PDE-constrained models with neural network terms: Optimization and global convergence

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, network parameters are calibrated available data. Calibration embedded can be performed optimizing over PDE. Motivated these applications, we rigorously study optimization class linear elliptic PDEs with terms. optimized using gradient descent, where evaluated an adjoint As number become large, converge non-local system. Using this limit system, able prove convergence network-PDE global minimum during optimization. Finally, use method train model for application fluid mechanics, which functions as closure Reynolds-averaged Navier–Stokes (RANS) equations. RANS trained on several datasets turbulent channel flow out-of-sample at different Reynolds numbers.