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In molecular, material, and process design, Gaussian mixture regression (GMR) can directly predict values of explanatory variables x such as experimental and process conditions from target values of objective variables y such as properties and activities of materials. In GMR, a prediction result of a Gaussian mixture distributions is provided as a probability density distribution. The weighted average of the means of Gaussians has been used as the representative value of Gaussian mixture. In this study, true GMR has been proposed, where the predicted value of the GMR model is calculated by solving a continuous optimization problem to obtain the maximum value of the probability density function (PDF). As the continuous optimization problem cannot be solved when variables contain discrete values, a genetic algorithm is used to search for a solution that satisfies the constraints, while maximizing the value of the PDF. Numerical simulations using nonlinear functions between y and x, and the case studies using real material datasets confirmed that the proposed methods produced solutions with drastically higher values of the PDF than the traditional methods, especially when x is constrained in the inverse analysis. The Python codes of the proposed methods are freely available. Direct inverse analysis with GMR using the proposed methods is expected to improve the efficiency of molecular, material, and process design.

True Gaussian mixture regression and genetic algorithm-based optimization with constraints for direct inverse analysis