Space-Time Continuous PDE Forecasting using Equivariant Neural Fields

• URL: http://arxiv.org/abs/2406.06660v1
• Date: Mon, 10 Jun 2024 11:49:11 GMT
• Title: Space-Time Continuous PDE Forecasting using Equivariant Neural Fields
• Authors: David M. Knigge, David R. Wessels, Riccardo Valperga, Samuele Papa, Jan-Jakob Sonke, Efstratios Gavves, Erik J. Bekkers,
• Abstract summary: Conditional Neural Fields (NeFs) have emerged as a powerful modelling paradigm for PDEs. We propose a space-time continuous NeF-based solving framework that 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.
• Score: 28.24612886348871
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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