Physics Informed Neural Network Framework for Unsteady Discretized Reduced Order System

TL;DR

This work develops a discretized reduced-order physics-informed neural network (DisPINN) framework by discretizing the governing PDEs, projecting onto a POD-based latent space, and using the discretized residual as a physics loss alongside data loss. It compares ANN-DisPINN and LSTM-DisPINN architectures across unsteady problems (mass-spring and Burgers'), demonstrates superior performance in data-sparse regimes, and introduces a robust method to couple PINNs with external forward solvers by importing residuals and Jacobians and updating the Jacobian periodically. Key contributions include integrating discretized ROMs with PINN training, implementing DEIM-based hyper-reduction, and enabling seamless external-solver coupling while preserving gradient-based optimization. The approach delivers data-efficient, physics-consistent predictions for unsteady problems and lays groundwork for coupling PINNs with complex solvers and geometries in practical computational science applications.

Abstract

This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent 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 physics-informed neural networks (PINN) as well. The potential of using discretized governing equations instead of continuous form lies in the flexibility of input to the PINN. Different sizes of data ranging from small, medium to big datasets are used to assess the potential of discretized-physics-informed neural networks when there is very sparse or no data available. The proposed methods are applied to a pitch-plunge airfoil motion governed by rigid-body dynamics and a one-dimensional viscous Burgers' equation. The current work also demonstrates the prediction capability of various discretized-physics-informed neural networks outside the domain where the data is available or governing equation-based residuals are minimized.

Figures (17)

• Figure 1: Input and output architecture of LSTM.
• Figure 2: Architecture of ANN-DisPINN.
• Figure 3: Architecture of a single LSTM-Cell and Discretized PDE-based LSTM-DisPINN.
• Figure 4: General framework for different reduced order models (i.e., projection-based approach, data-driven approach) and DisPINN, a bridge between these two aspects.
• Figure 5: Data-driven and physics-driven loss term with no. of epochs for different physics-informed neural networks.