Murugesh V, Priyadharshini M, Sharma Yogesh Kumar, Lilhore Umesh Kumar, Alroobaea Roobaea, Alsufyani Hamed, Baqasah Abdullah M, Simaiya Sarita
Department of CSE, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Guntur, AP, India.
Department of Computer Science and Engineering, Faculty of Science and Technology (ICFAI Tech), ICFAI Foundations for Higher Education, 501 203, Hyderabad, India.
Sci Rep. 2025 Mar 12;15(1):8456. doi: 10.1038/s41598-025-90556-5.
In this paper, the author introduces the Neural-ODE Hybrid Block Method, which serves as a direct solution for solving higher-order ODEs. Many single and multi-step methods employed in numerical approximations lose their stability when applied in the solution of higher-order ODEs with oscillatory and/or exponential features, as in this case. A new hybrid approach is formulated and implemented, which incorporates both the approximate power of neural networks and the stability and robustness of block numerical methods. In particular, it uses the ability of the neural networks to approximate the solution spaces, utilizes the block method for the direct solution of the higher-order ODEs and avoids the conversion of these equations into a system of the first-order ODEs. If used in the analysis, the method is capable of dealing with several dynamic behaviors, such as stiff equations and boundary conditions. This paper presents the mathematical formulation, the architecture of the employed neural network and the choice of its parameters for the proposed hybrid model. In addition, the results derived from the convergence and stability analysis agree that the suggested technique is more accurate compared to the existing solvers and can handle stiff ODEs effectively. Numerical experiments with ordinary differential equations indicate that the method is fast and has high accuracy with linear and nonlinear problems, including simple harmonic oscillators, damped oscillatory systems and stiff nonlinear equations like the Van der Pol equation. The advantages of this approach are thought to be generalized to all scientific and engineering disciplines, such as physics, biology, finance, and other areas in which higher-order ODEs demand more precise solutions. The following also suggests potential research avenues for future studies as well: prospects of the proposed hybrid model in the multi-dimensional systems, application of the technique to the partial differential equations (PDEs), and choice of appropriate neural networks for higher efficiency.
在本文中，作者介绍了神经常微分方程混合块方法，该方法可直接用于求解高阶常微分方程。在数值逼近中使用的许多单步和多步方法，在求解具有振荡和/或指数特征的高阶常微分方程时（如在这种情况下）会失去稳定性。本文提出并实现了一种新的混合方法，该方法结合了神经网络的近似能力以及块数值方法的稳定性和鲁棒性。具体而言，它利用神经网络逼近解空间的能力，使用块方法直接求解高阶常微分方程，并避免将这些方程转换为一阶常微分方程组。如果用于分析，该方法能够处理多种动态行为，如刚性方程和边界条件。本文给出了所提出混合模型的数学公式、所采用神经网络的架构及其参数选择。此外，收敛性和稳定性分析结果表明，与现有求解器相比，该技术更精确，能够有效处理刚性常微分方程。常微分方程的数值实验表明，该方法速度快，对于线性和非线性问题（包括简谐振子、阻尼振荡系统以及像范德波尔方程这样的刚性非线性方程）具有高精度。这种方法的优点被认为可以推广到所有科学和工程学科，如物理、生物、金融以及其他需要更精确求解高阶常微分方程的领域。以下内容还为未来研究提出了潜在的研究方向：所提出的混合模型在多维系统中的前景、该技术在偏微分方程中的应用以及选择合适的神经网络以提高效率。
