Additive manufacturing has enabled the production of functionally graded (FG) lattices with tailorable mechanical properties. Of all the lattice geometries, triply periodic minimal surfaces (TPMS) are especially attractive due to their unique geometric and mechanical features. To address this, a study on the nonlinear microscale mechanical behaviour of the Schwarz primitive (SP) TPMS is conducted using a systematic procedure covering micro-scale to full-scale modelling and simulation. The nonlinear microscale behaviour of the parameterized Schwarz primitive is investigated through theoretical and experimental methods. The microstructure framework entails simulating representative volume elements (RVEs) with different repeat unit cells to assess the RVE convergence and the size dependency of the effective behaviour. A parametric material model based on a physics-guided feed-forward neural network (FFNN) is employed to predict the constitutive model of the FG lattice structures at any arbitrary material point. Furthermore, the study examines the linear static behaviour of graded lattices made of SP-TPMS on the macroscale level to verify the method's reliability and efficiency. Homogeneous and graded lattice models are demonstrated using three different models: a full-scale model, a homogenized finite element model using Abaqus, and a linear 3D elastic theory using the differential quadrature method. Finally, the study discusses the convergence and accuracy of these methods for static mechanical behaviour at the finite deformation continuum level.