Correcting model misspecification in physics-informed neural networks (PINNs)

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.