A Framework for Objective-Driven Dynamical Stochastic Fields

Fields offer a versatile approach for describing complex systems composed of interacting and dynamic components. In particular, some of these dynamical and stochastic systems may exhibit goal-directed behaviors aimed at achieving specific objectives, which we refer to as $\textit{intelligent fields}$. However, due to their inherent complexity, it remains challenging to develop a formal theoretical description of such systems and to effectively translate these descriptions into practical applications. In this paper, we propose three fundamental principles -- complete configuration, locality, and purposefulness -- to establish a theoretical framework for understanding intelligent fields. Moreover, we explore methodologies for designing such fields from the perspective of artificial intelligence applications. This initial investigation aims to lay the groundwork for future theoretical developments and practical advances in understanding and harnessing the potential of such objective-driven dynamical stochastic fields.