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We propose a new method for spatio-temporal forecasting on arbitrarilydistributed points. Assuming that the observed system follows an unknownpartial differential equation, we derive a continuous-time model for thedynamics of the data via the finite element method. The resulting graph neuralnetwork estimates the instantaneous effects of the unknown dynamics on eachcell in a meshing of the spatial domain. Our model can incorporate priorknowledge via assumptions on the form of the unknown PDE, which induce astructural bias towards learning specific processes. Through this mechanism, wederive a transport variant of our model from the convection equation and showthat it improves the transfer performance to higher-resolution meshes on seasurface temperature and gas flow forecasting against baseline modelsrepresenting a selection of spatio-temporal forecasting methods. A qualitativeanalysis shows that our model disentangles the data dynamics into theirconstituent parts, which makes it uniquely interpretable.

Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks