Space-Time Continuous PDE Forecasting using Equivariant Neural Fields

David M Knigge, David Wessels, Riccardo Valperga, Samuele Papa, Jan-Jakob Sonke, Erik J Bekkers, Stratis Gavves

Keywords: pde solving, neural fields, equivariance, attention

TL;DR: We introduce an equivariant continuous PDE solving method based on Equivariant Neural Fields that preserves boundary conditions and known symmetries of the PDE.

Abstract: 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 -- such as 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 of the Conditional NeF - preserves 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. Furthermore, we validate 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.

Primary Area: Machine learning for physical sciences (for example: climate, physics)

Submission Number: 7270