  Large neural network models have been successful in learning functions of
importance in many branches of science, including physics, chemistry and
biology. Recent theoretical work has shown explicit learning bounds for wide
networks and kernel methods on some simple classes of functions, but not on
more complex functions which arise in practice. We extend these techniques to
provide learning bounds for analytic functions on the sphere for any kernel
method or equivalent infinitely-wide network with the corresponding activation
function trained with SGD. We show that a wide, one-hidden layer ReLU network
can learn analytic functions with a number of samples proportional to the
derivative of a related function. Many functions important in the sciences are
therefore efficiently learnable. As an example, we prove explicit bounds on
learning the many-body gravitational force function given by Newton's law of
gravitation. Our theoretical bounds suggest that very wide ReLU networks (and
the corresponding NTK kernel) are better at learning analytic functions as
compared to kernel learning with Gaussian kernels. We present experimental
evidence that the many-body gravitational force function is easier to learn
with ReLU networks as compared to networks with exponential activations.
