Physics-informed neural networks for inverse problems in supersonic flows

Abstract

Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we have data available for density gradients from Schlieren photography as well as data at the inflow and part of wall boundaries. These inverse problems are notoriously difficult and traditional methods may not be adequate to solve such ill-posed inverse problems. To this end, we employ the physics-informed neural networks (PINNs) and its extended version, extended PINNs (XPINNs), where domain decomposition allows to deploy locally powerful neural networks in each subdomain, which can provide additional expressivity in subdomains, where a complex solution is expected. Apart from the governing compressible Euler equations, we also enforce the entropy conditions in order to obtain viscosity solutions. Moreover, we enforce positivity conditions on density and pressure. We consider inverse problems involving two-dimensional expansion waves, two-dimensional oblique and bow shock waves. We compare solutions obtained by PINNs and XPINNs and invoke some theoretical results that can be used to decide on the generalization errors of the two methods.

Discontinuity Computing with Physics-Informed Neural Network

“…Jagtap et al [15] propose cPINN that splits the computing domain into several small subdomains with different NNs to solve Burgers equation and Euler equations. Jagtap et al [16] especially study inverse problems in supersonic flows. The above literature shows that PINN is really effective in handle inverse problems with prior information about the development of the flow structures, such as the density gradient.…”

Physics-aware deep learning framework for linear elasticity

“…The idea is to combine traditional scientific computational modeling with a data-driven ML framework to embed scientific knowledge into neural networks (NNs) to improve the performance of learning algorithms (Lagaris et al, 1998;Raissi and Karniadakis, 2018;Karniadakis et al, 2021). The Physics Informed Neural Networks (PINNs) (Lagaris et al, 1998;Raissi et al, 2019Raissi et al, , 2020 were developed for the solution and discovery of nonlinear PDEs leveraging the capabilities of deep neural networks (DNNs) as universal function approximators achieving considerable success in solving forward and inverse problems in different physical problems such as fluid flows (Sun et al, 2020;Jin et al, 2021), multi-scale flows (Lou et al, 2021), heat transfer (Cai et al, 2021;Zhu et al, 2021), poroelasticity (Haghighat et al, 2022), material identification (Shukla et al, 2021), geophysics (bin Waheed et al, 2021, 2022, supersonic flows (Jagtap et al, 2022), and various other applications (Waheed et al, 2020;Bekar et al, 2022). Contrary to traditional DL approaches, PINNs force the underlying PDEs and the boundary conditions in the solution domain ensuring the correct representation of governing physics of the problem.…”

Error estimates for physics informed neural networks approximating the Navier-Stokes equations

“…Since then, there has been an explosive growth in designing and applying PINNs for a variety of applications involving PDEs. A very incomplete list of references includes [36,28,33,45,12,13,14,16,29,30,31,2,40,15,11,41] and references therein.…”