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

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 PDEs 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.

A Multi-Scale Similarity Based Community Detection for Attributed Complex Networks

“…Another type of methods focuses on mapping the input complex signal to another one which is composed of lowfrequency components [30], [36], [45], [97], [99], [105], thus the original signal could be well learned. This mapping function is often implemented by introducing learnable hash tables between the input coordinate and the subsequent neural network, such as the single scale full-resolution hash table used in DINER [105], multi-scale pyramid hash tables in InstantNGP [45] and multiple shifting hash tables in PIXEL [97]. These methods achieve high performance for representing complex signals at the cost of losing ability for interpolation, often requiring additional regularizations [98].…”