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Convolutional Neural Networks in Phase Space and Inverse Problems

https://doi.org/10.1137/19M1294484

Abstract

We study inverse problems consisting of determining medium properties using the responses to probing waves from the machine learning point of view. Based on the analysis of propagation of waves and their nonlinear interactions, we construct a deep convolutional neural network to reconstruct the coefficients of nonlinear wave equations that model the medium properties. Furthermore, for given approximation accuracy, we obtain the depth and number of units of the network and their quantitative dependence on the complexity of the medium.

  • 1.  M. Beals, Propagation and Interaction of Singularities in Nonlinear Hyperbolic Problems, Progr. Math. 3, Birkhäuser Boston, Boston, MA, 1989. Google Scholar

  • 2.  J.-M. Bony , Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires , Ann. Sci. École Norm. Sup. (4) , 14 ( 1981 ), pp. 209 -- 246 . CrossrefGoogle Scholar

  • 3.  T. A. Bubba, M. Galinier, M. Lassas, M. Prato, L. Ratti, and S. Siltanen, Deep Neural Networks for Inverse Problems with Pseudodifferential Operators: An Application to Limited-Angle Tomography, preprint, https://arxiv.org/abs/2006.01620, 2020. Google Scholar

  • 4.  R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57, Société Mathématique de France, Paris, France, 1978. Google Scholar

  • 5.  C. Dafermos and  W. Hrusa , Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics , Arch. Ration. Mech. Anal. , 87 ( 1985 ), pp. 267 -- 292 . CrossrefISIGoogle Scholar

  • 6.  M. de Hoop,  H. Smith and  G. Uhlmann . van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective , Inverse Problems , 25 ( 2009 ), 025005 . CrossrefISIGoogle Scholar

  • 7.  M. de Hoop,  G. Uhlmann and  A. Vasy , Diffraction from conormal singularities , Ann. Sci. Éc. Norm. Supér. (4) , 48 ( 2015 ), pp. 351 -- 408 . CrossrefISIGoogle Scholar

  • 8.  M. de Hoop,  G. Uhlmann and  Y. Wang , Nonlinear interaction of waves in elastodynamics and an inverse problem , Math. Ann. , 376 ( 2020 ), pp. 765 -- 795 . CrossrefISIGoogle Scholar

  • 9.  J. J. Duistermaat, Fourier Integral Operators, Progr. Math. 130, Birkhäuser Boston, Boston, MA, 1996. Google Scholar

  • 10.  I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning, MIT Press, Cambridge, MA, 2016. Google Scholar

  • 11.  K. Hornik,  M. Stinchcombe and  H. White , Multilayer feedforward networks are universal approximators , Neural Networks , 2 ( 1989 ), pp. 359 -- 366 . CrossrefISIGoogle Scholar

  • 12.  K. Hornik,  M. Stinchcombe and  H. White , Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks , Neural Networks , 3 ( 1990 ), pp. 551 -- 560 . CrossrefISIGoogle Scholar

  • 13.  L. Hörmander, The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Classics Math., Springer-Verlag, Berlin, 2009. Google Scholar

  • 14.  A. Greenleaf and  G. Uhlmann , Estimates for singular Radon transforms and pseudodifferential operators with singular symbols , J. Funct. Anal. , 89 ( 1990 ), pp. 202 -- 232 . CrossrefISIGoogle Scholar

  • 15.  A. Kachalov, M. Lassas, and Y. Kurylev, Inverse Boundary Spectral Problems, Chapman and Hall/CRC, Boca Raton, FL, 2001. Google Scholar

  • 16.  K. Kothari, I. Dokmanić, and M. de Hoop, Learning the Geometry of Wave-Based Imaging, preprint, https://arxiv.org/abs/2006.05854v1, 2020. Google Scholar

  • 17.  Y. Kurylev,  M. Lassas and  G. Uhlmann , Inverse problems for Lorentzian manifolds and nonlinear hyperbolic equations , Invent. Math. , 212 ( 2018 ), pp. 781 -- 857 . CrossrefISIGoogle Scholar

  • 18.  Y. Kurylev, M. Lassas, and G. Uhlmann, Inverse Problems in Spacetime I: Inverse Problems for Einstein Equations, extended preprint, https://arxiv.org/abs/1405.4503, 2014. Google Scholar

