Understanding large systems is crucial for addressing challenges across science, engineering, economics, and social sciences. However, the inherent high dimensionality and complex dynamics of these systems pose difficulties for reductionist approaches. One key strategy to overcome the curse of dimensionality is to analyze the limiting behaviors of large systems at their asymptotics. Despite the valuable simplifications gained through this approach, deriving analytical solutions remains challenging at asymptotics due to the innate complexity. Machine learning and reinforcement learning techniques are hence employed to decipher the asymptotics of large systems.

This manuscript delves into both theoretical developments and practical implementations related to three types of large systems:large random graphs, mean field control games, and cognitive mean field control games. The corresponding numerical techniques utilize machine learning, reinforcement learning, and multi-agent reinforcement learning algorithms to learn the large system asymptotics.

In Chapter Learning Percolation in Large Random Graphs, we extensively analyze bootstrap percolation in random graphs to comprehend how an initial infection spreads throughout the entire system. The main theoretical result enables us to determine the asymptotic final fraction of infected vertices at the end of percolation by solving the fixed point problem of a nonlinear operator. We introduce a functional regulated neural network to efficiently learn the fixed point.

In Chapter Learning Mean Field Control Games, we explore a new framework to model the asymptotic behaviors of systems consisting of numerous competitive groups, each containing a large number of cooperative players. This framework is applied to study the flocking model, bank liquidity, and optimal trading strategy. We devise three-timescale mean field reinforcement learning algorithms to learn both the optimal strategy and the evolution of population distributions.

In Chapter Learning Cognitive Mean Field Control Games, we connect behavioral game theory with the mean field approach by introducing cognitive mean fields in large scale games to capture player heterogeneity with bounded rationalities and varied strategic levels. We propose multi-agent reinforcement learning algorithms based on actor-critic method to learn both the optimal strategies of all levels of players and variations of assorted cognitive mean fields.