We present an innovative solution to improve the efficiency of thermophotovoltaic (TPV) devices by tackling the problem of sub-bandgap photon losses. We propose an optimized design for thin-film mirrors using inverse electromagnetic design principles, thereby enhancing the average reflectivity and photon re-use. Our method surpasses the traditional Bragg mirror by employing a gradient-descent based optimization over Bragg mirror geometrical parameters, leveraging the transfer matrix method for derivative calculations. The optimized structure, based on continuously chirped distributed Bragg reflectors proposed herein demonstrates a remarkable increase in reflectivity beyond 98%, over an almost three-octaves bandwidth (0.1eV-0.74eV). We show that the incident power loss in InGaAs TPV cells at an emitter temperature of 1200°C is significantly reduced. While our work shows considerable promise, further exploration is needed to ascertain the practicability and robustness of these designs under various operational conditions. This study thus provides a major step forward in TPV technology, highlighting a new route towards more effective energy conversion systems.