In the preliminary phase of analyzing the thermoacoustic characteristics of a gas turbine combustor, implementing robust design principles is essential to minimize detrimental variations of its thermoacoustic performance under various sources of uncertainties. In this study, we systematically explore different aspects of robust design in thermoacoustic instability analysis, including risk analysis, control design, and inverse tolerance design. We simultaneously take into account multiple thermoacoustic modes and uncertainty sources from both the flame and acoustic boundary parameters. In addition, we introduce the concept of a “risk diagram” based on specific statistical descriptions of the underlying uncertain parameters, which allows practitioners to conveniently visualize the distribution of the modal instability risk over the entire parameter space. Throughout this study, a machine learning method called “Gaussian process” (GP) modeling approach is employed to efficiently tackle the challenge posed by the large parameter variational ranges, various statistical descriptions of the parameters, as well as the multifaceted nature of robust design analysis. For each of the investigated robust design tasks, we propose an efficient solution strategy and benchmark the accuracy of the results delivered by GP models. We demonstrate that GP models can be flexibly adjusted to various tasks while only requiring one-time training. Their adaptability and efficiency make this modeling approach very appealing for industrial practices.