We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of finite number of standard mathematical operations with finite (condition-independent) number of variables in a discrete or continous region of consideration. The general approach to obtain the compact-form solutions combines a number of methods in the fields of partial differential equations, enumerative combinatorics, and number theory, which we describe in detail. We discuss their advantages and perspectives for the various fields in physics, mathematics, and advanced calculus.