Delay Differential Equation

Delay differential equations (DDEs) model systems where the rate of change depends on past states, offering a powerful framework for diverse phenomena exhibiting time delays. Current research focuses on developing data-driven methods, particularly neural networks (including Neural DDEs and variations like those incorporating piecewise-constant delays or state-dependent delays), to efficiently solve both forward and inverse DDE problems, often addressing challenges like limited or noisy data. These advancements enable more accurate modeling of complex systems across various fields, from traffic forecasting and robotics to biological and physical processes, by incorporating the crucial aspect of time delays into model architectures. The resulting improved accuracy and efficiency in solving DDEs have significant implications for scientific understanding and practical applications.