Systematizing product design by latent-variable modeling – A unifying framework for the formulation and solution of PLS model inversion problems

Data-driven modeling has significantly transformed problem-solving in the process industry, especially in designing new products digitally by finding the process conditions that are required to manufacture a product with assigned quality. This can be achieved by utilizing historical process data via latent-variable model inversion, notably through the extensively used partial least-squares (PLS) regression model. Despite the development of numerous PLS model-inversion techniques, from straightforward algebraic manipulation of the model equations to the formulation and solution of complex nonlinear optimization problems, there lacks a comprehensive discussion on their comparative benefits and limitations. This paper offers a systematic analysis of PLS model inversion strategies, especially those based on optimization problems. We delve into aspects such as optimization in the latent or input variable spaces, the nature of constraints (soft vs. hard), and the feasibility of analytical solutions. We outline a clear hierarchical structure of the available methods based on the successive inclusion of constraints, and propose a general formulation of the PLS model inversion by optimization problem that encompasses all available methods. We support our theoretical analysis with a numerical case study, and provide the code to reproduce it and to solve general latent-variable model inversion problems according to the proposed formulation.