  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.
