Design optimization of adaptive systems requires a robust analysis method that can accommodate various changes in design and boundary conditions. In this work, physics-informed neural networks (PINNs) are used to approximate solutions to differential equations across a range of problem parameter values. This mesh-free method simply requires residual evaluation at sampling points within the analysis domain and along boundaries, and the training process does not require any reference problem to be solved through conventional solution methods. The trained model can be used to predict the solution field, conduct parameter space analysis and design optimization. Using automatic differentiation, the design objective and their derivatives can be computed as a post process for a gradient-based design optimization. The method is demonstrated in a 1D heat transfer problem governed by the steady-state heat equation. Use of the PINN model for design optimization is illustrated in a problem of finding a material transition location to minimize temperature at a specified location. The PINN model that does not include problem parameters as input can be trained to within 0.05% error. PINN models that involve problem parameters as inputs are more difficult to train, especially when the input-to-output relationship is complex.