in this talk, i will explore the rich interplay between differential equations and machine learning. i will highlight the use of collective dynamics and partial differential equations as powerful tools for improving machine learning algorithms and models. (i) in the first half of the talk, i will introduce a novel dynamical system that draws inspiration from collective intelligence observed in biology. this system offers a compelling alternative to gradient-based optimization. it enables gradient-free optimization to efficiently find global minimum in non-convex optimization problems. (ii) in the second half of the talk, i will build the connection between hamilton-jacobi-bellman equations and the multi-armed bandit (mab) problems. mab is a widely used paradigm for studying the exploration-exploitation trade-off in sequential decision making under uncertainty. this is the first work that establishes this connection in a general setting. i will present an efficient algorithm for solving mab problems based on this connection and demonstrate its practical applications.