Lean premixed combustion is the state-of-the-art technology to achieve ultra low NOx emissions in stationary gas turbines. However, lean premixed flames are susceptible to thermoacoustic instabilities, lean blowout, and flashback. The design of such a combustion system is thus always related to the balancing between the levels of emissions and flame stability. Data-driven optimization methods and the adaptation of models through artificial intelligence have experienced a surge in development in the past years. The goal of this study is to show the potential of these methods for gas turbine burner development. A special pilot burner that features 61 different positions of fuel injection, manufactured by means of selective laser melting is used to modify the gas mixture close to the flame anchoring position. Each of the injector lines is equipped with an individual valve, such that the distribution of fuel-air mixture can be modified variously. Installed into an industrial MGT6000 swirl combustor, a data-driven optimization method is used to find an optimal subset of injection locations by automated experiments. The method uses a surrogate model that is based on Gaussian processes regression. It is adopted for experimental optimization, keeping measurement efforts to a minimum. The optimizer controls the fuel valves and uses live measurements to find a distribution that generates minimal NOx emissions while ensuring flame stability. The solutions found by the optimization scheme are analyzed and advantages and limitations of the approach are discussed.