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Creating analytic representations of multiple potential energy surfaces for modeling electronically nonadiabatic processes is a major challenge being addressed in various ways by the chemical dynamics community. In this work, we introduce a new method that can achieve convenient learning of multiple potential energy surfaces (PESs) and their gradients (negatives of the forces) for a polyatomic system. This new method, called compatibilization by deep neural network (CDNN), is demonstrated to be accurate and, even more importantly, to be automatic. The only required input is a database with geometries and potential energies. The method produces a matrix, called the compatible potential energy matrix (CPEM), that may be interpreted as the electronic Hamiltonian in an implicit nonadiabatic basis, and the analytic adiabatic potential energy surfaces and their gradients are obtained by diagonalization and automatic differentiation. We show that the CPEM, which is neither adiabatic nor necessarily diabatic, can be discovered automatically during the learning procedure by the special design of a CDNN architecture. We believe that the CDNN method will be very useful in practice for learning coupled PESs for polyatomic systems because it is accurate and fully automatic.