In this work, we introduce Poisson Hamiltonian Neural Networks (PHNNs) as an extension of Hamiltonian Neural Networks to better capture the dynamics of Poisson-Hamiltonian systems. By incorporating structure-preserving numerical methods, PHNNs can learn a wider range of dynamical systems beyond traditional symplectic models. We explored different training strategies, comparing Explicit Euler (EE) and Poisson-Hamiltonian Integrators (PHI). Our results showed that, while Euler-trained models offer better short-term accuracy, PHI-trained models stand out for their long-term stability and preservation of geometric structures. A hybrid approach training with EE and testing with PHI proved to be the best balance between accuracy and stability. These results highlight the potential of combining machine learning with geometric numerical methods to model complex dynamic systems without the need for explicit governing equations.