IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v217y2024icp21-36.html
   My bibliography  Save this article

Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow

Author

Listed:
  • Bhaumik, Bivas
  • De, Soumen
  • Changdar, Satyasaran

Abstract

The present work introduces a deep learning approach to describe the perturbations of the pressure and radius in arterial blood flow. A mathematical model for the simulation of viscoelastic arterial flow is developed based on the assumption of long wavelength and large Reynolds number. Then, the reductive perturbation method is used to derive nonlinear evolutionary equations describing the resistance of the medium, the elastic properties, and the viscous properties of the wall. Using automatic differentiation, the solutions of nonlinear evolutionary equations at different time scales are represented using state-of-the-art physics-informed deep neural networks that are trained on a limited number of data points. The optimal neural network architecture for solving the nonlinear partial differential equations is found by employing Bayesian Hyperparameter Optimization. The proposed technique provides an alternate approach to avoid time-consuming numerical discretization methods such as finite difference or finite element for solving higher order nonlinear partial differential equations. Additionally, the capability of the trained model is demonstrated through graphs, and the solutions are also validated numerically. The graphical illustrations of pulse wave propagation can provide the correct interpretation of cardiovascular parameters, leading to an accurate diagnosis and successful treatment. Thus, the findings of this study could pave the way for the rapid development of emerging medical machine learning applications.

Suggested Citation

  • Bhaumik, Bivas & De, Soumen & Changdar, Satyasaran, 2024. "Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 21-36.
  • Handle: RePEc:eee:matcom:v:217:y:2024:i:c:p:21-36
    DOI: 10.1016/j.matcom.2023.10.011
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475423004469
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.matcom.2023.10.011?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Elhanafy, Ahmed & Guaily, Amr & Elsaid, Ahmed, 2019. "Numerical simulation of blood flow in abdominal aortic aneurysms: Effects of blood shear-thinning and viscoelastic properties," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 160(C), pages 55-71.
    2. Wu, Gang-Zhou & Fang, Yin & Kudryashov, Nikolay A. & Wang, Yue-Yue & Dai, Chao-Qing, 2022. "Prediction of optical solitons using an improved physics-informed neural network method with the conservation law constraint," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    3. Parand, K. & Aghaei, A.A. & Jani, M. & Ghodsi, A., 2021. "A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 114-128.
    4. Karthiga, R. & Narasimhan, K. & Amirtharajan, Rengarajan, 2022. "Diagnosis of breast cancer for modern mammography using artificial intelligence," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 316-330.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Nikolay A. Kudryashov, 2023. "Hamiltonians of the Generalized Nonlinear Schrödinger Equations," Mathematics, MDPI, vol. 11(10), pages 1-12, May.
    2. Gerasim Vladimirovich Krivovichev, 2021. "Comparison of Non-Newtonian Models of One-Dimensional Hemodynamics," Mathematics, MDPI, vol. 9(19), pages 1-16, October.
    3. Rahimkhani, P. & Ordokhani, Y., 2022. "Chelyshkov least squares support vector regression for nonlinear stochastic differential equations by variable fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    4. Ahadian, P. & Parand, K., 2022. "Support vector regression for the temperature-stimulated drug release," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    5. Krivovichev, Gerasim V., 2022. "Comparison of inviscid and viscid one-dimensional models of blood flow in arteries," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    6. Fang, Yin & Bo, Wen-Bo & Wang, Ru-Ru & Wang, Yue-Yue & Dai, Chao-Qing, 2022. "Predicting nonlinear dynamics of optical solitons in optical fiber via the SCPINN," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    7. Chen, Junchao & Song, Jin & Zhou, Zijian & Yan, Zhenya, 2023. "Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    8. Pakniyat, A. & Parand, K. & Jani, M., 2021. "Least squares support vector regression for differential equations on unbounded domains," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    9. Chen, Jing & Xiao, Min & Wu, Xiaoqun & Wang, Zhengxin & Cao, Jinde, 2022. "Spatiotemporal dynamics on a class of (n+1)-dimensional reaction–diffusion neural networks with discrete delays and a conical structure," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    10. Sun, Hongli & Lu, Yanfei, 2024. "A novel approach for solving linear Fredholm integro-differential equations via LS-SVM algorithm," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    11. Hajimohammadi, Zeinab & Parand, Kourosh, 2021. "Numerical learning approximation of time-fractional sub diffusion model on a semi-infinite domain," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:217:y:2024:i:c:p:21-36. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.