@inproceedings{
knigge2024spacetime,
title={Space-Time Continuous {PDE} Forecasting using Equivariant Neural Fields},
author={David M Knigge and David Wessels and Riccardo Valperga and Samuele Papa and Jan-Jakob Sonke and Erik J Bekkers and Stratis Gavves},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=wN5AgP0DJ0}
}
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.
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