Summary: Efficiently predicting the flow field and load in aerodynamic shape optimisation remains a highly challenging and relevant task. Deep learning methods have been of particular interest for such problems, due to their success in solving inverse problems in other fields. In the present study, U-net-based deep neural network (DNN) models are trained with high-fidelity datasets to infer flow fields, and then employed as surrogate models to carry out the shape optimisation problem, i.e. to find a minimal drag profile with a fixed cross-sectional area subjected to a two-dimensional steady laminar flow. A level-set method as well as Bézier curve method are used to parameterise the shape, while trained neural networks in conjunction with automatic differentiation are utilised to calculate the gradient flow in the optimisation framework. The optimised shapes and drag force values calculated from the flow fields predicted by the DNN models agree well with reference data obtained via a Navier-Stokes solver and from the literature, which demonstrates that the DNN models are capable not only of predicting flow field but also yielding satisfactory aerodynamic forces. This is particularly promising as the DNNs were not specifically trained to infer aerodynamic forces. In conjunction with a fast runtime, the DNN-based optimisation framework shows promise for general aerodynamic design problems.