Space-Time Continuous PDE Forecasting using Equivariant Neural Fields
<|reference_start|>Space-Time Continuous PDE Forecasting using Equivariant Neural Fields: 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.<|reference_end|>
@article{knigge2024space-time,
    title={Space-Time Continuous PDE Forecasting using Equivariant Neural Fields},
    author={David M. Knigge, David R. Wessels, Riccardo Valperga, Samuele Papa,
  Jan-Jakob Sonke, Efstratios Gavves, Erik J. Bekkers},
    journal={arXiv preprint arXiv:2406.06660},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.06660},
    primaryClass={cs.LG cs.AI cs.NE}
}