  • 19.  B. N. Kuvshinov,  T. J. H. Smit and  X. H. Campman , Non-linear interaction of elastic waves in rocks , Geophys. J. Internat. , 194 ( 2013 ), pp. 1920 -- 1940 . CrossrefISIGoogle Scholar

  • 20.  M. Lassas, T. Liimatainen, L. Potenciano-Machado, and T. Tyni, Uniqueness and Stability of an Inverse Problem for a Semi-Linear Wave Equation, preprint, https://arxiv.org/abs/2006.13193, 2020. Google Scholar

  • 21.  M. Lassas,  G. Uhlmann and  Y. Wang , Inverse problems for semilinear wave equations on Lorentzian manifolds , Comm. Math. Phys. , 360 ( 2018 ), pp. 555 -- 609 . CrossrefISIGoogle Scholar

  • 22.  M. Lassas, G. Uhlmann, and Y. Wang, Determination of Vacuum Space-Times from the Einstein-Maxwell Equations, preprint, https://arxiv.org/abs/1703.10704, 2017. Google Scholar

  • 23.  Y. LeCun,  Y. Bengio and  G. Hinton , Deep learning , Nature , 521 ( 2015 ), 436 . CrossrefISIGoogle Scholar

  • 24.  M. Leshno,  V. Y. Lin,  A. Pinkus and  S. Schocken , Multilayer feedforward networks with a non-polynomial activation function can approximate any function , Neural Networks , 6 ( 1993 ), pp. 861 -- 867 . CrossrefISIGoogle Scholar

  • 25.  Z. Long, Y. Lu, and B. Dong, PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network, J. Comput. Phys., 399 (2019), 108925. Google Scholar

  • 26.  L. Lu, P. Jin, and G. E. Karniadakis, DeepONet: Learning Nonlinear Operators for Identifying Differential Equations Based on the Universal Approximation Theorem of Operators, preprint, https://arxiv.org/abs/1910.03193, 2019. Google Scholar

  • 27.  S. Mallat , Group invariant scattering , Comm. Pure Appl. Math. , 65 ( 2012 ), pp. 1331 -- 1398 . CrossrefISIGoogle Scholar

  • 28.  S. Mallat , Understanding deep convolutional networks , Phil. Trans. R. Soc. A , 374 ( 2016 ), 20150203 . CrossrefISIGoogle Scholar

  • 29.  R. Melrose and  N. Ritter , Interaction of nonlinear progressing waves for semilinear wave equations , Ann. of Math. (2) , 121 ( 1985 ), pp. 187 -- 213 . CrossrefISIGoogle Scholar

  • 30.  R. Melrose and  G. Uhlmann , Lagrangian intersection and the Cauchy problem , Comm. Pure Appl. Math. , 32 ( 1979 ), pp. 483 -- 519 . CrossrefISIGoogle Scholar

  • 31.  G. Nakamura and  M. Watanabe , An inverse boundary value problem for a nonlinear wave equation , Inverse Probl. Imaging , 2 ( 2008 ), pp. 121 -- 131 . CrossrefISIGoogle Scholar

  • 32.  G. Nakamura and M. Vashisth, Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations, preprint, https://arxiv.org/abs/1712.09945, 2017. Google Scholar

  • 33.  M. Raissi,  P. Perdikaris and  G. E. Karniadakis , Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , J. Comput. Phys. , 378 ( 2019 ), pp. 686 -- 707 . CrossrefISIGoogle Scholar

  • 34.  J. Rauch and  M. Reed , Singularities produced by the nonlinear interaction of three progressing waves; examples , Comm. Partial Differential Equations , 7 ( 1982 ), pp. 1117 -- 1133 . CrossrefISIGoogle Scholar

  • 35.  L. Ruthotto and  E. Haber , Deep neural networks motivated by partial differential equations , J. Math. Imaging Vision , 62 ( 2020 ), pp. 352 -- 364 . CrossrefISIGoogle Scholar

  • 36.  M. E. Taylor, Pseudodifferential Operators and Nonlinear Partial Differential Equations, Birkhäuser Boston, Boston, MA, 1991. Google Scholar

  • 37.  M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, 2nd ed., Appl. Math. Sci. 116, Springer, New York, 2011. Google Scholar

  • 38.  G. Uhlmann and  Y. Wang , Determination of space-time structures from gravitational perturbations , Comm. Pure Appl. Math. , 73 ( 2020 ), pp. 1315 -- 1367 . CrossrefISIGoogle Scholar

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