Physics-informed neural networks for inverse problems in supersonic flows

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.

Consistent and symmetry preserving data-driven interface reconstruction for the level-set method

Recently, machine learning has been used to substitute parts of conventional computational fluid dynamics, e.g. the cell-face reconstruction in finite-volume solvers or the curvature computation in the Volume-of-Fluid (VOF) method. The latter showed improvements in terms of accuracy for coarsely resolved interfaces, however at the expense of convergence and symmetry. In this work, a combined approach is proposed, adressing the aforementioned shortcomings. We focus on interface reconstruction (IR) in the level-set method, i.e. the computation of the volume fraction and apertures. The combined model consists of a classification neural network, that chooses between the conventional (linear) IR and the neural network IR depending on the local interface resolution. The proposed approach improves accuracy for coarsely resolved interfaces and recovers the conventional IR for high resolutions, yielding first order overall convergence. Symmetry is preserved by mirroring and rotating the input level-set grid and subsequently averaging the predictions. The combined model is implemented into a CFD solver and demonstrated for two-phase flows. Furthermore, we provide details of floating point symmetric implementation and computational efficiency.