Precipitation data collected from sparse monitoring stations in numerous karst basins pose a challenge for hydrologic models to accurately capture spatial and temporal correlation between precipitation and karst spring discharge, hindering the development of robust simulation models. To address this data scarcity issue, this study employes a coupled deep learning model that integrates a variation autoencoder (VAE) for augmenting precipitation data and a long short-term memory (LSTM) network for karst spring discharge prediction. The VAE contributes by generating synthetic precipitation data through an encoding-decoding process. This process generalizes the observed precipitation data by deriving joint latent distributions with improved preservation of temporal and spatial correlations in the data. The combined VAE-generated precipitation and observation data are used to train and test the LSTM for predicting the spring discharge. Applied to Niangziguan spring catchment in northern China, our coupled VAE/LSTM model demonstrated significantly higher predictive accuracy compared to a LSTM model using only field observations. We also explored temporal and spatial correlations in the observed data and the impact of different ratios of VAE-generated precipitation data to actual data on model performances. Additionally, our study evaluated the effectiveness of VAE-augmented data on various deep learning models and compared VAE with other data augmentation techniques. Our study demonstrates that the VAE offers a novel approach to address data scarcity and uncertainty, improving learning generalization and predictive capability of various hydrological models.

We develop a deep learning approach to learn the boundary-to-solution operator, i.e., to establish the boundary to steady solution mapping, in the Toth basin of arbitrary top and bottom topographies and two types of prescribed boundary conditions. The machine-learned mapping is represented by a DeepOnet, which takes the geometrical data and boundary conditions as the inputs and produces the steady state solution as the output. In this approach, we approximate the top and bottom boundaries by either truncated Fourier series or piecewise linear representations. The DeepOnet maps directly the finite dimensional representations of the boundaries to the steady state solution of the ground water transport equation in the Toth basin. We present two different implementations of the DeepOnet: 1) the Toth basin is embedded in a rectangular computational domain, and 2) the Toth basin with arbitrary top and bottom boundaries is mapped into a rectangular computational domain via a nonlinear transformation. We implement the DeepOnet with respect to the Dirichlet and Robin boundary condition at the top and the Neumann boundary condition at the impervious bottom boundary, respectively. Both implementations yield the same results, showcasing a new deep learning approach to study ground water transport phenomena.