A space-time least-squares support vector regression scheme for inverse source problem of the time-fractional wave equation

Document Type : Research Article

Authors

• A. Mohammadi
• A. Tari Marzabad

Department of Mathematics, Shahed University, Tehran, Iran.

10.22067/ijnao.2024.86883.1392

Abstract

The inverse problems in various fields of applied sciences and industrial design are concerned with the estimation of parameters that cannot be directly measured. In this work, we present a novel numerical approach for addressing the fractional inverse source problem by a machine learning algorithm and considering the ideas behind the spectral methods. The introduced algorithm utilizes a space-time Galerkin type of least-squares support vector regression to approximate the unknown source in a finitedimensional space, providing a stable and efficient solution. With the proposed machine learning method, we overcome the limitations of classical numerical methods and offer a promising alternative for tackling inverse source problems while avoiding overfitting by carefully selecting regularization parameters. To validate the effectiveness of our approach and illustrate an exponential convergence, we present some test problems along with the corresponding numerical results. The proposed method’s superior accuracy compared to the existing methods is also illustrated.  

Keywords

• Machine learning
• Support vector machines
• Inverse Source problem
• Time fractional wave equation
• Space-time Galerkin

Main Subjects

• Numerical Analysis

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References

[1] Adler, A., Araya-Polo, M., and Poggio T., Deep learning for seismic inverse problems: Toward the acceleration of geophysical analysis workflows, IEEE Signal Process. Mag. 38(2) (2021), 89–119.
[2] Arridge, S.R. and Schotland, J.C., Optical tomography: forward and inverse problems, Inverse Probl. 25(12) (2009), 123010.
[3] Ashurov, R.R. and Faiziev, Y.É., Inverse problem for finding the order of the fractional derivative in the wave equation, Math.l Notes, 110(5-6) (2021), 842–852.
[4] Bourquin, F. and Nassiopoulos, A. Inverse reconstruction of initial and boundary conditions of a heat transfer problem with accurate final state, Int. J. Heat Mass Transf. 54(15-16) (2011), 3749–3760.
[5] Campbell, C. and Ying, Y. Learning with support vector machines, Springer Nature, 2022.
[6] Cui, M., Mei, J., Zhang, B-W, Xu, B.-B., Zhou, L., and Zhang, Y., Inverse identification of boundary conditions in a scramjet combustor with a regenerative cooling system, Appl. Therm. Eng. 134 (2018), 555–563.
[7] Del Aguila Pla, P. and Jaldén, J. Cell detection by functional inverse diffusion and non-negative group sparsity—part ii: Proximal optimiza tion and performance evaluation, IEEE Trans. Signal Process. 66(20) (2018), 5422–5437.
[8] Diethelm, K., The analysis of fractional dierential equations: An application-oriented exposition using dierential operators of Caputo type, Springer, 2010.
[9] Ernst, F. and Schweikard, A. Fundamentals of Machine Learning: Support Vector Machines Made Easy, Utb GmbH, 2020.
[10] Guo, B. Spectral methods and their applications, World Scientific, 1998. [11] Hu, W., Gu, Y., and Fan, C.M., A meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materials, Eng. Anal. Bound. Elem. 114 (2020), 1–7.
[12] Jiang, D., Liu, Y., and Yamamoto, M., Inverse source problem for a wave equation with final observation data, Mathematical Analysis of Continuum Mechanics and Industrial Applications, 26 (2017), 153–164.
[13] Kinash, N. and Janno, J., An inverse problem for a generalized fractional derivative with an application in reconstruction of time-and spacedependent sources in fractional diffusion and wave equations, Mathematics, 7(12) (2019), 1138.
[14] Li, X. and Xu, C., A space-time spectral method for the time-fractional diffusion equation, SIAM J. Numer. Anal. 47(3) (2009), 2108–2131.
[15] Liu, Y. and Xie, M., Rebooting data-driven soft-sensors in process industries: A review of kernel methods, J. Process Control, 89 (2020), 58–73.
[16] Lloyd, S., Schaal, C. and Jeong, C., Inverse modeling and experimental validation for reconstructing wave sources on a 2D solid from surficial measurement, Ultrasonics, 128 (2023), 106880.
[17] Mainardi, F., Fractional calculus in wave propagation problems, In Forum der Berliner Mathematischer Gesellschaft, vol. 19, pp. 20-52. 2011.

