Physics-informed neural network (PINN) is a data-driven solver for partial
and ordinary differential equations(ODEs/PDEs). It provides a unified framework
to address both forward and inverse problems. However, the complexity of the
objective function often leads to training failures. This issue is particularly
prominent when solving high-frequency and multi-scale problems. We proposed
using transfer learning to boost the robustness and convergence of training
PINN, starting training from low-frequency problems and gradually approaching
high-frequency problems. Through two case studies, we discovered that transfer
learning can effectively train PINN to approximate solutions from low-frequency
problems to high-frequency problems without increasing network parameters.
Furthermore, it requires fewer data points and less training time. We
elaborately described our training strategy, including optimizer selection, and
suggested guidelines for using transfer learning to train neural networks for
solving more complex problems.

提供了使用转移学习来增强 PINN 的鲁棒性和收敛性的训练方法，通过两个案例研究发现转移学习可以有效训练 PINN 在低频问题到高频问题的近似解，同时减少了网络参数，所需数据点和训练时间。同时提供了优化器选择和使用转移学习解决更复杂问题的指南。

基于物理知识的神经网络在高频和多尺度问题上的应用及迁移学习

Physics-Informed Neural Networks for High-Frequency and Multi-Scale  Problems using Transfer Learning

Physics-informed neural networks (PINN) combine deep neural networks with the
solution of partial differential equations (PDEs), creating a new and promising
research area for numerically solving PDEs. Faced with a class of multi-scale
problems that include loss terms of different orders of magnitude in the loss
function, it is challenging for standard PINN methods to obtain an available
prediction. In this paper, we propose a new framework for solving multi-scale
problems by reconstructing the loss function. The framework is based on the
standard PINN method, and it modifies the loss function of the standard PINN
method by applying different numbers of power operations to the loss terms of
different magnitudes, so that the individual loss terms composing the loss
function have approximately the same order of magnitude among themselves. In
addition, we give a grouping regularization strategy, and this strategy can
deal well with the problem which varies significantly in different subdomains.
The proposed method enables loss terms with different magnitudes to be
optimized simultaneously, and it advances the application of PINN for
multi-scale problems.

使用物理信息神经网络（PINN）来解决具有多尺度问题的新框架，通过重新构建损失函数并应用不同数量的幂运算到损失项上以使损失函数中的项具有相近的数量级，并且引入分组正则化策略以解决不同子域中变化显著的问题，该方法使得具有不同数量级的损失项可以同时优化，推动了 PINN 在多尺度问题中的应用。

基于信息受控神经网络的多尺度深度学习框架

A deep learning framework for multi-scale models based on  physics-informed neural networks

While significant progress has been made on Physics-Informed Neural Networks
(PINNs), a comprehensive comparison of these methods across a wide range of
Partial Differential Equations (PDEs) is still lacking. This study introduces
PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a
diverse dataset, comprising over 20 distinct PDEs from various domains
including heat conduction, fluid dynamics, biology, and electromagnetics. These
PDEs encapsulate key challenges inherent to real-world problems, such as
complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality.
PINNacle also offers a user-friendly toolbox, incorporating about 10
state-of-the-art PINN methods for systematic evaluation and comparison. We have
conducted extensive experiments with these methods, offering insights into
their strengths and weaknesses. In addition to providing a standardized means
of assessing performance, PINNacle also offers an in-depth analysis to guide
future research, particularly in areas such as domain decomposition methods and
loss reweighting for handling multi-scale problems and complex geometry. While
PINNacle does not guarantee success in all real-world scenarios, it represents
a significant contribution to the field by offering a robust, diverse, and
comprehensive benchmark suite that will undoubtedly foster further research and
development in PINNs.

该研究介绍了一种名为 PINNacle 的基准测试工具，用于评估物理学为基础的神经网络（PINNs）在各种领域中处理各种偏微分方程的性能，该工具包括大约 10 种最先进的 PINN 方法，以实现系统评估和比较。