In this paper I present an analysis of methods for automated solution of heterogeneous agent models with continuously distributed heterogeneity and aggregate shocks based on linearizing the equilibrium conditions with respect to a projection approximation of the cross-sectional distribution and individual agent decision rules. I show that for a broad class of stan- dard dynamic models with sufficient smoothly distributed heterogeneity, nesting those described in Arellano and Bonhomme [2016], the equilibrium conditions can be represented as infinitely wide neural networks based on compositions of pointwise nonlinearities and linear integral operators. This representation ensures commutativity of discretization and differen- tiation and so enables a method based on linearization with respect to the discretized functions that provably converges as the discretization is refined to the true functional derivatives of the original model and so en- sures convergence of the model solution under the conditions in Childers [2018]. Optimal convergence rates are shown depending on the combi- nation of model features and choice of basis function. The method and principles for model building are illustrated with an application to a ver- sion of the incomplete markets model of Huggett [1993] with continuously distributed idiosyncratic and aggregate income risk.