Learning Neural PDE Solvers with Convergence Guarantees

TL;DR

This paper introduces a neural network-based method to learn domain-specific PDE solvers that maintain convergence guarantees and outperform traditional solvers in speed.

Contribution

It presents a novel approach to adapt existing PDE solvers with neural networks while ensuring convergence and correctness.

Findings

Achieves 2-3 times speedup over state-of-the-art solvers.

Generalizes to various geometries and boundary conditions.

Maintains strong convergence guarantees.

Abstract

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be sub-optimal for specific classes of problems. In contrast to existing hand-crafted solutions, we propose an approach to learn a fast iterative solver tailored to a specific domain. We achieve this goal by learning to modify the updates of an existing solver using a deep neural network. Crucially, our approach is proven to preserve strong correctness and convergence guarantees. After training on a single geometry, our model generalizes to a wide variety of geometries and boundary conditions, and achieves 2-3 times speedup compared to state-of-the-art solvers.