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

• URL: http://arxiv.org/abs/2203.08852v1
• Date: Wed, 16 Mar 2022 18:19:43 GMT
• Title: Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks
• Authors: Marten Lienen, Stephan G\"unnemann
• Abstract summary: 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. A qualitative analysis shows that our model disentangles the data into their constituent parts, which makes it uniquely interpretable.
• Score: 2.538209532048867
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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