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

2022/03/16 by Lienen, Marten, Günnemann, Stephan · 2 citations
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2203.08852

Abstract

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.

Cited by

• Data-free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs
• Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamics

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