From curve fitting to scientific machine learning: A critical synthesis of solute transport modeling and breakthrough curve interpretation

Solute breakthrough curves serve as fundamental diagnostic metrics for characterizing mass transport mechanisms in hydrogeological systems. Traditional interpretations relying exclusively on boundary outlet sampling and deterministic inversion inherently suffer from severe mathematical ill-posedness and equifinality. Resolving these structural limitations necessitates a methodological transition toward high-dimensional internal sensing and probabilistic parameterization. This review presents a critical synthesis of how advanced observational constraints and scientific machine learning are fundamentally reshaping solute transport modeling. We evaluate the progression from classical iterative algorithms to ensemble-based data assimilation frameworks capable of reconstructing complex spatial heterogeneity. Furthermore, we critically assess the operational trade-offs of modern artificial intelligence paradigms. This evaluation analyzes the capacity of probabilistic generative surrogates to ensure robust uncertainty quantification and examines the role of physics-informed neural networks in enforcing mass conservation, while explicitly addressing their optimization instabilities when deployed on noisy field datasets. To organize model and algorithm selection, we develop a structured Complexity–Model–Data framework that relates diagnostic evidence for process complexity to candidate transport formulations and the observational support available for inversion. Dimensionless indicators and BTC morphology are used to narrow the set of plausible models, whereas data density, computational demand, and uncertainty requirements inform the selection of deterministic, probabilistic, data-assimilation, or physics-informed methods. A one-dimensional synthetic case illustrates how this reasoning can be implemented under controlled conditions. The framework provides a practical basis for transparent, site-specific model evaluation while preserving the roles of hydrogeological interpretation, measurement quality, management objectives, and expert judgment.