Physics informed neural network framework for unsteady discretized reduced order system

Abstract

This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent full-order and reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas any other discretization technique can be adopted), then projected on a reduced or latent space using the Proper Orthogonal Decomposition (POD)-Galerkin approach, and next, the residual arising from discretized reduced order equation is considered as an additional loss penalty term alongside the data-driven loss term using different variants of deep learning method such as Artificial neural network (ANN), Long Short-Term Memory based neural network (LSTM). The LSTM neural network has been proven to be very effective for time-dependent problems in a purely data-driven environment. The current work demonstrates the LSTM network's potential over ANN networks in PINN as well. The major difficulties in coupling PINN with external forward solvers often arise from the inability to access the discretized forms of the governing equation directly through the PINN solver and also to include those forms in the computational graph of the network. This poses a significant challenge, especially when a gradient-based optimization approach is considered in the neural network. Therefore, we propose an additional step in the PINN algorithm to overcome these difficulties. The proposed methods are applied to a pitch-plunge airfoil motion governed by rigid-body dynamics and a one-dimensional viscous Burgers' equation. The potential of using discretized governing equations instead of a continuous form lies in the flexibility of input to the PINN. The current work also demonstrates the prediction capability of various discretized-physics-informed neural networks outside the domain where the data is available or where the governing equation-based residuals are minimized.