Bridging Hydrological Ensemble Simulation and Learning Using Deep Neural Operators

Ensemble-based simulation and learning (ESnL) has long been used in hydrology for parameter inference, but computational demands of process-based ESnL can be quite high. To address this issue, we propose a deep neural operator learning approach. Neural operators are generic machine learning algorithms that can learn functional mappings between infinite-dimensional spaces, providing a highly flexible tool for scientific machine learning. Our approach is built upon DeepONet, a specific deep neural operator, and is designed to address several common problems in hydrology, namely, model parameter estimation, prediction at ungaged locations, and uncertainty quantification. Here we demonstrate the effectiveness of our DeepONet-based workflow using an existing large model ensemble created for an eastern U.S. watershed that is instrumented with 10 streamflow gages. Results suggest DeepONet achieves high efficiency in learning an ML surrogate model from the model ensemble, with the modified Kling-Gupta Efficiency exceeding 0.9 on holdout test sets. Parameter inference, carried out using the trained DeepONet surrogate model and genetic algorithm, also yields robust results. Additionally, we formulate and train a separate DeepONet model for physics-informed, seq-to-seq streamflow forecasting, which further reduces biases in the pre-trained DeepONet surrogate model. While this study focuses primarily on a single watershed, our approach is general and may be extended to enable learning from model ensembles across multiple basins or models. Thus, this research represents a significant contribution to the application of hybrid machine learning in hydrology.