×

zbMATH — the first resource for mathematics

An artificial neural network as a troubled-cell indicator. (English) Zbl 1415.65229
Summary: High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by the objective to construct a universal troubled-cell indicator that can be used for general conservation laws, we propose a new approach to detect discontinuities using artificial neural networks (ANNs). In particular, we construct a multilayer perceptron (MLP), which is trained offline using a supervised learning strategy, and thereafter used as a black-box to identify troubled-cells. The proposed MLP indicator can accurately identify smooth extrema and is independent of problem-dependent parameters, which gives it an advantage over traditional limiter-based indicators. Several numerical results are presented to demonstrate the robustness of the MLP indicator in the framework of Runge-Kutta discontinuous Galerkin schemes, and its performance is compared with the minmod limiter and the minmod-based TVB limiter.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
68T05 Learning and adaptive systems in artificial intelligence
Software:
AlexNet; Adam; TensorFlow
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dafermos, C. M., Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, vol. 325, (2010), Springer-Verlag Berlin · Zbl 1196.35001
[2] van Leer, B., Towards the ultimate conservative difference scheme, V: a second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[3] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, III: one-dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093
[4] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws, V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[5] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 131, 3-47, (1997) · Zbl 0866.65058
[6] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200-212, (1994) · Zbl 0811.65076
[7] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, 325-432, (1998), Springer Berlin, Heidelberg · Zbl 0927.65111
[8] Hesthaven, J. S.; Warburton, T., Nodal discontinuous Galerkin methods, Texts in Applied Mathematics, vol. 54, (2008), Springer New York · Zbl 1134.65068
[9] Jameson, A.; Schmidt, W.; Turkel, E., Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time stepping schemes, (1981), AIAA Paper 81-1259
[10] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 512, 451-512, (2004) · Zbl 1046.65078
[11] Qiu, J.; Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci. Comput., 26, 907-929, (2005) · Zbl 1077.65109
[12] Qiu, J.; Shu, C.-W., A comparison of troubled-cell indicators for Runge-Kutta discontinuous Galerkin methods using weighted essentially nonoscillatory limiters, SIAM J. Sci. Comput., 27, 995-1013, (2005) · Zbl 1092.65084
[13] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws, II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083
[14] Vuik, M. J.; Ryan, J. K., Automated parameters for troubled-cell indicators using outlier detection, SIAM J. Sci. Comput., 38, A84-A104, (2016) · Zbl 1330.65155
[15] Tukey, J. W., Exploratory data analysis, (1977), Addison-Wesley · Zbl 0409.62003
[16] Gao, Z.; Wen, X.; Don, W. S., Enhanced robustness of the hybrid compact-WENO finite difference scheme for hyperbolic conservation laws with multi-resolution analysis and Tukey’s boxplot method, J. Sci. Comput., 73, 736-752, (2017) · Zbl 1381.65065
[17] Fu, G.; Shu, C.-W., A new troubled-cell indicator for discontinuous Galerkin methods for hyperbolic conservation laws, J. Comput. Phys., 347, 305-327, (2017) · Zbl 1380.65262
[18] Haykin, S., Neural networks: A comprehensive foundation, (1998), Prentice Hall PTR Upper Saddle River, NJ, USA · Zbl 0828.68103
[19] Cristea, P. D., Application of neural networks, (Image Processing and Visualization, (2009), Springer Netherlands, Dordrecht), 59-71
[20] Dede, G.; Sazli, M. H., Speech recognition with artificial neural networks, Digit. Signal Process., 20, 763-768, (2010)
[21] Lagaris, I. E.; Likas, A.; Fotiadis, D. I., Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw., 9, 987-1000, (1998)
[22] Golak, S., A MLP solver for first and second order partial differential equations, 789-797, (2007), Springer Berlin, Heidelberg
[23] Rudd, K.; Ferrari, S., A constrained integration (cint) approach to solving partial differential equations using artificial neural networks, Neurocomputing, 155, 277-285, (2015)
[24] Cybenko, G., Continuous valued neural networks with two hidden layers are sufficient, (1988), Department of Computer Science, Tufts University Medford, MA, Technical Report
[25] Cybenko, G., Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst., 2, 303-314, (1989) · Zbl 0679.94019
[26] Guliyev, N. J.; Ismailov, V. E., A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function, Neural Comput., 28, 1289-1304, (2016)
[27] Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 89-112, (2001), (electronic) · Zbl 0967.65098
[28] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261, (2001) · Zbl 1065.76135
[29] Kriesel, D., A brief introduction to neural networks, (2007)
[30] McCulloch, W. S.; Pitts, W., A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5, 115-133, (1943) · Zbl 0063.03860
[31] Schmidhuber, J., Deep learning in neural networks: an overview, Neural Netw., 61, 85-117, (2015)
[32] Nair, V.; Hinton, G. E., Rectified linear units improve restricted Boltzmann machines, (Fürnkranz, J.; Joachims, T., Proceedings of the 27th International Conference on Machine Learning, ICML-10, (2010), Omnipress), 807-814
[33] Krizhevsky, A.; Sutskever, I.; Hinton, G. E., Imagenet classification with deep convolutional neural networks, (Pereira, F.; Burges, C. J.C.; Bottou, L.; Weinberger, K. Q., Advances in Neural Information Processing Systems, vol. 25, (2012), Curran Associates, Inc.), 1097-1105, (2012)
[34] Maas, A. L.; Hannun, A. Y.; Ng, A. Y., Rectifier nonlinearities improve neural network acoustic models, (Proceedings of the 30th International Conference on Machine Learning, (2013))
[35] Kullback, S.; Leibler, R. A., On information and sufficiency, Ann. Math. Stat., 22, 79-86, (1951) · Zbl 0042.38403
[36] Bengio, Y., Practical recommendations for gradient-based training of deep architectures, 437-478, (2012), Springer Berlin, Heidelberg
[37] Nowlan, S. J.; Hinton, G. E., Simplifying neural networks by soft weight-sharing, Neural Comput., 4, 473-493, (1992)
[38] TensorFlow, Large-scale machine learning on heterogeneous systems, (2015)
[39] Bartholomew-Biggs, M.; Brown, S.; Christianson, B.; Dixon, L., Automatic differentiation of algorithms, J. Comput. Appl. Math., 124, 171-190, (2000) · Zbl 0994.65020
[40] Kingma, D. P.; Ba, J., Adam: a method for stochastic optimization, (2014), CoRR
[41] Qiu, J.; Shu, C.-W., On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J. Comput. Phys., 183, 187-209, (2002) · Zbl 1018.65106
[42] LeVeque, R., Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, (2002), Cambridge University Press · Zbl 1010.65040
[43] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31, (1978) · Zbl 0387.76063
[44] Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math., 7, 159-193, (1954) · Zbl 0055.19404
[45] Linde, T.; Roe, P., Robust Euler codes, (1987), AIAA Paper-97-2098
[46] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83, 32-78, (1989) · Zbl 0674.65061
[47] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173, (1984) · Zbl 0573.76057
[48] Zhang, X.; Shu, C.-W., On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 3091-3120, (2010) · Zbl 1187.65096
[49] Zhang, X.; Shu, C.-W., On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 8918-8934, (2010) · Zbl 1282.76128
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.