Title: Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations
Primary area: learning on graphs and other geometries & topologies
Keywords: Geometric Deep Learning, PDE Identification, Any-dimensional Learning

Abstract: Machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a fixed number of spatial dimensions and a particular choice of coordinates in which the data have been collected. This dependence prevents the learned equation from generalizing to other spaces. In this work, we reformulate the problem in terms of coordinate- and dimension-independent representations, paving the way toward what we might call “spatially liberated" PDE learning. In this work, we propose an approach to learning differential equations by expressing them in a way that is independent of the coordinate system and even the underlying manifold where the equation is defined. This allows us to learn a differential equation in low-dimensional spaces and generalize to higher-dimensional spaces with different geometric properties. We provide extensive numerical experiments that demonstrate that our approach allows for seamless transitions across various geometric contexts. We show that the dynamics learned in one space can be used to make accurate predictions in other spaces with different dimensions, coordinate systems, boundary conditions, and curvatures.