Data and code underlying the publication: Neural oscillators for generalization of physics-informed machine learning

### Research Objective<br>The primary objective of this research is to enhance the generalization of physics-informed machine learning (PIML) models by integrating them with neural oscillators. The goal is to improve the accuracy of these models in predicting solutions to partial differential equations (PDEs) beyond the training domain.<br>### Type of Research<br>This research is applied and experimental. It focuses on developing and validating a new methodological approach to enhance the generalization capabilities of PIML models through a series of numerical experiments on various nonlinear and high order PDEs.<br>### Method of Data Collection<br>The data used for validating numerical experiments are closed form analytic solution and physics-informed method is utilized to simulate the dataset. Both are explicitly mention in the python notebooks. The experiments are conducted on time-dependent nonlinear PDEs, including the viscous Burgers equation, Allen-Cahn equation, nonlinear Schrödinger equation, Euler-Bernoulli beam equation, and a 2D Kovasznay flow.<br>### Type of Data/codes<br>1. All implementation are done using jupyter notebooks (.ipynb) or .py.2. .mat files are analytical solution generated using PINN simulation.3. (.jpeg), (.pdf) are figures which are used in the main manuscript.<br>

### 研究目标
本研究的核心目标为将物理知情机器学习（physics-informed machine learning, PIML）模型与神经振荡器（neural oscillators）相结合，旨在提升其泛化能力，进而优化此类模型在训练定义域之外预测偏微分方程（partial differential equations, PDEs）解的精度。
### 研究类型
本研究属于应用实验类研究，聚焦于开发并验证一种全新的方法论框架，通过针对各类非线性高阶偏微分方程开展一系列数值实验，增强PIML模型的泛化能力。
### 数据采集方法
本研究用于数值实验验证的数据包含闭式解析解与通过物理知情方法生成的仿真数据集，上述两类数据的相关细节均已在Python代码笔记本中明确说明。本次实验选取含时非线性偏微分方程作为测试对象，具体包括粘性伯格斯方程（viscous Burgers equation）、艾伦-钱方程（Allen-Cahn equation）、非线性薛定谔方程（nonlinear Schrödinger equation）、欧拉-伯努利梁方程（Euler-Bernoulli beam equation）以及二维科兹奈流（2D Kovasznay flow）。
### 数据与代码类型
1. 所有实现均通过Jupyter笔记本（.ipynb）或Python脚本（.py）完成。
2. .mat格式文件为基于物理知情神经网络（physics-informed neural networks, PINN）仿真生成的解析解。
3. .jpeg与.pdf格式文件为本研究主文稿中使用的可视化图表。