Space-Time Continuous PDE Forecasting using Equivariant Neural Fields
This work addresses the challenge of imposing constraints like symmetries in PDE forecasting for researchers in computational physics and machine learning, representing an incremental improvement over existing neural field methods.
The paper tackles the problem of forecasting partial differential equations (PDEs) with space-time continuous models by proposing a framework that respects known symmetries, improving generalization and data-efficiency. It shows that this approach generalizes to unseen spatial and temporal locations and geometric transformations, outperforming baselines in challenging geometries.
Recently, Conditional Neural Fields (NeFs) have emerged as a powerful modelling paradigm for PDEs, by learning solutions as flows in the latent space of the Conditional NeF. Although benefiting from favourable properties of NeFs such as grid-agnosticity and space-time-continuous dynamics modelling, this approach limits the ability to impose known constraints of the PDE on the solutions -- e.g. symmetries or boundary conditions -- in favour of modelling flexibility. Instead, we propose a space-time continuous NeF-based solving framework that - by preserving geometric information in the latent space - respects known symmetries of the PDE. We show that modelling solutions as flows of pointclouds over the group of interest $G$ improves generalization and data-efficiency. We validated that our framework readily generalizes to unseen spatial and temporal locations, as well as geometric transformations of the initial conditions - where other NeF-based PDE forecasting methods fail - and improve over baselines in a number of challenging geometries.