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

Scientific Reports, Journal Year: 2025, Volume and Issue: 15(1)

Published: March 12, 2025

Abstract In this paper, the author introduces 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 of ODEs with oscillatory and/or exponential features, case. A new hybrid approach is formulated implemented, incorporates both approximate power neural networks robustness block methods. particular, it uses ability to spaces, utilizes method avoids conversion these equations into system first-order If used analysis, capable dealing several dynamic behaviors, such stiff boundary conditions. This paper presents mathematical formulation, architecture network choice its parameters proposed model. addition, results derived from convergence analysis agree that suggested technique more accurate compared existing solvers can handle effectively. Numerical experiments ordinary differential indicate fast has high accuracy linear nonlinear problems, including simple harmonic oscillators, damped systems like Van der Pol equation. The advantages are thought be generalized all scientific engineering disciplines, physics, biology, finance, other areas demand precise solutions. following also suggests potential research avenues future studies well: prospects model multi-dimensional systems, application partial (PDEs), appropriate higher efficiency.

Language: Английский

Mathematical approach for rapid determination of pull-in displacement in MEMS devices

Frontiers in Physics, Journal Year: 2025, Volume and Issue: 13

Published: April 7, 2025

Introduction Microelectromechanical systems (MEMS) are pivotal in diverse fields such as telecommunications, healthcare, and aerospace. A critical challenge MEMS devices is accurately determining the pull-in displacement voltage, which significantly impacts device performance. Existing methods, including variational iteration method homotopy perturbation method, often fall short providing precise estimations of these parameters. Methods This study introduces a novel mathematical approach that combines physical insights into phenomenon with theory. The begins definition device's model. By uniquely applying principle incorporating custom-designed functional, set equations derived. These transformed an iterative algorithm for calculating displacement, nonlinear terms addressed through approximation techniques tailored to system’s characteristics. Results Validation using specific examples demonstrates method's accuracy voltage. For instance, oscillator case, exact results were achieved computation time 0.015 s. Compared traditional this yields values rather than approximations, showcasing superior precision efficiency. Discussion proposed offers significant advantages, enhanced accuracy, reduced computational time, minimized error accumulation by solving algebraic instead iterating differential equations. It also exhibits robustness variations initial conditions system Limitations include need modifying criterion when formulation unattainable exclusion environmental factors like temperature pressure fluctuations. Future research should focus on refining models incorporate integrating Galerkin technology. Conclusion advances understanding behavior holds substantial potential design optimization across various applications, further driving progression

Language: Английский