Partial differential equations (PDEs) are widely used across scientific disciplines to model the space-time evolution of diverse physical phenomena. A PDE that is of special interest is the drift-diffusion PDE that models the transfer of a physical quantity due to both drift (arising from velocity fields) and diffusion (arising from collisions). This PDE manifests in a well-known form in score-matching diffusion generative models where the drift term is the learned score function. It also arises in dynamic density functional theory (DDFT) to model non-equilibrium fluid dynamics, where the drift-term is the unknown free-energy of the fluid. We will work on developing methods to learn this PDE from data using neural networks for improvements in speed, accuracy, and in-domain transfer. This holds potential in shape analysis leading to improved methods for shape interpolation and generation. Additionally, it holds potential in polymer science for the enhanced modelling of special polymers used in biodegradable plastics and carbon-capture membranes.