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.
翻译:设计专门的航空航天器往往需要准确解决反超声压缩流问题。特别是,我们考虑的问题是,我们拥有Scherieren摄影的密度梯度数据以及流入和部分墙界的数据。这些反面问题臭名昭著地困难,传统方法可能不足以解决这种不正确反向问题。为此,我们使用物理知情神经网络(PINNs)及其扩展版、扩展的PINNs(XPINNs),在其中,域分解允许在每个子域部署强大的当地神经网络,这些网络可以在预期有复杂解决办法的子域内提供额外的直观度。除了可压缩 Euler方程式之外,我们还执行恒温条件,以获得反向的解决方案。此外,我们还在密度和压力上实施负感性条件。我们考虑了与两维扩张波、二维反调和弓形冲击波有关的反向问题。我们比较了PINNs和XPINNSs获得的解决方案,并引用了用于决定一般化的两种方法的理论性结果。