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: this https URL

提出一种将经典数值方法和基于学习的方法巧妙结合的数据驱动 PDE 求解器 ——Physics-informed cell representations (PIXEL)，以解决 PINNs 算法在科学和工程领域中应用广泛性受限的问题，并在多个具有挑战性的 PDE 问题上得到了快速收敛速度和高精度的实验结果。

PIXEL: 快速准确的 PDE 的基于物理学的单元表示法

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

Language models are defined over a finite set of inputs, which creates a
vocabulary bottleneck when we attempt to scale the number of supported
languages. Tackling this bottleneck results in a trade-off between what can be
represented in the embedding matrix and computational issues in the output
layer. This paper introduces PIXEL, the Pixel-based Encoder of Language, which
suffers from neither of these issues. PIXEL is a pretrained language model that
renders text as images, making it possible to transfer representations across
languages based on orthographic similarity or the co-activation of pixels.
PIXEL is trained to reconstruct the pixels of masked patches, instead of
predicting a distribution over tokens. We pretrain the 86M parameter PIXEL
model on the same English data as BERT and evaluate on syntactic and semantic
tasks in typologically diverse languages, including various non-Latin scripts.
We find that PIXEL substantially outperforms BERT on syntactic and semantic
processing tasks on scripts that are not found in the pretraining data, but
PIXEL is slightly weaker than BERT when working with Latin scripts.
Furthermore, we find that PIXEL is more robust to noisy text inputs than BERT,
further confirming the benefits of modelling language with pixels.

该文章介绍了 PIXEL，一种基于像素的预训练语言模型，通过将文本渲染为图像，PIXEL 可以处理不同语言，特别是非拉丁文字。实验发现 PIXEL 在语法和语义处理方面表现优异，且对于噪声文本的输入更为稳健。