MAgNet: Mesh Agnostic Neural PDE Solver

• URL: http://arxiv.org/abs/2210.05495v1
• Date: Tue, 11 Oct 2022 14:52:20 GMT
• Title: MAgNet: Mesh Agnostic Neural PDE Solver
• Authors: Oussama Boussif, Dan Assouline, Loubna Benabbou, Yoshua Bengio
• Abstract summary: Climate predictions require fine-temporal resolutions to resolve all turbulent scales in the fluid simulations. Current numerical model solveers PDEs on grids that are too coarse (3km to 200km on each side) We design a novel architecture that predicts the spatially continuous solution of a PDE given a spatial position query.
• Score: 68.8204255655161
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The computational complexity of classical numerical methods for solving Partial Differential Equations (PDE) scales significantly as the resolution increases. As an important example, climate predictions require fine spatio-temporal resolutions to resolve all turbulent scales in the fluid simulations. This makes the task of accurately resolving these scales computationally out of reach even with modern supercomputers. As a result, current numerical modelers solve PDEs on grids that are too coarse (3km to 200km on each side), which hinders the accuracy and usefulness of the predictions. In this paper, we leverage the recent advances in Implicit Neural Representations (INR) to design a novel architecture that predicts the spatially continuous solution of a PDE given a spatial position query. By augmenting coordinate-based architectures with Graph Neural Networks (GNN), we enable zero-shot generalization to new non-uniform meshes and long-term predictions up to 250 frames ahead that are physically consistent. Our Mesh Agnostic Neural PDE Solver (MAgNet) is able to make accurate predictions across a variety of PDE simulation datasets and compares favorably with existing baselines. Moreover, MAgNet generalizes well to different meshes and resolutions up to four times those trained on.

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