Physics-informed neural networks for high-speed flows

By Zhiping Mao, Ameya Dilip Jagtap, G. Em Karniadakis. 2020

1,237 citations recorded by the source corpus.

Abstract In this work we investigate the possibility of using physics-informed neural networks (PINNs) to approximate the Euler equations that model high-speed aerodynamic flows. In particular, we solve both the forward and inverse problems in one-dimensional and two-dimensional domains. For the forward problem, we utilize the Euler equations and the initial/boundary conditions to formulate the loss function, and solve the one-dimensional Euler equations with smooth solutions and with solutions that have a contact discontinuity as well as a two-dimensional oblique shock wave problem. We demonstrate that we can capture the solutions with only a few scattered points clustered randomly around the discontinuities. For the inverse problem, motivated by mimicking the Schlieren photography experimental technique used traditionally in high-speed aerodynamics, we use the data on density gradient ∇ ρ ( x , t ) , the pressure p ( x ∗ , t ) at a specified point x = x ∗ as well as the conservation laws to infer all states of interest (density, velocity and pressure fields). We present illustrative benchmark examples for both the problem with smooth solutions and Riemann problems (Sod and Lax problems) with PINNs, demonstrating that all inferred states are in good agreement with the reference solutions. Moreover, we show that the choice of the position of the point x ∗ plays an important role in the learning process. In particular, for the problem with smooth solutions we can randomly choose the position of the point x ∗ from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point x ∗ from the domain between the initial discontinuous point and the shock position of the final time. We also solve the inverse problem by combining the aforementioned data and the Euler equations in characteristic form, showing that the results obtained by using the Euler equations in characteristic form are better than that obtained by using the Euler equations in conservative form. Furthermore, we consider another type of inverse problem, specifically, we employ PINNs to learn the value of the parameter γ in the equation of state for the parameterized two-dimensional oblique wave problem by using the given data of the density, velocity and the pressure, and we identify the parameter γ accurately. Taken together, our results demonstrate that in the current form, where the conservation laws are imposed at random points, PINNs are not as accurate as traditional numerical methods for forward problems but they are superior for inverse problems that cannot even be solved with standard techniques.

Subfields: Aerospace Engineering, Computational Mechanics, Statistical and Nonlinear Physics

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Citations in the corpus

Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions (2021, background): “…applied to various problems in engineering and biology with applications including but not limited to heat transfer (Cai et al., 2021), solution of Navier-Stokes equations in fluid mechanics (Jin et al., 2021), high speed fluid flow (Mao et al., 2020) and solid mechanics (Haghighat et al., 2021).”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “Jin et al. [13] applied FC-PINNs to solve Navier–Stokes equations, ranging from laminar to turbulent regimes, while Mao et al. [14] further showed their effectiveness on high-speed flow problems.”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “[13], here we only use the density gradients in a sub-domain D of the computational domain , i.”
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, methodology): “This is seen in the robustness and accuracy of the results (see [44,53,54] and references therein).”
Machine Learning in Aerodynamic Shape Optimization (2022, methodology): “[282] showed satisfactory results on both forward and inverse problems using PINN-approximated Euler equations for high-speed aerodynamic flows.”
DeepXDE: A Deep Learning Library for Solving Differential Equations (2019, background): “, randomly distributed points, or clustered points in the domain [37]), i.”
CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method (2021, methodology): “PINNs have been demonstrated for various physics phenomena, including heat transfer [23,24], fluid dynamics [25–27] and electromagnetic propagation [28–30].”
B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data (2020, neutral): “for a wide range of PDEs [9, 10, 11, 12, 13].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “could be utilized directly by plugging it into any differential problem simplified the learning curve for beginning to use such, and it was the first step for many researchers who wanted to solve their problems with a Neural network approach (Markidis, 2021). The success of the PINNs can be seen fro”
hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition (2020, methodology): “The PINN formulation has been recently successfully employed in many physical problems such as discovering turbulence models from scattered/noisy measurements [28], high speed flows [4], stochastic differential equation by generative adversarial networks [24], fractional differential equations [29],”
Physics-informed neural networks for inverse problems in supersonic flows (2022, methodology): “Instead, we use the information of density gradient ∇ρ and a small number of pressure data points on the surface of the wedge, following the one-dimensional problem [3].”
Parallel Physics-Informed Neural Networks via Domain Decomposition (2021, background): “Due to their simplicity, PINNs have led to progress in many areas of computational science, see for examples [7, 8, 9, 10, 11, 12, 13, 14, 15].”
A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks (2022, background): “[36] for high-speed flows, but this approach is highly problem-dependent and usually tedious and time-consuming.”