  We present a collection of explicit formulas for the minimum volume of
Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric
Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an
infinite class of 4d N=1 supersymmetric gauge theories, which are realized as
worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under
the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is
inversely proportional to the central charge of the corresponding 4d N=1
superconformal field theories. The presented formulas for the minimum volume
are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These
explicit results are derived by implementing machine learning regularization
techniques that advance beyond previous applications of machine learning for
determining the minimum volume. Moreover, the use of machine learning
regularization allows us to present interpretable and explainable formulas for
the minimum volume. Our work confirms that, even for extensive sets of toric
Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with
remarkable accuracy.
