  We present a continuous formulation of machine learning, as a problem in the
calculus of variations and differential-integral equations, in the spirit of
classical numerical analysis. We demonstrate that conventional machine learning
models and algorithms, such as the random feature model, the two-layer neural
network model and the residual neural network model, can all be recovered (in a
scaled form) as particular discretizations of different continuous
formulations. We also present examples of new models, such as the flow-based
random feature model, and new algorithms, such as the smoothed particle method
and spectral method, that arise naturally from this continuous formulation. We
discuss how the issues of generalization error and implicit regularization can
be studied under this framework.
