  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.
