Integration of neural networks with numerical solution of PDEs for closure models development

Modeling often requires closures to account for the multiscale/multiphysics nature of certain phenomena. Recently, there has been interest in the application of machine learning (ML) for their development. Most of the applications are purely data-driven; however, incorporation of the knowledgebase is an opportunity to enhance flexibility and predictive capability of ML models. This paper presents a PDE-integrated ML framework. PDEs are solved using convolutional operators and integrated with neural networks (NNs). Such integration allows one to train the NNs directly on observed field variables. To demonstrate the framework's viability, NNs are integrated with heat conduction, Navier-Stokes, and RANS equations. (C) 2021 Elsevier B.V. All rights reserved.