Evolution of chromatographic modeling: From mechanistic models to hybrid models with physics-based deep learning

Hybrid modeling based on physics-based deep learning (PBDL) represents a transformative approach that unifies mechanistic understanding and data-driven learning, offering a pathway beyond the limitations of traditional chromatographic models. This review systematically summarizes the evolution of PBDL methods for chromatography across three generations. The first generation, surrogate-model-based solvers, accelerates simulations through mechanistic up-sampling and fast inference but remains constrained by indirect physical coupling, reflecting "data-assisted physics". The second generation, physics-informed neural networks, embeds governing equations into the loss function, enabling simultaneous learning from physics and data, while facing challenges in loss balancing and numerical integration, representing "physics-constrained data". The third generation, differentiable numerical simulations of physical systems, integrates neural networks within numerical solvers, achieving high-fidelity modeling and gradient-based optimization, achieving "mutual feedback between physics and data". Collectively, these advances empower chromatographic models with the ability to self-learn complex adsorption behaviors under physical constraints, paving the way toward real-time digital twins and intelligent bioprocess modeling for the next generation of chromatographic engineering.