  In this paper, we present a physics-constrained deep neural network (PCDNN)
method for parameter estimation in the zero-dimensional (0D) model of the
vanadium redox flow battery (VRFB). In this approach, we use deep neural
networks (DNNs) to approximate the model parameters as functions of the
operating conditions. This method allows the integration of the VRFB
computational models as the physical constraints in the parameter learning
process, leading to enhanced accuracy of parameter estimation and cell voltage
prediction. Using an experimental dataset, we demonstrate that the PCDNN method
can estimate model parameters for a range of operating conditions and improve
the 0D model prediction of voltage compared to the 0D model prediction with
constant operation-condition-independent parameters estimated with traditional
inverse methods. We also demonstrate that the PCDNN approach has an improved
generalization ability for estimating parameter values for operating conditions
not used in the DNN training.
