Variational Methods for Solving Problems Associated with Artificial Intelligence

• Published in: Differential equations 2023-07, Vol.59 (7), p.919-932
• Main Authors: Babushkin, M. V., Tertychny-Dauri, V. Yu
• Format: Article
• Language: English
• Subjects: Adaptation; Adaptive control; Adaptive systems; Algorithms; Analysis; Artificial intelligence; Calculus of variations; Control systems; Control Theory; Data mining; Difference and Functional Equations; Machine learning; Mathematical analysis; Mathematics; Mathematics and Statistics; Methods; Ordinary Differential Equations; Partial Differential Equations; Problem solving; Separation; Unsupervised learning; Variational methods
• ISSN: 0012-2661; 1608-3083
• DOI: 10.1134/S0012266123070066

We introduce the concepts of a learning algorithm, an objective function, a recognition system, a class of patterns, a training set, a reward algorithm, a finitely convergent algorithm, an adaptive control system, a control goal, control tactics, adaptation time, etc., related to the problem of artificial intelligence in the processes of learning and adaptation. The general problem of self-learning (unsupervised learning)—about the separation of sets—in terms of the classical calculus of variations is posed. The generality of the problem is due to the introduction of an additional time variable into the analysis. The problem is solved by determining extremal conditions under which the minimization of the overall average risk functional is achieved. Problems corresponding to nonfixed and fixed time intervals are considered. For these two cases, expressions are found for calculating variations in cost functionals. Necessary conditions are indicated for determining the extremal values of the self-learning process (separation of classes of a set of patterns) in time.