We present a highly scalable strategy for developing mesh-free neuro-symbolic partial differential equation solvers from existing numerical discretizations found in scientific computing. This strategy is unique in that it can be used to efficiently train neural network surrogate models for the solution functions and the differential operators, while retaining the accuracy and convergence properties of state-of-the-art numerical solvers. This neural bootstrapping method is based on minimizing residuals of discretized differential systems on a set of random collocation points with respect to the trainable parameters of the neural network, achieving unprecedented resolution and optimal scaling for solving physical and biological systems.