TaylorPDENet: Learning PDEs from non-grid Data
Paul Heinisch Andrzej Dulny Anna Krause Andreas Hotho
Modeling data obtained from dynamical systems has gained attention in recent
years as a challenging task for machine learning models. Previous approaches
assume the measurements to be distributed on a grid. However, for real-world
applications like weather prediction, the observations are taken from arbitrary
locations within the spatial domain. In this paper, we propose TaylorPDENet - a
novel machine learning method that is designed to overcome this challenge. Our
algorithm uses the multidimensional Taylor expansion of a dynamical system at
each observation point to estimate the spatial derivatives to perform
predictions. TaylorPDENet is able to accomplish two objectives simultaneously:
accurately forecast the evolution of a complex dynamical system and explicitly
reconstruct the underlying differential equation describing the system. We
evaluate our model on a variety of advection-diffusion equations with different
parameters and show that it performs similarly to equivalent approaches on
grid-structured data while being able to process unstructured data as well.