Time-domain Adjoint Method for Energy and Signal Routing in Complex Coupled Systems

The time-domain adjoint method provides a powerful framework for gradient-based optimization in complex dynamical systems. The key advantage lies in computational efficiency that, regardless of the number of design parameters, the sensitivity of an objective function with respect to parameters can be obtained through only one forward propagation followed by one backward propagation. This property makes the method particularly suitable for optimizing large parameter spaces and high dimensional systems where conventional optimization or finite difference approaches are computationally expensive. When applied to complex communication systems, where numerous interacting variables and even time varying or moving objects coexist, the time-domain adjoint method enables effective routing and control of signals or energy in the space-time domain. To demonstrate this capability, theoretical, numerical, and experimental studies were performed on representative electronic systems. The results confirm that the time-domain adjoint method not only ensures accurate gradient optimization but also offers a robust and scalable optimization strategy, revealing optimal pathways for energy transfer and signal manipulation in dynamically evolving real world environments.