Physics-informed non-intrusive reduced-order modeling of parameterized dynamical systems

In this study, we present a new framework of physics-informed non-intrusive reduced-order modeling (ROM) of dynamical systems modeled by parametric, partial differential equations (PDEs). Given new time and parameter values of a PDE, the framework utilizes trained physics-informed ML models to quickly estimate high-fidelity solutions while simultaneously observing the constraints and dynamics of the system. In the offline training phase, proper orthogonal decomposition (POD) decomposes a training database of high-fidelity solutions into POD modes and POD coefficients. A feed-forward neural network is trained to map time-parameter values to the few dominant POD coefficients. The loss function is composed of two terms: (1) error between original data and reconstructed data and (2) PDE residuals where each term of the PDE is expressed using Galerkin expansion on the reduced basis composed of the most dominant POD modes. The PDE residuals are not evaluated using POD–Galerkin (reduced-order) equations. The novelty of this work lies in the construction of PDE residual term and an a priori analysis that allows one to select weighting factor (or Lagrange multiplier) ahead of it. It has been found that a physics-informed ROM minimizing the two terms generates new solutions orders-of-magnitude accurate than a vanilla ROM that minimizes only the first error term. Besides estimating reconstruction error on a database, the framework also allows estimation of reconstruction quality of different terms such as advection and diffusion in the PDE. This is expected to promote better integration and interpretation of ML in reduced-order modeling of dynamical systems. During the online prediction phase, given new values of time and parameters, the generalized coordinates are quickly estimated and used in reconstruction. High-fidelity solutions are thus obtained orders-of-magnitude faster than a conventional numerical simulation. The framework is demonstrated on 1D and 2D Burgers’ equations and an incompressible flow over a backward facing step.