Runge Kutta

Runge-Kutta methods are a family of iterative numerical techniques for approximating solutions to differential equations, crucial for modeling diverse physical systems. Current research focuses on enhancing their accuracy and efficiency, particularly through integration with neural networks, such as Physics-Informed Neural Networks (PINNs) and novel architectures like Graph Neural Runge-Kutta (GNRK) methods, to solve both ordinary and partial differential equations. These advancements improve the accuracy of solutions, particularly for stiff or chaotic systems, and enable error estimation, leading to more reliable simulations in fields ranging from power systems to Hamiltonian dynamics. The development of adaptive algorithms and the application to non-Euclidean spaces further broaden the applicability and robustness of these methods.