The application of Extreme Learning Machines (ELMs) to the numerical solution of differential equations has experienced remarkable growth in recent years. Starting from the pioneering Physics-Informed Extreme Learning Machine (PIELM) framework, an increasingly broad family of methods has emerged, extending the original formulation to more complex differential models, inverse problems, domain decomposition strategies and adaptive techniques. At the same time, several approaches originally developed under different names and motivations can be interpreted within the same random projection paradigm, highlighting a common mathematical structure that deserves further investigation. This mini-symposium aims to bring together researchers working on ELM-based and related random projection methods for differential equations, with the objective of discussing their theoretical foundations, computational properties, and practical applications. Particular attention will be devoted to the convergence behavior of these methods. Although they are not high-order methods in the classical polynomial approximation sense, growing theoretical and numerical evidence indicates that they often exhibit spectral-like convergence, placing them among the closest physics-informed learning approaches to traditional spectral methods. This distinctive behavior originates from the use of a fixed, randomly generated hidden layer that defines a rich approximation space while preserving the computational efficiency characteristic of ELMs. The mini-symposium welcomes contributions addressing theoretical analysis, numerical algorithms, and applications of ELM-based methods for ordinary, partial, stochastic and integro-differential equations. Contributions employing ELMs or related random feature techniques as components of broader numerical frameworks are equally encouraged. The goal is to provide a forum where researchers can exchange ideas, identify common principles underlying existing methods, and discuss future directions for this rapidly evolving research area.