  Tame functions are a class of nonsmooth, nonconvex functions, which feature
in a wide range of applications: functions encountered in the training of deep
neural networks with all common activations, value functions of mixed-integer
programs, or wave functions of small molecules. We consider approximating tame
functions with piecewise polynomial functions. We bound the quality of
approximation of a tame function by a piecewise polynomial function with a
given number of segments on any full-dimensional cube. We also present the
first mixed-integer programming formulation of piecewise polynomial regression.
Together, these can be used to estimate tame functions. We demonstrate
promising computational results.
