving PDEs, they are infeasible for the nonhomogeneous PDEs with
unmeasurable source terms. To this end, a coupled PINN (CPINN) with a recurrent
prediction (RP) learning strategy (CPINN- RP) is proposed. First, CPINN
composed of NetU and NetG is proposed. NetU is for approximating PDEs solutions
and NetG is for regularizing the training of NetU. The two networks are
integrated into a data-physics-hybrid loss function. Then, we theoretically
prove that the proposed CPINN has a satisfying approximation capability for
solutions to nonhomogeneous PDEs with unmeasurable source terms. Besides the
theoretical aspects, we propose a hierarchical training strategy to optimize
and couple NetU and NetG. Secondly, NetU-RP is proposed for compensating
information loss in data sampling to improve the prediction performance, in
which RP is the recurrently delayed outputs of well-trained CPINN and hard
sensors. Finally, the artificial and practical datasets are used to verify the
feasibility and effectiveness of CPINN-RP for soft sensors.
