Over the years, various methods have been developed to attain these desirable features. For example, the potential energy invariance with respect to translation and rotation and invariance with respect to permutation of identical nuclei is often achieved by coordinate transformation, that is, one represents the molecular geometry by appropriate internal coordinates instead of atomic Cartesian coordinates, and by using permutationally invariant polynomials of internuclear distances, or constraining permutation invariance in equivariant message-passing graph neural networks. Proper asymptotic behavior can be challenging because one often has insufficient data for training because the asymptotic space is infinite, but it can achieved by constraining the form of the fit, for example by using a parametrically managed activation function. Correct topology of the PESs in the vicinity of conical intersections can be enforced by transforming the learning target from adiabatic PESs and nonadiabatic couplings into diabatic potential energy matrices (DPEMs). We have developed a series of methods to achieve semiautomatic learning of coupled adiabatic PESs and their couplings. The first generation of these semiautomatic methods is called diabatization by deep neural network (DDNN); it automatically discovers the DPEMs on-the-fly during the learning of adiabatic PESs, and therefore it greatly reduces the human effort by eliminating the need to construct DPEMs as a pre-fitting step, for example by finding diabatic molecular orbitals or diabatic configuration state functions. Next we presented permutationally restrained DDNN (PR-DDNN) in which we added approximate permutation invariance by adding a permutational-invariance term in the cost function. Our next version, called parametrically managed DDNN (PM-DDNN), enforces physical asymptotic behavior of PESs by introducing a parametrically managed activation function in the architecture of PR-DDNN. With these developments, the coupled PESs learned by PM-DDNN include all three of the desirable features mentioned above.