Beyond black-box models: Measuring physics-driven and data-driven contributions in hybrid process models

In this work, we introduce a practical hybrid methodology that fuses first-principles knowledge with measured process data to improve model prediction and extrapolation. Using sequential-orthogonalized partial least squares (SO-PLS), we (i) remove variation already explained by physics so the data-driven component learns only from unexplained residual structures, and (ii) decompose each prediction into mechanistic and empirical contributions. This decomposition yields a simple diagnostic – the mechanistic-to-measured (M2M) ratio – that quantifies the relative influence of physics versus data and flags when the fit is dominated by empirical corrections, signaling potential degradation under extrapolation. We demonstrate the methodology on a simulated two-stage batch reactor with both well-specified and misspecified kinetics. Even with model misspecification, the mechanistic component adds useful information; however, the contribution analysis shows substantial empirical adjustment remains necessary. The M2M ratio enables practitioners to compare candidate models and select those more likely to remain reliable beyond the training domain.