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.