IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v488y2025ics0096300324005782.html
   My bibliography  Save this article

Deep-time neural networks: An efficient approach for solving high-dimensional PDEs

Author

Listed:
  • Aghapour, Ahmad
  • Arian, Hamid
  • Seco, Luis

Abstract

This paper presents the Deep-Time Neural Network (DTNN), an efficient and novel deep-learning approach for solving partial differential equations (PDEs). DTNN leverages the power of deep neural networks to approximate the solution for a class of quasi-linear parabolic PDEs. We demonstrate that DTNN significantly reduces the computational cost and speeds up the training process compared to other models in the literature. The results of our study indicate that DTNN architecture is promising for the fast and accurate solution of time-dependent PDEs in various scientific and engineering applications. The DTNN architecture addresses the pressing need for enhanced time considerations in the deeper layers of Artificial Neural Networks (ANNs), thereby improving convergence time for high-dimensional PDE solutions. This is achieved by integrating time into the hidden layers of the DTNN, demonstrating a marked improvement over existing ANN-based solutions regarding efficiency and speed.

Suggested Citation

  • Aghapour, Ahmad & Arian, Hamid & Seco, Luis, 2025. "Deep-time neural networks: An efficient approach for solving high-dimensional PDEs," Applied Mathematics and Computation, Elsevier, vol. 488(C).
    • Handle: RePEc:eee:apmaco:v:488:y:2025:i:c:s0096300324005782
    DOI: 10.1016/j.amc.2024.129117
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300324005782
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2024.129117?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. Boris Hanin, 2019. "Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations," Mathematics, MDPI, vol. 7(10), pages 1-9, October.
    2. Yajie Yu & Narayan Ganesan & Bernhard Hientzsch, 2023. "Backward Deep BSDE Methods and Applications to Nonlinear Problems," Risks, MDPI, vol. 11(3), pages 1-16, March.
    3. Stéphane Crépey & Rémi Gerboud & Zorana Grbac & Nathalie Ngor, 2013. "Counterparty Risk And Funding: The Four Wings Of The Tva," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(02), pages 1-31.
    4. Maximilien Germain & Mathieu Laurière & Huyên Pham & Xavier Warin, 2022. "DeepSets and their derivative networks for solving symmetric PDEs ," Post-Print hal-03154116, HAL.
    5. Christian Bender & Nikolaus Schweizer & Jia Zhuo, 2017. "A Primal–Dual Algorithm For Bsdes," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 866-901, July.
    6. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    8. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2023. "Deep stochastic optimization in finance," Digital Finance, Springer, vol. 5(1), pages 91-111, March.
    9. Maximilien Germain & Mathieu Lauri`ere & Huy^en Pham & Xavier Warin, 2021. "DeepSets and their derivative networks for solving symmetric PDEs," Papers 2103.00838, arXiv.org, revised Jan 2022.
    10. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    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. Christian Bender & Christian Gärtner & Nikolaus Schweizer, 2018. "Pathwise Dynamic Programming," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 965-965, August.
    2. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    3. Yuecai Han & Chunyang Liu, 2018. "Asian Option Pricing under Uncertain Volatility Model," Papers 1808.00656, arXiv.org.
    4. repec:dau:papers:123456789/5374 is not listed on IDEAS
    5. Joel Vanden, 2006. "Exact Superreplication Strategies for a Class of Derivative Assets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(1), pages 61-87.
    6. Antoine Jacquier & Zan Zuric, 2023. "Random neural networks for rough volatility," Papers 2305.01035, arXiv.org.
    7. Bizid, Abdelhamid & Jouini, Elyès, 2005. "Equilibrium Pricing in Incomplete Markets," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 40(4), pages 833-848, December.
    8. Wang, Song, 2024. "Pricing European call options with interval-valued volatility and interest rate," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    9. Timothy C. Johnson, 2012. "Ethics and Finance: the role of mathematics," Papers 1210.5390, arXiv.org.
    10. Hyeong-Ohk Bae & Seunggu Kang & Muhyun Lee, 2024. "Option Pricing and Local Volatility Surface by Physics-Informed Neural Network," Computational Economics, Springer;Society for Computational Economics, vol. 64(5), pages 3143-3159, November.
    11. Fabien Le Floc'h, 2021. "Pricing American options with the Runge-Kutta-Legendre finite difference scheme," Papers 2106.12049, arXiv.org.
    12. Mykland, Per Aslak, 2019. "Combining statistical intervals and market prices: The worst case state price distribution," Journal of Econometrics, Elsevier, vol. 212(1), pages 272-285.
    13. Halidias Nikolaos, 2024. "A novel portfolio optimization method and its application to the hedging problem," Monte Carlo Methods and Applications, De Gruyter, vol. 30(3), pages 249-267.
    14. Maxim Bichuch & Jiahao Hou, 2024. "Deep PDE solution to BSDE," Digital Finance, Springer, vol. 6(4), pages 727-758, December.
    15. Vorbrink, Jörg, 2014. "Financial markets with volatility uncertainty," Journal of Mathematical Economics, Elsevier, vol. 53(C), pages 64-78.
    16. Wei Chen, 2013. "Fractional G-White Noise Theory, Wavelet Decomposition for Fractional G-Brownian Motion, and Bid-Ask Pricing Application to Finance Under Uncertainty," Papers 1306.4070, arXiv.org.
    17. Daniel Sevcovic & Magdalena Zitnanska, 2016. "Analysis of the nonlinear option pricing model under variable transaction costs," Papers 1603.03874, arXiv.org.
    18. Hyungsok Ahn & Adviti Muni & Glen Swindle, 1999. "Optimal hedging strategies for misspecified asset price models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 197-208.
    19. Lin, Lisha & Li, Yaqiong & Wu, Jing, 2018. "The pricing of European options on two underlying assets with delays," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 495(C), pages 143-151.
    20. Balder, Sven & Brandl, Michael & Mahayni, Antje, 2009. "Effectiveness of CPPI strategies under discrete-time trading," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 204-220, January.
    21. Mahayni, Antje & Schlögl, Erik, 2003. "The Risk Management of Minimum Return Guarantees," Bonn Econ Discussion Papers 18/2003, University of Bonn, Bonn Graduate School of Economics (BGSE).

    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:apmaco:v:488:y:2025:i:c:s0096300324005782. 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: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.