The paper presents a generalized Drude-Lorentz (GDL) model for dielectric permittivity that complies with the singularity expansion method (SEM). The SEM is applied to expand the permittivity as a transfer function of a physical system, which is then recast into the GDL form. This model includes both Drude and Lorentz terms, with the additional imaginary terms in the generalized Lorentz terms allowing for a more accurate representation of the permittivity in the complex frequency plane. The parameters of the GDL model are retrieved using experimental data and optimized through auto-differentiation, demonstrating high accuracy for a wide range of materials, including metals, 2D materials, and dielectrics. The distribution of the poles in the complex frequency plane is used to characterize the material behavior, with Drude singularities indicating metallic behavior and Lorentz poles marking transitions between different material regimes. The GDL model outperforms the classical Drude-Lorentz model, especially in non-metallic media, due to its ability to capture the temporal dependencies of the electric field and displacement field.The paper presents a generalized Drude-Lorentz (GDL) model for dielectric permittivity that complies with the singularity expansion method (SEM). The SEM is applied to expand the permittivity as a transfer function of a physical system, which is then recast into the GDL form. This model includes both Drude and Lorentz terms, with the additional imaginary terms in the generalized Lorentz terms allowing for a more accurate representation of the permittivity in the complex frequency plane. The parameters of the GDL model are retrieved using experimental data and optimized through auto-differentiation, demonstrating high accuracy for a wide range of materials, including metals, 2D materials, and dielectrics. The distribution of the poles in the complex frequency plane is used to characterize the material behavior, with Drude singularities indicating metallic behavior and Lorentz poles marking transitions between different material regimes. The GDL model outperforms the classical Drude-Lorentz model, especially in non-metallic media, due to its ability to capture the temporal dependencies of the electric field and displacement field.