Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. Beginning with a neural network model of Touboul and Robert, the basic results on exchangeable systems are introduced and then applied to new results on stochastic Allen-Cahn equations with boundary conditions. New material is joint with Dan Crisan and Chris Janjigian. Earlier work with Peter Donnelly, Phil Protter, Jie Xiong, Yoonjung Lee, and Peter Kotelenez is also discussed.