A machine learning approach for solving inverse Stefan problem

[1] J. Adler and O. Oktem, ¨ Solving ill-posed inverse problems using iterative deep neural networks, Inverse Probl. 33
(2017), no. 12, 124007.
[2] L. Bar and N. Sochen, Unsupervised deep learning algorithm for pde-based forward and inverse problems, 2019.
[3] J. Berg and K. Nystr¨om, Neural network augmented inverse problems for pdes, arXiv, 2018.
[4] J.P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.
[5] P. Dadvand, R. Lopez, and E. Onate, Artificial neural networks for the solution of inverse problems, Proc. Int.
Conf. Design Optim. Meth. Appl. ERCOFTAC, vol. 2006, 2006.
[6] A. Hajiollow, Y. Lotfi, K. Parand, A.H. Hadian, K. Rashedi, and J.A. Rad, Recovering a moving boundary from
Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi) linearization idea
combined with radial basis functions (RBFs) approximation, Engin. Comput. (2020), 1–15.
[7] J. Ji, C. Zhang, Y. Gui, Q. L¨u, and J. Kodikara, New observations on the application of LS-SVM in slope system
reliability analysis, J. Comput. Civil Engin. 31 (2017), no. 2, 06016002.
[8] M.T. Jin, K.H.and McCann, E. Froustey, and M. Unser, Deep convolutional neural network for inverse problems
in imaging, IEEE Trans. Image Process. 26 (2017), no. 9, 4509–4522.
[9] B.T. Johansson, D. Lesnic, and T. Reeve, A method of fundamental solutions for the one-dimensional inverse
Stefan problem, Appl. Math. Modell. 35 (2011), no. 9, 4367–4378.
[10] A. Karami, S. Abbasbandy, and E. Shivanian, Meshless Local Petrov–Galerkin Formulation of Inverse Stefan
Problem via Moving Least Squares Approximation, Math. Comput. Appl. 24 (2019), no. 4, 101.
[11] Y. Khoo and L. Ying, Switchnet: a neural network model for forward and inverse scattering problems, SIAM J.
Sci. Comput. 41 (2019), no. 5, A3182–A3201.
[12] A. Kirsch, An introduction to the mathematical theory of inverse problems, vol. 120, Springer Science & Business
Media, 2011.
[13] A. Kirsch and A. Rieder, Inverse problems for abstract evolution equations with applications in electrodynamics
and elasticity, Inverse Probl. 32 (2016), no. 8, 085001.
[14] L. Liu, W. Huang, and C. Wang, Hyperspectral image classification with kernel-based least-squares support vector
machines in sum space, IEEE J. Select. Topics Appl. Earth Observ. Remote Sens. 11 (2017), no. 4, 1144–1157.
[15] Y. Lotfi, K. Parand, K. Rashedi, and J. Amani Rad, Numerical study of temperature distribution in an inverse
moving boundary problem using a meshless method, Engin. Comput. (2019), 1–15.
[16] X. Lu, W. Zou, and M. Huang, A novel spatiotemporal LS-SVM method for complex distributed parameter systems
with applications to curing thermal process, IEEE Trans. Ind. Inf. 12 (2016), no. 3, 1156–1165.
[17] A. Lucas, M. Iliadis, R. Molina, and A.K. Katsaggelos, Using deep neural networks for inverse problems in
imaging: beyond analytical methods, IEEE Signal Process. Magazine 35 (2018), no. 1, 20–36.
[18] D. Lv, Q. Zhou, J.K. Choi, J. Li, and X. Zhang, Nonlocal TV-Gaussian prior for Bayesian inverse problems with
applications to limited CT reconstruction, Inverse Probl. Imaging 14 (2020), no. 1, 117.
[19] J. Matlak, D. S lota, and A. Zielonka, Reconstruction of the heat transfer coefficient in the inverse Stefan problem,
Hutnik, Wiadomo´sci Hutnicze 85 (2018), no. 1, 6–9.
[20] S. Mehrkanoon, T. Falck, and J.A. Suykens, Approximate solutions to ordinary differential equations using least
squares support vector machines, IEEE Trans. Neural Networks Learn. Syst. 23 (2012), no. 9, 1356–1367.
[21] S. Mehrkanoon and J.A.K. Suykens, Learning solutions to partial differential equations using LS-SVM, Neurocomput. 159 (2015), 105–116.[22] S. Ozer, C.H. Chen, and H.A. Cirpan, A set of new chebyshev kernel functions for support vector machine pattern
classification, Pattern Recogn. 44 (2011), no. 7, 1435–1447.
[23] L.C. Padierna, M. Carpio, A. Rojas-Dom´ınguez, H. Puga, and H. Fraire, A novel formulation of orthogonal
polynomial kernel functions for SVM classifiers: the Gegenbauer family, Pattern Recog. 84 (2018), 211–225.
[24] M. Parhizi and A. Jain, Solution of the Phase Change Stefan Problem With Time-Dependent Heat Flux Using
Perturbation Method, J. Heat Transfer 141 (2019), no. 2, 024503.
[25] L. Pilozzi, F.A. Farrelly, G. Marcucci, and C. Conti, Machine learning inverse problem for topological photonics,
Commun. Phys. 1 (2018), no. 1, 1–7.
[26] J.A. Rad, K. Rashedi, K. Parand, and H. Adibi, The meshfree strong form methods for solving one dimensional
inverse Cauchy-Stefan problem, Engin. Comput. 33 (2017), no. 3, 547–571.
[27] G.M.M. Reddy, M. Vynnycky, and J.A. Cuminato, An efficient adaptive boundary algorithm to reconstruct Neumann boundary data in the MFS for the inverse Stefan problem, J. Comput. Appl. Math. 349 (2019), 21–40.
[28] S. Sarabadan, K. Rashedi, and H. Adibi, Boundary determination of the inverse heat conduction problem in one
and two dimensions via the collocation method based on the satisfier functions, Iran. J. Sci. Technol. Trans. A:
Sci. 42 (2018), no. 2, 827–840.
[29] M.M. Sarsengeldin, S.N.N. Kharin, S. Kassabek, and Z. Mukambetkazin, Exact Solution of the One Phase Inverse
Stefan Problem, Filomat 32 (2018), no. 3, 985–990.
[30] J. Shen, T. Tang, and L.-L. Wang, Spectral methods: algorithms, analysis and applications, vol. 41, Springer
Science & Business Media, 2011.
[31] C.B. Smith and E.M. Hernandez, Non-negative constrained inverse eigenvalue problems–Application to damage
identification, Mech. Syst. Signal Process. 129 (2019), 629–644.
[32] J.A. Suykens, T. Van Gestel, J. De Brabanter, B. De Moor, and J.P. Vandewalle, Least squares support vector
machines, World Scientific, 2002.
[33] J.A.K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural Process. Lett. 9
(1999), no. 3, 293–300.
[34] Y.-H. Wu and H. Shen, Grey-related least squares support vector machine optimization model and its application
in predicting natural gas consumption demand, J. Comput. Appl. Math. 338 (2018), 212–220.
[35] N. Ye, R. Sun, Y. Liu, and L. Cao, Support vector machine with orthogonal chebyshev kernel, 18th Int. Conf.
Pattern Recog. (ICPR’06), vol. 2, IEEE, 2006, pp. 752–755.
[36] N. Zabaras and K. Yuan, Dynamic programming approach to the inverse Stefan design problem, Numerical Heat
Transfer 26 (1994), no. 1, 97–104.