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.

翻译：非齐次偏微分方程（PDEs）是软测量建模中描述具有不可测源项时空工业系统的适用模型，现有物理信息神经网络（PINNs）无法有效求解该类方程。为此，本文提出一种耦合物理信息神经网络（CPINN）结合循环预测（RP）学习策略的CPINN-RP方法，用于振动位移等时空工业过程的软测量建模。首先，构建包含NetU与NetG的CPINN：NetU用于逼近待求解PDEs的解，NetG用于正则化NetU的训练过程，两网络通过数据-物理混合损失函数实现融合。其次，从理论上证明所提CPINN对PDEs解具有满意的逼近能力。在理论分析基础上，提出分层训练策略以优化并耦合两网络，最终获取CPINN参数。进一步，通过RP补偿项（即CPINN的循环延迟输出）构建NetU-RP架构以提升软测量性能。最后，仿真与实验验证了CPINN-RP的有效性与实际应用价值。