  Data-driven discovery of governing equations in computational science has
emerged as a new paradigm for obtaining accurate physical models and as a
possible alternative to theoretical derivations. The recently developed
physics-informed neural networks (PINNs) have also been employed to learn
governing equations given data across diverse scientific disciplines. Despite
the effectiveness of PINNs for discovering governing equations, the physical
models encoded in PINNs may be misspecified in complex systems as some of the
physical processes may not be fully understood, leading to the poor accuracy of
PINN predictions. In this work, we present a general approach to correct the
misspecified physical models in PINNs for discovering governing equations,
given some sparse and/or noisy data. Specifically, we first encode the assumed
physical models, which may be misspecified, then employ other deep neural
networks (DNNs) to model the discrepancy between the imperfect models and the
observational data. Due to the expressivity of DNNs, the proposed method is
capable of reducing the computational errors caused by the model
misspecification and thus enables the applications of PINNs in complex systems
where the physical processes are not exactly known. Furthermore, we utilize the
Bayesian PINNs (B-PINNs) and/or ensemble PINNs to quantify uncertainties
arising from noisy and/or gappy data in the discovered governing equations. A
series of numerical examples including non-Newtonian channel and cavity flows
demonstrate that the added DNNs are capable of correcting the model
misspecification in PINNs and thus reduce the discrepancy between the physical
models and the observational data. We envision that the proposed approach will
extend the applications of PINNs for discovering governing equations in
problems where the physico-chemical or biological processes are not well
understood.
