PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers

TL;DR

PIXEL introduces a novel physics-informed cell representation method that combines classical numerical techniques with machine learning to solve PDEs more accurately and efficiently than traditional PINNs.

Contribution

The paper presents PIXEL, a new PDE solver that improves convergence speed and accuracy by integrating grid-based structures with neural networks, overcoming PINNs' limitations.

Findings

PIXEL achieves faster convergence than PINNs.

PIXEL provides higher accuracy on challenging PDEs.

PIXEL effectively enforces PDE constraints with automatic differentiation.

Abstract

With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded in various forward and inverse PDE problems thanks to their excellent properties, such as flexibility, mesh-free solutions, and unsupervised training. However, their slower convergence speed and relatively inaccurate solutions often limit their broader applicability in many science and engineering domains. This paper proposes a new kind of data-driven PDEs solver, physics-informed cell representations (PIXEL), elegantly combining classical numerical methods and learning-based approaches. We adopt a grid structure from the numerical methods to improve accuracy and convergence speed and overcome the spectral bias presented in PINNs. Moreover, the proposed method enjoys the same benefits in PINNs, e.g., using the same optimization frameworks to solve both forward and inverse PDE problems and readily enforcing PDE constraints with modern automatic differentiation techniques. We provide experimental results on various challenging PDEs that the original PINNs have struggled with and show that PIXEL achieves fast convergence speed and high accuracy. Project page: https://namgyukang.github.io/PIXEL/