Space-Time Continuous PDE Forecasting using Equivariant Neural Fields

TL;DR

This paper introduces a space-time continuous neural field framework for PDE forecasting that preserves geometric symmetries, leading to better generalization and data efficiency in modeling solutions across various geometries.

Contribution

It proposes a novel NeF-based framework that incorporates known PDE symmetries into the latent space, enhancing generalization and respecting physical constraints.

Findings

Improved generalization to unseen locations and transformations

Enhanced data efficiency in PDE solution modeling

Outperforms existing NeF-based methods on challenging geometries

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 -- 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.