Elastostatics of axisymmetric Kirchhoff nanoplates is investigated exploiting a stress-drivennonlocal theory to capture size-dependent mechanical behaviours. Physics-Informed NeuralNetworks (PINNs) are applied as a cutting-edge machine learning tool to solve the governingsixth-order differential problem, offering a powerful alternative to traditional methods. Several case studies involving graphene nanoplates are analyzed and solved. Notably, PINNs are employed to approximate the plate displacement field by minimizing a composite loss function that accounts for the residuals of the governing differential equation as well as both standard and non-standard boundary conditions. Numerical outcomes demonstrate a perfect agreement with numerically obtained solutions available in literature. The proposed methodology integrates advanced nonlocal modeling of scale effects provided by the stress-driven nonlocal theory with the versatility of PINN approaches in handling higher-order derivatives compared to conventional numerical methods. This synergy provides an efficient and reliable tool for tackling challenging nanomechanical problems.