Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations
This work addresses a foundational limitation in PDE learning for scientific computing, enabling more flexible and generalizable models, though it is incremental in applying existing formalism to a new context.
The authors tackled the problem of machine learning methods for PDE identification being tied to specific spatial dimensions and coordinates, which limits generalization; they introduced a coordinate- and dimension-independent approach using exterior calculus, demonstrating accurate predictions across different spaces, dimensions, and conditions in models like FitzHugh-Nagumo and Patlak-Keller-Segel.
The machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a given number of spatial dimensions and a choice of coordinates the data have been collected in. This dependence prevents the learned evolution 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 call ``spatially liberated" PDE learning. To this end, we employ a machine learning approach to predict the evolution of scalar field systems expressed in the formalism of exterior calculus, which is coordinate-free and immediately generalizes to arbitrary dimensions by construction. We demonstrate the performance of this approach in the FitzHugh-Nagumo and Barkley reaction-diffusion models, as well as the Patlak-Keller-Segel model informed by in-situ chemotactic bacteria observations. We provide extensive numerical experiments that demonstrate that our approach allows for seamless transitions across various spatial contexts. We show that the field dynamics learned in one space can be used to make accurate predictions in other spaces with different dimensions, coordinate systems, boundary conditions, and curvatures.