Space-Time Continuous PDE Forecasting using Equivariant Neural Fields
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911912375091200 |
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| author | Knigge, David M. Wessels, David R. Valperga, Riccardo Papa, Samuele Sonke, Jan-Jakob Gavves, Efstratios Bekkers, Erik J. |
| author_facet | Knigge, David M. Wessels, David R. Valperga, Riccardo Papa, Samuele Sonke, Jan-Jakob Gavves, Efstratios Bekkers, Erik J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_06660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Space-Time Continuous PDE Forecasting using Equivariant Neural Fields Knigge, David M. Wessels, David R. Valperga, Riccardo Papa, Samuele Sonke, Jan-Jakob Gavves, Efstratios Bekkers, Erik J. Machine Learning Artificial Intelligence Neural and Evolutionary Computing 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. |
| title | Space-Time Continuous PDE Forecasting using Equivariant Neural Fields |
| topic | Machine Learning Artificial Intelligence Neural and Evolutionary Computing |
| url | https://arxiv.org/abs/2406.06660 |