Physics-informed neural networks for inverse problems in supersonic flows (2202.11821v1)

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.

• The paper demonstrates that integrating PINNs with the Euler equations and entropy conditions yields accurate state inference in supersonic flow inverse problems.
• The paper applies domain decomposition and adaptive activation functions to effectively capture shock and expansion wave dynamics.
• The paper shows that XPINNs achieve lower generalization errors compared to traditional PINNs, enhancing predictive modeling for high-speed aerodynamics.

Physics-Informed Neural Networks for Inverse Problems in Supersonic Flows

The paper presented in "Physics-informed neural networks for inverse problems in supersonic flows" focuses on addressing the challenge of solving inverse problems associated with supersonic compressible flows, which have significant implications in aerospace vehicle design. Traditional methods have proven inadequate to resolve such ill-posed problems, particularly when faced with the complexities of shock waves and expansion waves. The authors employ physics-informed neural networks (PINNs) and extended PINNs (XPINNs) to provide solutions by leveraging domain decomposition and enhancing neural network capabilities within specific subdomains.

Key to this approach is the integration of PINNs with the governing compressible Euler equations, alongside entropy conditions to ensure viscosity solutions, and enforcing positivity conditions on physical quantities like density and pressure. This framework allows for the inference of states in problems such as two-dimensional expansion waves, and oblique and bow shock waves—complex scenarios where traditional numerical solvers falter.

The methodology involves using data from density gradients obtained through Schlieren photography, alongside partial boundary data. The PINNs recast the original inverse problem into an optimization problem through the loss function where various constraints and data types are seamlessly incorporated. The paper distinctly highlights the comparative examination of PINNs and XPINNs through computational examples, showcasing the performance differences based on the deployment of dynamic weights and adaptive activation functions.

Numerical results illustrate that both PINNs and XPINNs provide accurate predictions, especially when utilizing adaptive activation functions or dynamic weights. The XPINNs show slightly superior accuracy due to the localized power of neural networks, which are tailored to handle complex solutions around discontinuities like shock waves. For the expansion wave problem, XPINNs demonstrated improved generalization with a reduced generalization error, attributed to the decomposition of complex solutions into simpler parts across multiple subdomains.

Implications are far-reaching for both theoretical advancements and practical applications in engineering. The approach sets a precedent for employing machine learning techniques to solve inverse problems traditionally deemed unsolvable, thereby enabling enhanced predictive modeling for high-speed aerodynamics. Enhanced generalization capacities of XPINNs, pointed out in theoretical bounds referenced within the paper, suggest that future developments could further capitalize on neural network-based methodologies, optimizing alongside computational complexity reductions.

In conclusion, the paper contributes substantially to the field by demonstrating practical applications of neural networks in complex fluid dynamics scenarios and provides a clear path towards leveraging AI for solving real-world engineering challenges where conventional methods reach their limits.