A novel hybrid framework for efficient higher order ODE solvers using neural networks and block methods

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.