Abstract
This paper focuses on how to approximate traveling wave solutions for various kinds of partial differential equations via artificial neural networks. A traveling wave solution is hard to obtain with traditional numerical methods when the corresponding wave speed is unknown in advance. We propose a novel method to approximate both the traveling wave solution and the unknown wave speed via a neural network and an additional free parameter. We proved that under a mild assumption, the neural network solution converges to the analytic solution and the free parameter accurately approximates the wave speed as the corresponding loss tends to zero for the Keller–Segel equation. We also demonstrate in the experiments that reducing loss through training assures an accurate approximation of the traveling wave solution and the wave speed for the Keller–Segel equation, the Allen–Cahn model with relaxation, and the Lotka–Volterra competition model.
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References
Ali, A., Shah, K., Khan, R.A.: Numerical treatment for traveling wave solutions of fractional whitham-broer-kaup equations. Alex. Eng. J. 57(3), 1991–1998 (2018)
Bramburger, J.J.: Exact minimum speed of traveling waves in a keller-segel model. Appl. Math. Lett. 111, 106594 (2020)
Chen, X., Liu, G., Qi, Y.: The existence of minimum speed of traveling wave solutions to a non-kpp isothermal diffusion system. J. Differ. Equ. 263(3), 1695–1707 (2017)
Guo, J.-S., Lin, Y.-C.: The sign of the wave speed for the lotka-volterra competition-diffusion system. Commun. Pure Appl. Anal. 12(5), 2083 (2013)
Hagan, P.S.: Traveling wave and multiple traveling wave solutions of parabolic equations. SIAM J. Math. Anal. 13(5), 717–738 (1982)
Hagstrom, T., Keller, H.B.: The numerical calculation of traveling wave solutions of nonlinear parabolic equations. SIAM J. Sci. Stat. Comput. 7(3), 978–988 (1986)
He, J.-H., Xu-Hong, W.: Exp-function method for nonlinear wave equations. Chaos, Solitons Fractals 30(3), 700–708 (2006)
Hirsch, M.W., Smale, S., Devaney, R.L.: Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, Cambridge (2012)
Hwang, H.J., Jang, J.W., Jo, H., Lee, J.Y.: Trend to equilibrium for the kinetic fokker-planck equation via the neural network approach, J. Comput. Phys. 419, 109665 (2020)
Jo, H., Son, H., Hwang, H.J., Kim, E.H.: Deep neural network approach to forward-inverse problems. Netw. Heterogen. Media 15(2), 247–259 (2020)
Kan-On, Y.: Parameter dependence of propagation speed of travelling waves for competition-diffusion equations. SIAM J. Math. Anal. 26(2), 340–363 (1995)
Kaya, D., Inan, I.E.: Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation. Appl. Math. Comput. 151(3), 775–787 (2004)
Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30(2), 235–248 (1971)
Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980 (2014)
Larson, D.A.: Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of fisher type. SIAM J. Appl. Math. 34(1), 93–104 (1978)
Lattanzio, C., Mascia, C., Plaza, R.G., Simeoni, C.: Analytical and numerical investigation of traveling waves for the allen-cahn model with relaxation. Math. Models Methods Appl. Sci. 26(05), 931–985 (2016)
Li, T., Wang, Z.-A.: Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J. Differ. Equ. 250(3), 1310–1333 (2011)
Li, T., Wang, Z.-A.: Steadily propagating waves of a chemotaxis model. Math. Biosci. 240(2), 161–168 (2012)
Li, X.: Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Neurocomputing 12(4), 327–343 (1996)
Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: Deepxde: A deep learning library for solving differential equations. SIAM Rev. 63(1), 208–228 (2021)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)
Mansour, M.B.A.: Traveling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Appl. Math. Model. 32(2), 240–247 (2008)
Méndez, V., Fort, J., Farjas, J.: Speed of wave-front solutions to hyperbolic reaction-diffusion equations. Phys. Rev. E 60(5), 5231 (1999)
Murray, J.D.: Mathematical Biology: I. An Introduction, vol. 17. Springer, Berlin (2007)
Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., Lerer, A.: Automatic differentiation in pytorch, paper was published in 2017 by NeurIPS, Autodiff Workshop
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high-performance deep learning library, arXiv preprint arXiv:1912.01703 (2019)
Qin, C.-Y., Tian, S.-F., Wang, X.-B., Zhang, T.-T., Li, J.: Rogue waves, bright-dark solitons and traveling wave solutions of the (3+ 1)-dimensional generalized kadomtsev-petviashvili equation. Comput. Math. Appl. 75(12), 4221–4231 (2018)
Rosu, H.C., Cornejo-Pérez, O.: Supersymmetric pairing of kinks for polynomial nonlinearities. Phys. Rev. E 71(4), 046607 (2005)
Salako, R., Shen, W.: Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on r\(^{\wedge }\) n, arXiv preprint arXiv:1609.05387 (2016)
Sirignano, J., Spiliopoulos, K.: Dgm: a deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375, 1339–1364 (2018)
Tariq, H., Akram, G.: New traveling wave exact and approximate solutions for the nonlinear cahn-allen equation: evolution of a nonconserved quantity. Nonlinear Dyn. 88(1), 581–594 (2017)
Wang, Q., Chen, Y., Zhang, H.: A new riccati equation rational expansion method and its application to (2+ 1)-dimensional burgers equation. Chaos, Solitons Fractals 25(5), 1019–1028 (2005)
Wang, Z.-C., Li, W.-T., Ruan, S.: Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J. Differ. Equ. 238(1), 153–200 (2007)
Wazwaz, A.-M.: The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. Comput. 188(2), 1467–1475 (2007)
Shi-Liang, W., Chen, G.: Uniqueness and exponential stability of traveling wave fronts for a multi-type sis nonlocal epidemic model. Nonlinear Anal. Real World Appl. 36, 267–277 (2017)
Xin, X.: Existence and stability of traveling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Diff. Equat. 3(4), 541–573 (1991)
Yang, C., Rodriguez, N.: A numerical perspective on traveling wave solutions in a system for rioting activity. Appl. Math. Comput. 364, 124646 (2020)
Acknowledgements
This work was supported by National Research Foundation of Korea (NRF) Grants funded by the Korean government (MSIT) (Nos. NRF-2017R1E1A1A03070105 and NRF-2019R1A5A1028324)
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Cho, S.W., Hwang, H.J. & Son, H. Traveling Wave Solutions of Partial Differential Equations Via Neural Networks. J Sci Comput 89, 21 (2021). https://doi.org/10.1007/s10915-021-01621-w
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DOI: https://doi.org/10.1007/s10915-021-01621-w