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Partial differential equations (PDEs) are a model candidate for soft sensorsin industrial processes with spatiotemporal dependence. Althoughphysics-informed neural networks (PINNs) are a promising machine learningmethod for solving PDEs, they are infeasible for the nonhomogeneous PDEs withunmeasurable source terms. To this end, a coupled PINN (CPINN) with a recurrentprediction (RP) learning strategy (CPINN- RP) is proposed. First, CPINNcomposed of NetU and NetG is proposed. NetU is for approximating PDEs solutionsand NetG is for regularizing the training of NetU. The two networks areintegrated into a data-physics-hybrid loss function. Then, we theoreticallyprove that the proposed CPINN has a satisfying approximation capability forsolutions to nonhomogeneous PDEs with unmeasurable source terms. Besides thetheoretical aspects, we propose a hierarchical training strategy to optimizeand couple NetU and NetG. Secondly, NetU-RP is proposed for compensatinginformation loss in data sampling to improve the prediction performance, inwhich RP is the recurrently delayed outputs of well-trained CPINN and hardsensors. Finally, the artificial and practical datasets are used to verify thefeasibility and effectiveness of CPINN-RP for soft sensors.

Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction for Soft Sensors