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

$\mathcal{P}\mathcal{S}$ bent functions constructed from finite pre-quasifield spreads