Deep Learning-Based Numerical Solutions for Nonlinear Partial Differential Equations in Fluid Dynamics
DOI:
https://doi.org/10.64137/3108-2637/IJMAR-V2I1P104Keywords:
Deep Learning, Fluid Dynamics, Nonlinear Partial Differential Equations, Physics-Informed Neural Networks (PINNs), Numerical Simulation, Navier Stokes Equations, Scientific Computing, Fourier Neural Operators, Deep Operator Networks, Computational Fluid Dynamics (CFD)Abstract
Nonlinear partial differential equations (PDEs) play a vital role in modeling complex fluid dynamics phenomena such as turbulence, heat transfer, and flow instability. Traditional numerical approaches, including finite difference and finite element methods, often require extensive computational resources and may struggle with large-scale or real-time simulations. Recent advancements in deep learning have introduced efficient alternatives for solving nonlinear PDEs through data-driven and physics-informed approaches. This paper presents a comprehensive study of deep learning-based numerical solutions for nonlinear PDEs in fluid dynamics. The proposed framework integrates neural network architectures with physics informed constraints to accurately approximate fluid behavior while reducing computational complexity. Various deep learning models, including Physics-Informed Neural Networks (PINNs), Deep Operator Networks, and Fourier Neural Operators, are analyzed and compared with conventional numerical techniques. Experimental results demonstrate improved prediction accuracy, faster convergence, and enhanced scalability in solving complex fluid flow problems. The study further discusses practical applications, current limitations, and future research opportunities in AI-driven scientific computing for fluid dynamics.
References
[1] “Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-Informed Neural Networks A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics, 378, 686-707. - References - Scientific Research Publishing,” Scirp.org, 2019. https://www.scirp.org/reference/referencespapers?referenceid=4219158 (accessed May 16, 2026).
[2] “Brunton, S.L., Noack, B.R. and Koumoutsakos, P. (2020) Machine Learning for Fluid Mechanics. Annual Review of Fluid Mechanics, 52, 477-508. - References - Scientific Research Publishing,” Scirp.org, 2020. https://www.scirp.org/reference/referencespapers?referenceid=2886121 (accessed May 16, 2026).
[3] J. Sirignano and K. Spiliopoulos, “DGM: A deep learning algorithm for solving partial differential equations,” Journal of Computational Physics, vol. 375, pp. 1339–1364, Dec. 2018, doi: https://doi.org/10.1016/j.jcp.2018.08.029.
[4] Z. Li et al., “Fourier Neural Operator for Parametric Partial Differential Equations,” arXiv:2010.08895 [cs, math], Oct. 2020, Available: https://arxiv.org/abs/2010.08895
[5] L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis, “Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators,” Nature Machine Intelligence, vol. 3, no. 3, pp. 218–229, Mar. 2021, doi: https://doi.org/10.1038/s42256-021-00302-5.
[6] J. Han, A. Jentzen, and W. E, “Solving high-dimensional partial differential equations using deep learning,” Proceedings of the National Academy of Sciences, vol. 115, no. 34, pp. 8505–8510, Aug. 2018, doi: https://doi.org/10.1073/pnas.1718942115.
[7] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,” Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, Jun. 2021, doi: https://doi.org/10.1038/s42254-021-00314-5.
[8] “Goodfellow, I., et al. (2016) Deep Learning. MIT Press, Cambridge, MA. - References - Scientific Research Publishing,” www.scirp.org. https://www.scirp.org/reference/referencespapers?referenceid=2791883
[9] Y. LeCun, Y. Bengio, and G. Hinton, “Deep Learning,” Nature, vol. 521, no. 7553, pp. 436–444, May 2015, doi: https://doi.org/10.1038/nature14539.
[10] D. Muthuramakrishnan, K. Bhagyasri, “One Modulo N Mean Labeling of Graphs,” International Journal of Scientific Research and Review, vol. 7, no. 8, pp. 345-354, 2018.
[11] H. Aref, “Computational Techniques for Fluid Dynamics, Vols. I, II, Second Edn. By C. A. J. FLETCHER. Springer, 1991. Vol. I, 401 pp., DM65; Vol. II, 493 pp., DM75.,” Journal of Fluid Mechanics, vol. 258, pp. 371–371, Jan. 1994, doi: https://doi.org/10.1017/s002211209421337x.
[12] K. Bhagyasri, D. Muthuramakrishnan, “On One Modulo N Mean Graphs,” International Journal of Scientific Research and Review, vol. 7, no. 8, pp. 355-363, 2018.
[13] D. Muthuramakrishnan, K. Bhagyasri, “Properties of one modulo N mean labeling of Graphs,” International Journal of Research in Advent Technology, vol. 6, no. 8, pp. 1883-1887, 2018.
[14] J. H. Ferziger, M. Perić, and R. L. Street, Computational Methods for Fluid Dynamics. Cham: Springer International Publishing, 2020. doi: https://doi.org/10.1007/978-3-319-99693-6.
[15] Versteeg, An Introduction to Computational Fluid Dynamics The Finite Volume Method, 2/e. Pearson Education India, 2007.
[16] “Chorin, A.J. and Marsden, J. E. (1993) A Mathematical Introduction to Fluid Mechanics. Springer, New York. - References - Scientific Research Publishing,” Scirp.org, 2016. https://www.scirp.org/reference/referencespapers?referenceid=1858967 (accessed May 16, 2026).
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