Differentiable, Learnable MILC: Balancing Predictive Skill and Physical Interpretability

Deep learning models, particularly LSTMs, have transformed large-sample hydrology by achieving high streamflow predictive performance, yet they remain largely black-box approaches with limited physical interpretability and no explicit representation of multiphysical hydrological processes. Differentiable, learnable process-based models (or δ-models) overcome these limitations by embedding neural networks within differentiable physics frameworks. While existing benchmarks like HBV-δ have proven this concept across 671 US basins, they rely on conceptual foundations (e.g., empirical beta-functions) that approximate, rather than resolve, underlying soil physics. This study introduces MILC-δ (Modular Differentiable Physic-Informed Learning), designed to bridge this gap. The MILC model utilizes continuous soil water retention curves and physically derived drainage laws, which can aid in more accurate hydrological flux simulation. Thus, we developed a MILC-δ - a hydrologic model embedded with neural networks and trained in a differentiable programming framework. Consequently, MILC-δ is anticipated to match or exceed HBV-δ by leveraging neural networks to map static catchment attributes directly to physically measurable properties (e.g., pore size distribution, hydraulic conductivity) rather than abstract calibration parameters. Initial testing of the developed model shows that the model performs at par in some basins and better than HBV-δ in other basins. This approach gives LSTM-level accuracy and generalizability as well as the clear physical story stakeholders actually need to explain the decline in baseflow, threats to the groundwater recharge, etc.