Dynamical system neural network for hydrological modelling

Neural networks are efficient and effective in predicting system states in hydrology. However, most current approaches lack hydrological flow partitioning, do not allow for training on measurements of multiple variables, or lack capability to tightly integrate physically-based components. To address these shortcomings I propose and evaluate an approach referred to below as Dynamical System Neural Network (DSNN). DSNN is a feedforward neural network with an architecture that resembles the organisation in components of the real-world system it represents. In hydrology, the DSNN represents each water flow (e.g. seepage, snow melt) by a collection of input, hidden, and output neural layers, where each input is the state of a hydrological storage (e.g. groundwater storage influencing seepage) or other variable (e.g. air temperature influencing snow melt). These components are interconnected to form a single neural network of the complete dynamical system considered, where all storages and flows are explicitly quantified. If physical understanding of a flow and its parameterization is available, a known formulation can be used as a replacement of a neural network component. The DSNN is applied forward in time, backpropagating gradients over all timesteps. It can be run in spatially lumped or semi-distributed mode. To demonstrate the approach, a DSNN is presented of the Austrian Dorfertal (Kals) Alpine catchment containing snow and subsurface water storages and associated flows including streamflow. The DSNN is trained, validated, and tested on daily streamflow over ~40 years. To explore the capability of the DSNN in estimating the magnitude and dynamics of internal system storages (snow water equivalent, subsurface water storage) and flows (evapotranspiration, sublimation, snowmelt, seepage), the DSNN is first trained and tested with streamflow data generated by a conceptual model. The DSNN turns out to be capable of reproducing - with a satisfactory level of precision - the system states and fluxes calculated by the conceptual model, with decreasing performance when measurement error is added to the artificially generated streamflow data before training. To explore its predictive performance, the DSNN is applied on measured streamflow data for the Dorfertal, comparing multiple DSNN setups that represent all flows as neural network components or only a subset of flows where remaining flows are represented with a standard conceptual model (e.g. linear reservoir). Preliminary results indicate that in predictive performance, in most setups, the DSNN outperforms a standard conceptual model trained on the same streamflow data, with NSE values for testing of 0.74 and 0.71, respectively. This preliminary result indicates DSNN to be a promising approach for blending process-based and neural network based modelling as well as for training (i.e. calibration) of neural network models on measurements of multiple hydrological variables as these are all explicitly represented by the DSNN and can thus be incorporated in the loss function (e.g. streamflow, snow depth, groundwater, evapotranspiration).