PDE-constrained optimization is an exceedingly difficult task due to the curse of dimensionality. This is particularly true in aerodynamics optimization as remeshing or deformation of existing meshes is often required. In this work, we aim to overcome this challenge by proposing a physics-based machine learning framework that can handle complex optimization problems efficiently. The proposed approach is able to simultaneous make predictions and perform optimizations. Specifically, the framework consists of a surrogate model to generate high-fidelity solutions and a gradient-based algorithm to address high-dimensional optimization problems. The starting point is to use physics-informed neural network (PINN) to construct surrogate models that output flow fields associated with varied design configurations. More importantly, we build the architecture of surrogate models and include design variables as inputs to PINN. A trained surrogate model can represent solutions on unseen design cases from both seen and unseen categories. In the optimization process, a quasi-Newton algorithm is used and further accelerated by automatic differentiation. This approach combines two strengths of deep neural networks: the ability to approximate a high dimensional function well, and the automatic differentiability of that function with respect to its inputs. The proposed method is particularly appealing to optimization problems where labelled data is scarce or expensive.