  Bent functions are of great importance in both mathematics and information
science. The $\mathcal{P}\mathcal{S}$ class of bent functions was introduced by
Dillon in 1974, but functions belonging to this class that can be explicitly
represented are only the $\mathcal{P}\mathcal{S}_{\text{ap}}$ functions, which
were also constructed by Dillon after his introduction of the
$\mathcal{P}\mathcal{S}$ class. In this paper, a technique of using finite
pre-quasifield spread from finite geometry to construct
$\mathcal{P}\mathcal{S}$ bent functions is proposed. The constructed functions
are in similar styles with the $\mathcal{P}\mathcal{S}_{\text{ap}}$ functions.
To explicitly represent them in bivariate forms, the main task is to compute
compositional inverses of certain parametric permutation polynomials over
finite fields of characteristic 2. Concentrated on the Dempwolff-M\"uller
pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield, three new
subclasses of the $\mathcal{P}\mathcal{S}$ class are obtained. They are the
only sub-classes that can be explicitly constructed more than 30 years after
the $\mathcal{P}\mathcal{S}_{\text{ap}}$ subclass was introduced.
