Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks
This work addresses forecasting challenges in physical systems like climate and fluid dynamics, offering an interpretable method, though it is incremental as it builds on existing graph neural network and PDE-based approaches.
The paper tackles the problem of spatio-temporal forecasting from sparse observations by assuming an unknown partial differential equation and using a finite element method to derive a continuous-time model, resulting in improved transfer performance to higher-resolution meshes on sea surface temperature and gas flow forecasting against baseline models.
We propose a new method for spatio-temporal forecasting on arbitrarily distributed points. Assuming that the observed system follows an unknown partial differential equation, we derive a continuous-time model for the dynamics of the data via the finite element method. The resulting graph neural network estimates the instantaneous effects of the unknown dynamics on each cell in a meshing of the spatial domain. Our model can incorporate prior knowledge via assumptions on the form of the unknown PDE, which induce a structural bias towards learning specific processes. Through this mechanism, we derive a transport variant of our model from the convection equation and show that it improves the transfer performance to higher-resolution meshes on sea surface temperature and gas flow forecasting against baseline models representing a selection of spatio-temporal forecasting methods. A qualitative analysis shows that our model disentangles the data dynamics into their constituent parts, which makes it uniquely interpretable.