zbMATH — the first resource for mathematics

Physics-informed neural networks for high-speed flows. (English) Zbl 1442.76092
Summary: 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 \(\nabla \rho(x, t)\), the pressure \(p(x^\ast, t)\) at a specified point \(x = x^\ast\) 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 \(x^\ast\) 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 \(x^\ast\) from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point \(x^\ast\) 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 \(\gamma\) 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 \(\gamma\) 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.

76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Full Text: DOI
[1] Courant, R.; Friedrichs, K. O., Supersonic Flow and Shock Waves, Vol. 21 (1999), Springer Science & Business Media
[2] Liepmann, H. W.; Roshko, A., Elements of Gasdynamics (2001), Courier Corporation · Zbl 0078.39901
[3] Zucker, R. D.; Biblarz, O., Fundamentals of Gas Dynamics (2002), John Wiley & Sons
[4] Dafermos, C. M., (Hyperbolic Conservation Laws in Continuum Physics. Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325 (2016), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1364.35003
[5] LeVeque, R. J., (Finite Volume Methods for Hyperbolic Problems. Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics (2002), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1010.65040
[6] Lomtev, I.; Karniadakis, G., Discontinuous Galerkin methods in CFD, (APS Division of Fluid Dynamics Meeting Abstracts (1998))
[7] Lomtev, I.; Karniadakis, G. E., A discontinuous Galerkin method for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 29, 5, 587-603 (1999) · Zbl 0951.76041
[8] Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2007), Springer Science & Business Media
[9] Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications, Vol. 11 (2012), Springer Science & Business Media
[10] Shu, C.-W., A brief survey on discontinuous Galerkin methods in computational fluid dynamics, Adv. Mech., 43, 6, 541-553 (2013)
[11] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformaly high order essentially non-oscillatory schemes III, Math. Comput., 193, 563-594 (2004)
[12] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228 (1996) · Zbl 0877.65065
[13] Wang, Z., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Prog. Aerosp. Sci., 43, 1-3, 1-41 (2007)
[14] Pirozzoli, S., Numerical methods for high-speed flows, (Annual Review of Fluid Mechanics, Volume 43. Annual Review of Fluid Mechanics, Volume 43, Annu. Rev. Fluid Mech., vol. 43 (2011), Annual Reviews: Annual Reviews Palo Alto, CA), 163-194 · Zbl 1299.76103
[15] Johnsen, E.; Larsson, J.; Bhagatwala, A. V.; Cabot, W. H.; Moin, P.; Olson, B. J.; Rawat, P. S.; Shankar, S. K.; Sjögreen, B.; Yee, H. C.; Zhong, X.; Lele, S. K., Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J. Comput. Phys., 229, 4, 1213-1237 (2010) · Zbl 1329.76138
[16] Raissi, M.; Karniadakis, G. E., Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys., 357, 125-141 (2018) · Zbl 1381.68248
[17] Raissi, M.; Perdikaris, P.; Karniadakis, G. E., Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707 (2019) · Zbl 1415.68175
[18] Pang, G.; Yang, L.; Karniadakis, G. E., Neural-net-induced Gaussian process regression for function approximation and PDE solution, J. Comput. Phys., 384, 270-288 (2019)
[19] K.O. Lye, S. Mishra, D. Ray, Deep learning observables in computational fluid dynamics, arXiv preprint arXiv:1903.03040.
[20] J. Magiera, D. Ray, J.S. Hesthaven, C. Rohde, Constraint-aware neural networks for Riemann problems, arXiv preprint arXiv:1904.12794.
[21] Pang, G.; Lu, L.; Karniadakis, G. E., FPINNs: fractional physics-informed neural networks, SIAM J. Sci. Comput., 41, 4, A2603-A2626 (2019) · Zbl 1420.35459
[22] L. Yang, D. Zhang, G.E. Karniadakis, Physics-informed generative adversarial networks for stochastic differential equations, arXiv preprint arXiv:1811.02033. · Zbl 1440.60065
[23] Meng, X.; Karniadakis, G. E., A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems, J. Comput. Phys., 401, 109020 (2020)
[24] Wong, J. S.; Darmofal, D. L.; Peraire, J., The solution of the compressible Euler equations at low Mach numbers using a stabilized finite element algorithm, Comput. Methods Appl. Mech. Engrg., 190, 43-44, 5719-5737 (2001) · Zbl 1044.76036
[25] Nazarov, M.; Larcher, A., Numerical investigation of a viscous regularization of the Euler equations by entropy viscosity, Comput. Methods Appl. Mech. Engrg., 317, 128-152 (2017)
[26] Monthe, L.; Benkhaldoun, F.; Elmahi, I., Positivity preserving finite volume Roe schemes for transport-diffusion equations, Comput. Methods Appl. Mech. Engrg., 178, 3-4, 215-232 (1999) · Zbl 0967.76063
[27] Guermond, J.-L.; Nazarov, M., A maximum-principle preserving \(C^0\) finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Engrg., 272, 198-213 (2014) · Zbl 1296.65133
[28] C. Michoski, M. Milosavljevic, T. Oliver, D. Hatch, Solving irregular and data-enriched differential equations using deep neural networks, arXiv preprint arXiv:1905.04351.
[29] M. Raissi, A. Yazdani, G.E. Karniadakis, Hidden fluid mechanics: A Navier-Stokes informed deep learning framework for assimilating flow visualization data, arXiv preprint arXiv:1808.04327.
[30] Anderson, J. D., Computational Fluid Dynamics: The Basics with Applications (1995), McGraw-Hill
[31] Bayliss, A.; Turkel, E., Far field boundary conditions for compressible flows, J. Comput. Phys., 48, 2, 182-199 (1982) · Zbl 0494.76072
[32] Coclici, C.; Wendland, W. L., Domain decomposition methods and far-field boundary conditions for 2D compressible viscous flows, (Recent Advances in Numerical Methods and Applications, II (Sofia, 1998) (1999), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 429-437 · Zbl 1119.76360
[33] Sanders, R.; Weiser, A., A high order staggered grid method for hyperbolic systems of conservation laws in one space dimension, Comput. Methods Appl. Mech. Engrg., 75, 1-3, 91-107 (1989) · Zbl 0694.65042
[34] Tang, Y.-H.; Lee, S.-T.; Yang, J.-Y., A high-order pathline Godunov scheme for unsteady one-dimensional equilibrium flows, Comput. Methods Appl. Mech. Engrg., 161, 3-4, 257-288 (1998) · Zbl 0949.76055
[35] Alves, M.; Cruz, P.; Mendes, A.; Magalhaes, F.; Pinho, F.; Oliveira, P., Adaptive multiresolution approach for solution of hyperbolic PDEs, Comput. Methods Appl. Mech. Engrg., 191, 36, 3909-3928 (2002) · Zbl 1010.65042
[36] Banks, J. W.; Hittinger, J. A.F.; Connors, J. M.; Woodward, C. S., Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport, Comput. Methods Appl. Mech. Eng., 213/216, 1-15 (2012) · Zbl 1243.65102
[37] Gryngarten, L. D.; Menon, S., A generalized approach for sub-and super-critical flows using the Local Discontinuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 253, 169-185 (2013) · Zbl 1297.76099
[38] Guermond, J.-L.; Popov, B.; Tomov, V., Entropy-viscosity method for the single material Euler equations in Lagrangian frame, Comput. Methods Appl. Mech. Engrg., 300, 402-426 (2016) · Zbl 1423.76235
[39] Nogueira, X.; Ramírez, L.; Clain, S.; Loubère, R.; Cueto-Felgueroso, L.; Colominas, I., High-accurate SPH method with multidimensional optimal order detection limiting, Comput. Methods Appl. Mech. Engrg., 310, 134-155 (2016)
[40] Ji, Z.; Fu, L.; Hu, X. Y.; Adams, N. A., A new multi-resolution parallel framework for SPH, Comput. Methods Appl. Mech. Engrg., 346, 1156-1178 (2019)
[41] Baydin, A. G.; Pearlmutter, B. A.; Radul, A. A.; Siskind, J. M., Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res., 18, 1-43 (2018) · Zbl 06982909
[42] Bergstra, J. S.; Bardenet, R.; Bengio, Y.; Kégl, B., Algorithms for hyper-parameter optimization, (Advances in Neural Information Processing Systems (2011)), 2546-2554
[43] Snoek, J.; Larochelle, H.; Adams, R. P., Practical bayesian optimization of machine learning algorithms, (Advances in Neural Information Processing Systems (2012)), 2951-2959
[44] Snoek, J.; Rippel, O.; Swersky, K.; Kiros, R.; Satish, N.; Sundaram, N.; Patwary, M.; Prabhat, M.; Adams, R., Scalable bayesian optimization using deep neural networks, (International Conference on Machine Learning (2015)), 2171-2180
[45] X. Chen, J. Duan, G.E. Karniadakis, Learning and meta-learning of stochastic advection-diffusion-reaction systems from sparse measurements, arXiv preprint arXiv:1910.09098.
[46] Jagtap, A. D.; Kawaguchi, K.; Karniadakis, G. E., Adaptive activation functions accelerate convergence in deep and physics-informed neural networks, J. Comput. Phys., 109136 (2019)
[47] A.D. Jagtap, K. Kawaguchi, G.E. Karniadakis, Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks, arXiv preprint arXiv:1909.12228.
[48] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1, 1-31 (1978) · Zbl 0387.76063
[49] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2013), Springer Science & Business Media
[50] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods for Computational Fluid Dynamics (2013), Oxford University Press · Zbl 1256.76003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.