In recent years, applications of neural networks to the modeling of physical phenomena have attracted much attention. This study proposes a method for learning systems that are described by the GENERIC formalism, which is a combination of analytical mechanics and non-equilibrium thermodynamics. GENERIC systems admit the energy conservation law and the law of increasing entropy under certain conditions. However, designing neural network models that satisfy these conditions is difficult. In this study, we introduce a relaxation model of the GENERIC form, thereby introducing an equivalence class into the set of models. Because the equivalence class of the target model includes a model that can be learned by neural networks, the learned model has the energy conservation law and the law of increasing entropy in high accuracy with respect to the true energy and the true entropy.