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

Physics-informed neural networks (PINNs) have recently emerged and succeeded in various PDEs problems with their mesh-free properties, flexibility, and unsupervised training. However, their slower convergence speed and relatively inaccurate solutions often limit their broader applicability. 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. The various challenging PDE experiments show that the original PINNs have struggled and that PIXEL achieves fast convergence speed and high accuracy.