[18] Mehrkanoon, S. and Suykens, J.A., Learning solutions to partial differential equations using LS-SVM, Neurocomputing, 159 (2015), 105–116.
[19] Ohe, T., Real-time reconstruction of moving point/dipole wave sources from boundary measurements, Inverse Probl. Sci. Eng. 28(8) (2020), 1057–1102.
[20] Ongie, G., Jalal, A., Metzler, C.A., Baraniuk, R.G., Dimakis, A.G., and Willett, R., Deep learning techniques for inverse problems in imaging, IEEE J. Sel. Areas Inf. Theory, 1(1) (2020), 39–56.
[21] Pakniyat, A., Parand, K., and Jani, M., Least squares support vector regression for differential equations on unbounded domains, Chaos Solit. Fractals. 151 (2021), 111232.
[22] Parand, K., Aghaei, A.A., Jani, M. and Sahleh, R., Solving integral equations by ls-svr. In Learning with Fractional Orthogonal Kernel Classifiers in Support Vector Machines: Theory, Algorithms and Applications, Singapore: Springer Nature Singapore, 2023.
[23] Parand, K., Aghaei, A.A., Kiani, S., Zadeh, T.I. and Khosravi, Z., A neural network approach for solving nonlinear differential equations of Lane–Emden type, Eng. Comput. (2023), 1–7.
[24] Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.
[25] Qiu, L., Hu, C. and Qin, Q.H., A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation, Appl. Math. Lett.109 (2020), 106554.
[26] Reddy, B.D., Introductory functional analysis: with applications to boundary value problems and finite elements, Springer Science and Business Media, 2013.
[27] Ren, K. and Triki, F., A Global stability estimate for the photo-acoustic inverse problem in layered media, Eur. J. Appl. Math. 30(3) (2019), 505–528.

[28] Sakamoto, K. and Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382(1) (2011), 426–447.
[29] Seliga, L. and Slodicka, M., An inverse source problem for a damped wave equation with memory, J. Inverse Ill-Posed Probl. 24(2) (2016), 111–122.
[30] Shen, J. and Sheng, C.-T., An efficient space–time method for timefractional diffusion equation, J. Sci. Comput. 81 (2019), 1088–1110.
[31] Shen J., Tang T., and Wang L.L., Spectral methods: algorithms, analysis and applications, Springer, 2011.
[32] Smith, R.C. and Demetriou, M.A., Research directions in distributed parameter systems, SIAM, 2003
[33] Suykens, J.A.K, Gestel, T.V., De Brabanter, J., De Moor, B., and Vandewalle, J., Least squares support vector machines, Singapore: World Scientific Publishing Company, 2002.
[34] Suykens, J.A.K. and Vandewalle, J., Least squares support vector machine classifiers, Neural Process. Lett. 9(3) (1999), 293–300.
[35] Taheri, T., Aghaei, A.A., and Parand, K., Bridging machine learning and weighted residual methods for delay differential equations of fractional order, Appl. Soft Comput. (2023) 110936.
[36] Yang, J.P. and Hsin, W.-C., Weighted reproducing kernel collocation method based on error analysis for solving inverse elasticity problems, Acta Mech. 230 (2019), 3477–3497.
[37] Zhu, X., and Law, S.-S., Recent developments in inverse problems of vehicle–bridge interaction dynamics, J. Civ. Struct. Health Monit. 6 (2016), 107–128.