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Exploring the Efficiency of Neural Networks for Solving Dynamic Process Problems: The Fisher Equation Investigation

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Biologically Inspired Cognitive Architectures 2023 (BICA 2023)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 1130))

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Abstract

The numerical solution of problems based on ordinary and partial differential equations has been a subject of extensive research. While several methods, such as the finite difference, finite element, and finite volume methods, have been developed, each has its own strengths and limitations. This paper presents a different approach that utilizes feedforward neural networks to approximate functions, resulting in a differentiable analytical expression. Compared to other methods, this approach requires significantly fewer model parameters, leading to reduced computational requirements. The study examines the influence of a neural network’s loss function configuration on the accuracy and convergence rate of solving partial differential equations for functions of two variables: coordinate and time, using the example of the Fisher’s equation.

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References

  1. Samarskii, A.A., Volosevich, P.P., Volchinskaya, M.I., Kurdyumov, S.P.: A finite-difference method for the solution of one-dimensional non-stationary problems in magneto-hydrodynamics. USSR Comput. Math. Math. Phys. 8(5), 117–134 (1968)

    Article  Google Scholar 

  2. Bressloff, P.C., Coombes, S.: Dynamics of strongly coupled spiking neurons. Neural Comput. 12(1), 91–129 (2000)

    Article  Google Scholar 

  3. Greenspan, H.P.: Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. 51(4), 317–340 (1972)

    Article  Google Scholar 

  4. Samarskii, A.A., Nikolaev, E.S.: Methods for solving grid equations (1978)

    Google Scholar 

  5. Zienkiewicz, O.C., Morice, P.B.: The Finite Element Method in Engineering Science, vol. 1977. McGraw-Hill, London (1971)

    Google Scholar 

  6. Kovenya, V.M., Chirkov, D.V.: Methods of Finite Differences and Finite Volumes for Solving Problems of Mathematical Physics, pp. 7–8. Novosibirsk State University, Novosibirsk (2013)

    Google Scholar 

  7. Lagaris, I.E., Likas, A., Fotiadis, D.I.: Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Networks 9(5), 987–1000 (1998)

    Article  Google Scholar 

  8. Ladygin, S.A., Karachurin, R.N., Ryabov, P.N.: Numerical approach for studying problems for differential equations based on neural network method. In: IX International Conference “Laser, Plasma Research and Technologies” LaPlaz-2023: Proceedings, Conference Series LaPlaz, Publisher National Research Nuclear University MEPhI (Moscow), Abstracts, p. 147 (2023)

    Google Scholar 

  9. Ladygin, S.A., Karachurin, R.N., Ryabov, P.N., Kudryashov, N.A.: On the features of a numerical approach based on neural networks with direct communication for solving problems for differential equations. Phys. Atomic Nuclei (in press)

    Google Scholar 

  10. Wu, G., Wang, F., Qiu, L.: Physics-informed neural network for solving Hausdorff derivative Poisson equations. Fractals, 2340103 (2023)

    Google Scholar 

  11. Uddin, Z., Ganga, S., Asthana, R., Ibrahim, W.: Wavelets based physics informed neural networks to solve non-linear differential equations. Sci. Rep. 13(1), 2882 (2023)

    Article  Google Scholar 

  12. Fang, Y., Wu, G.Z., Kudryashov, N.A., Wang, Y.Y., Dai, C.Q.: Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method. Chaos Solitons Fractals 158, 112118 (2022)

    Article  MathSciNet  Google Scholar 

  13. Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7(4), 355–369 (1937)

    Article  Google Scholar 

  14. Kolmogorov, A.: Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application á un problème biologique. Moscow Univ. Bull. Math. 1, 1–25 (1937)

    Google Scholar 

  15. Ketkar, N., Moolayil, J., Ketkar, N., Moolayil, J.: Automatic differentiation in deep learning. In: Learn Best Practices of Deep Learning Models with PyTorch, Deep Learning with Python, pp. 133–145 (2021)

    Google Scholar 

  16. Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. National Research Nuclear University MEPhI, 31, Kashirskoe shosse, 115409, Moscow, Russia

    Raul Karachurin, Stanislav Ladygin, Pavel Ryabov, Kirill Shilnikov & Nikolay Kudryashov

  2. Department of Computational Physics, Moscow Institute of Physics and Technology (MIPT), 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation

    Kirill Shilnikov

Authors
  1. Raul Karachurin
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  2. Stanislav Ladygin
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  3. Pavel Ryabov
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  4. Kirill Shilnikov
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  5. Nikolay Kudryashov
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Corresponding author

Correspondence to Stanislav Ladygin .

Editor information

Editors and Affiliations

  1. Cybernetics Department, Moscow Engineering Physics Institute, Moscow, Russia

    Alexei V. Samsonovich

  2. College of Science and Technology, Ningbo University, Ningbo, China

    Tingting Liu

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Karachurin, R., Ladygin, S., Ryabov, P., Shilnikov, K., Kudryashov, N. (2024). Exploring the Efficiency of Neural Networks for Solving Dynamic Process Problems: The Fisher Equation Investigation. In: Samsonovich, A.V., Liu, T. (eds) Biologically Inspired Cognitive Architectures 2023. BICA 2023. Studies in Computational Intelligence, vol 1130. Springer, Cham. https://doi.org/10.1007/978-3-031-50381-8_53

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