Hybrid modeling with attribution-guided symbolic enhancements of known physics: Applications to complex biological and engineering systems

Hybrid mechanistic-machine learning models combine first-principles equations with data-driven components to compensate for incomplete system knowledge, promising to bridge mechanistic understanding and data-driven flexibility. In practice, however, model discrepancy is often absorbed into black-box corrections, limiting interpretability and scientific insight. In this work, we propose HATS (Hybrid modeling with ATtribution-guided Symbolic regression), a framework that replaces opaque data-driven corrections with compact, interpretable symbolic representations. HATS iteratively trains a hybrid mechanistic–data-driven model and uses shapley additive explanation (SHAP) based attribution analysis to prune weakly influential inputs, thereby localizing the dominant sources of model mismatch. Symbolic regression is then applied to recover closed-form expressions for the unknown dynamics under explicit complexity constraints. When direct symbolic recovery is infeasible, HATS learns a low-dimensional latent representation of the uncertainty and performs symbolic regression in the latent space, yielding simple and interpretable models. Unlike existing hybrid modeling approaches, HATS adaptively determines where, why , and how uncertainty should be represented, combining feature attribution, symbolic regression, and latent learning within a unified framework. We demonstrate the approach through an in silico rediscovery of the TNF- α signaling pathway and an industrial-scale fermentation case study, thereby highlighting HATS as a novel pathway toward transparent, discovery-oriented hybrid modeling of complex systems.