Physics-informed neural networks for inverse problems in supersonic flows

TL;DR

This paper explores the use of physics-informed neural networks (PINNs and XPINNs) to solve complex inverse problems in supersonic flows, incorporating domain decomposition, entropy, and positivity constraints for improved accuracy.

Contribution

It introduces the application of PINNs and XPINNs with domain decomposition and physical constraints to inverse supersonic flow problems, advancing solution accuracy and generalization understanding.

Findings

PINNs and XPINNs effectively solve inverse supersonic flow problems.

XPINNs provide enhanced expressivity in complex solution regions.

Theoretical insights help assess generalization errors of the methods.

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 deploying 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.