Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction in Soft Sensor Modeling

Nonhomogeneous partial differential equations (PDEs) are an applicable model in soft sensor modeling for describing spatiotemporal industrial systems with unmeasurable source terms, which cannot be well solved by existing physics-informed neural networks (PINNs). To this end, a coupled PINN (CPINN) with a recurrent prediction (RP) learning strategy (CPINN-RP) is proposed for soft sensor modeling in spatiotemporal industrial processes, such as vibration displacement. First, CPINN containing NetU and NetG is proposed. NetU is used to approximate the solutions to PDEs under study and NetG is used to regularize 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 capacity to the PDEs solutions. Besides the theoretical aspects, we propose a hierarchical training strategy to optimize and couple the two networks to achieve the parameters of CPINN. Secondly, NetU-RP is achieved by NetU compensated by RP, the recurrently delayed output of CPINN, to further improve the soft sensor performance. Finally, simulations and experiment verify the effectiveness and practical applications of CPINN-RP.