Abstract
The aim of this paper is to provide a deep learning based method that can solve high-dimensional Fredholm integral equations. A deep residual neural network is constructed at a fixed number of collocation points selected randomly in the integration domain. The loss function of the deep residual neural network is defined as a linear least-square problem using the integral equation at the collocation points in the training set. The training iteration is done for the same set of parameters for different training sets. The numerical experiments show that the deep learning method is efficient with a moderate generalization error at all points. And the computational cost does not suffer from “curse of dimensionality” problem.





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References
Michalski, K.A., Mosig, J.R.: Multilayered media Green’s functions in integral equation formulations. IEEE T. Antenn. Propag. 45(3), 508–519 (2002)
Yang, P., Liou, K.N.: Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals. Appl. Opt. 35, 6568–6584 (1996)
De Bonis, M.C., Stanić, M.P., Tomović Mladenović, T.V.: Nystörm methods for approximating the solutions of an integral equation arising from a problem in mathematical biology. Appl. Numer. Math. 171, 193–211 (2022)
Baleanu, D., Zibaei, S., Namjoo, M., Jajarmi, A.: A nonstandard finite difference scheme for the modelling and nonidentical synchronization of a novel fractional chaotic system. Adv. Differ. Equ. 2021, 308 (2021)
Baleanu, D., Hassan Abadi, M., Jajarmi, A., Zarghami Vahid, K., Nieto, J.J.: A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects. Alex. Eng. J. 61(6), 4779–4791 (2022)
Erturk, V.S., Godwe, E., Baleanu, D., Kumar, P., Asad, J., Jajarmi, A.: Novel fractional-order Lagrangian to describe motion of beam on nanowire. Acta Phys. Pol. A. 140(3), 265–272 (2021)
Jajarmi, A., Baleanu, D., Zarghami Vahid, K., Mohammadi Pirouz, H., Asad, J.H.: A new and general fractional Lagrangian approach: a capacitor microphone case study. Results Phys. 31, 104950 (2021)
Atkinson, K.: Iterative variants of the Nyström method for the numerical solution of integral equations. Numer. Math. 22(1), 17–31 (1974)
Khorrami, N., Shamloo, A.S., Parsa Moghaddam, B.: Nyström method for solution of Fredholm integral equations of the second kind under interval data. J. Intell. Fuzzy Syst. 36(3), 2807–2816 (2019)
Han, G., Wang, R.: Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations. J. Comput. Appl. Math. 139(1), 49–63 (2002)
Babolian, E., Shahsavaran, A.: Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets. J. Comput. Appl. Math. 225(1), 87–95 (2009)
Asady, B., Hakimzadegan, F., Nazarlue, R.: Utilizing artificial neural network approach for solving two-dimensional integral equations. Math. Sci. 8, 117 (2014)
Effati, S., Buzhabadi, R.: A neural network approach for solving Fredholm integral equations of the second kind. Neural Comput. Appl. 21(5), 843–852 (2012)
Wang, K., Wang, Q.: Taylor collocation method and convergence analysis for the Volterra-Fredholm integral equations. J. Comput. Appl. Math. 260, 294–300 (2014)
Jin, C., Ding, J.: Solving Fredholm integral equations via a piecewise linear maximum entropy method. J. Comput. Appl. Math. 304, 130–137 (2016)
Saeed, T., Sabir, Z., Alhodaly, M.S., Alsulami, H.H., Guerrero Sánchez, Y.: An advanced heuristic approach for a nonlinear mathematical based medical smoking model. Results Phys. 32, 105137 (2022)
Sabir, Z., Wahab, H.A., Javeed, S., Baskonus, H.M.: An efficient stochastic numerical computing framework for the nonlinear higher order singular models. Fractal Fract. 5, 176 (2021)
Sabir, Z.: Stochastic numerical investigations for nonlinear three-species food chain system. Int. J. Biomath. 2250005, (2021)
Sabir, Z., Wahab, H.A.: Evolutionary heuristic with Gudermannian neural networks for the nonlinear singular models of third kind. Phys. Scr. 96, 125261 (2021)
Sabir, Z., Ali, M.R., Sadat, R.: Gudermannian neural networks using the optimization procedures of genetic algorithm and active set approach for the three-species food chain nonlinear model. J. Ambient Intell. Human. Comput. (2022). https://doi.org/10.1007/s12652-021-03638-3
Wang, B., Wang, Y., Gómez-Aguilar, J.F., Sabir, Z., Raja, M.A.Z., Jahanshahi, H., Alassafi, M.O., Alsaadi, F.E.: Gudermannian neural networks to investigate the Liénard differential model. Fractals (2021). https://doi.org/10.1142/S0218348X22500505
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. IEEE Conference on Computer Vision and Pattern Recognition, (2016), Las Vegas, USA
Sun, Y., Chen, Y., Wang, X., Tang, X.: Deep learning face representation by joint identification-verification. NIPS 27, (2014)
Graves, A., Mohamed, A. R., Hinton, G.: Speech recognition with deep recurrent neural networks. IEEE International Conference on Acoustics, Speech and Signal Processing, May 2013, Vancouver, Canada
Hinton, G., Deng, L., Yu, D., Dahl, G.E., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T.N., Kingsbury, B.: Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Proc. Mag. 29(6), 82–97 (2012)
Tom, Y., Devamanyu, H., Soujanya, P., Erik, C.: Recent trends in deep learning based natural language processing. IEEE Comput. Intell. Mag. 13(3), 55–75 (2018)
Beck, C.E.W., Jentzen, A.: Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. J. Nonlinear Sci. 29, 1563–1619 (2019)
Han, J., Jentzen, A.: Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5, 349–380 (2017)
Yu, B.: The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat. 6, 1–12 (2018)
Zhang, L., Han, J., Wang, H., Car, R.E.W.: Deep potential molecular dynamics: a scalable model with the accuracy of quantum mechanics. Phys. Rev. Lett. 120(14), 143001 (2018)
Kingma, D. P., Ba, J. L.: Adam: A Method for stochastic Optimization. The International Conference on Learning Representations (ICLR), May 2015, San Diego, USA
Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE T. Inform. Theory. 39(3), 930–945 (1993)
Barron, A. R., Klusowski, J. M.: Approximation and estimation for high-dimensional deep learning networks. arXiv:1809.03090v2, (2018)
Ma, C., Wu, L., Wojtowytsch, S., Wu, L.: Towards a mathematical understanding of neural network-based machine learning: what we know and what we don’t. arXiv:2009.10713v2, (2020)
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The research of Congming Jin was supported by the National Natural Science Foundation of China (No. 11571314).
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Guan, Y., Fang, T., Zhang, D. et al. Solving Fredholm Integral Equations Using Deep Learning. Int. J. Appl. Comput. Math 8, 87 (2022). https://doi.org/10.1007/s40819-022-01288-3
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DOI: https://doi.org/10.1007/s40819-022-01288-3