TaylorPDENet: Learning PDEs from non-grid Data

TL;DR

TaylorPDENet is a machine learning approach that accurately models dynamical systems from unstructured data by estimating derivatives via Taylor expansion, enabling both forecasting and PDE reconstruction.

Contribution

It introduces a novel method that handles non-grid data for PDE learning, extending capabilities beyond traditional grid-based approaches.

Findings

Performs comparably to grid-based methods on structured data

Successfully processes unstructured data for PDE modeling

Accurately reconstructs underlying differential equations

Abstract

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.