MAgNet: Mesh Agnostic Neural PDE Solver

TL;DR

MAgNet is a novel neural PDE solver that leverages implicit neural representations and graph neural networks to achieve mesh-agnostic, accurate, and physically consistent predictions across various resolutions and meshes.

Contribution

The paper introduces MAgNet, a mesh-agnostic neural PDE solver that generalizes across different meshes and resolutions using INR and GNN architectures.

Findings

MAgNet achieves accurate PDE predictions on various datasets.

It generalizes well to different meshes and resolutions.

MAgNet outperforms existing baselines in accuracy and generalization.

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.