Multivariate forecasting with physically connected time series

Develop a forecasting method that jointly predicts multiple groundwater-level time series while incorporating the physical connections among those series.

Background

The paper introduces Backcast Forecast Physics Informed RNN (BFcastPIRNN), a recurrent architecture that incorporates partially differentiable groundwater equations and estimates unobservable physical variables for individual time series. Its experiments focus on separate groundwater-level series associated with individual piezometers, although the studied hydrological systems may exhibit connections through shared or interacting physical processes.

The authors explicitly identify joint forecasting of multiple physically connected time series as a direction for future work. This remains unresolved in the paper: the proposed experiments do not establish how BFcastPIRNN or a related architecture should represent cross-series physical dependencies, share information among piezometers, or preserve the relevant equations in a multivariate forecast.