Soft sensors have been extensively used to monitor key variables using easy-to-measure variables and mathematical models. Partial differential equations (PDEs) are model candidates for soft sensors in industrial processes with spatiotemporal dependence. However, gaps often exist between idealized PDEs and practical situations. Discovering proper structures of PDEs, including the differential operators and source terms, can remedy the gaps. To this end, a coupled physics-informed neural network with Akaike's criterion information (CPINN-AIC) is proposed for PDE discovery of soft sensors. First, CPINN is adopted for obtaining solutions and source terms satisfying PDEs. Then, we propose a data-physics-hybrid loss function for training CPINN, in which undetermined combinations of differential operators are involved. Consequently, AIC is used to discover the proper combination of differential operators. Finally, the artificial and practical datasets are used to verify the feasibility and effectiveness of CPINN-AIC for soft sensors. The proposed CPINN-AIC is a data-driven method to discover proper PDE structures and neural network-based solutions for soft sensors.

翻译：软传感器通过易于测量的变量和数学模型广泛应用于关键变量的监测。偏微分方程（PDE）是工业过程中具有时空依赖性的软传感器模型候选。然而，理想化PDE与实际工况之间往往存在差异。通过发现包含微分算子和源项在内的PDE适当结构，可弥合这种差距。为此，本文提出一种基于赤池信息准则的耦合物理信息神经网络（CPINN-AIC）用于软传感器的PDE发现。首先，采用CPINN求解满足PDE的解和源项；其次，提出包含微分算子未定组合的数据-物理混合损失函数训练CPINN；进而，利用AIC准则发现最优微分算子组合；最后，通过人工与实测数据集验证了CPINN-AIC用于软传感器的可行性与有效性。所提CPINN-AIC是一种数据驱动方法，能够为软传感器发现恰当的PDE结构及基于神经网络的解。