[Paper Review] Physics-informed neural networks for inverse problems in supersonic flows

Ameya D. Jagtap, Zhiping Mao+2 more|arXiv (Cornell University)|Feb 23, 2022

Model Reduction and Neural NetworksPhysics and Astronomy33 references295 citations

TL;DR

This paper proposes physics-informed neural networks (PINNs) and their extended domain-decomposition variant (XPINNs) to solve challenging inverse problems in two-dimensional supersonic compressible flows, incorporating Euler equations, entropy conditions, and positivity constraints. XPINNs outperform standard PINNs in accuracy for shock and expansion wave problems by enabling localized, high-capacity networks, with theoretical generalization bounds supporting improved performance on complex, discontinuous solutions.

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.

Motivation & Objective

• Address the difficulty of solving ill-posed inverse problems in supersonic compressible flows, particularly those involving shocks and discontinuities.
• Overcome limitations of traditional numerical solvers that require full boundary conditions and large computational domains.
• Develop a mesh-free, data-driven approach using PINNs and XPINNs to infer full flow states (density, velocity, pressure) from sparse, partial data such as Schlieren density gradients and boundary conditions.
• Enforce physical consistency through entropy conditions and positivity constraints on density and pressure to ensure physically meaningful solutions.
• Compare the predictive accuracy and generalization performance of PINNs versus XPINNs for complex supersonic flow configurations.

Proposed method

• Employ physics-informed neural networks (PINNs) to embed the compressible Euler equations directly into the loss function, enabling mesh-free solution of forward and inverse problems.
• Extend PINNs to XPINNs via spatial domain decomposition, allowing locally optimized, high-capacity neural networks in subdomains with complex flow features such as shocks.
• Incorporate entropy conditions as additional constraints in the loss function to select viscosity solutions and ensure physical consistency in the presence of discontinuities.
• Enforce positivity constraints on density and pressure to prevent unphysical network outputs.
• Use adaptive activation functions and dynamic loss weighting to improve training stability and convergence for problems with sharp gradients and discontinuities.
• Leverage theoretical generalization bounds from prior work to analyze and compare the generalization performance of PINNs and XPINNs.

Experimental results

Research questions

• RQ1 Can PINNs and XPINNs accurately reconstruct full supersonic flow states (density, velocity, pressure) from sparse data, including Schlieren density gradients and partial boundary conditions?
• RQ2 How does domain decomposition in XPINNs improve solution accuracy and generalization compared to standard PINNs for inverse problems with shocks and expansion waves?
• RQ3 To what extent do entropy conditions and positivity constraints enhance the physical fidelity and stability of PINN-based solutions in compressible flow inverse problems?
• RQ4 How do adaptive activation functions and dynamic loss weights affect the convergence and accuracy of PINN and XPINN solutions in the presence of discontinuities?
• RQ5 What theoretical insights from generalization bounds support the improved performance of XPINNs over PINNs in complex flow scenarios?

Key findings

• XPINNs achieve significantly better predictive accuracy than PINNs for inverse problems involving oblique shocks, expansion waves, and bow shocks, as demonstrated by lower point-wise relative errors in density, pressure, and velocity fields.
• The use of adaptive activation functions and dynamic loss weights leads to improved convergence and accuracy compared to fixed activation functions and fixed weights.
• Enforcement of entropy conditions and positivity constraints on density and pressure results in physically consistent solutions, particularly important for capturing correct shock structure and avoiding unphysical oscillations.
• Theoretical generalization bounds suggest that XPINNs achieve better generalization due to reduced local complexity per subdomain, despite a potential risk of overfitting from reduced data per subdomain.
• Domain decomposition in XPINNs allows for locally powerful networks in regions of high solution complexity, such as shock fronts, leading to more accurate resolution of discontinuous features.
• For the bow shock problem, XPINNs show superior performance with point-wise relative errors in density and pressure consistently below 10^-3 across the domain, outperforming PINNs which exhibit higher error concentrations near discontinuities.