通过深度学习从稀缺的数据中发现Lax对
Shuning Lin1, Yong Chen2,3
1School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science, Fudan University, Shanghai 200433, China.
Chaos (Woodbury, N.Y.)
|November 14, 2025
概括
这项研究介绍了Lax-Pair-FIND,这是一种用于从数据中发现Lax对的新算法. 它有效地从稀疏或杂的数据中识别线性进化运算子,在物理系统中推进方程发现.
科学领域:
- 计算物理 计算物理
- 数据科学数据科学数据科学
- 机器学习 机器学习
- 应用数学 应用数学 应用数学
背景情况:
- 数据科学和机器学习的重大进展使得在解决局部微分方程 (PDEs) 和发现物理系统方程方面取得了进展.
- 数据驱动的发现方法越来越多地用于揭示科学数据中的复杂数学关系.
研究的目的:
- 提出并验证一个新的数据驱动算法,Lax-Pair-FIND,用于发现Lax对.
- 使用稀疏或杂的数据和已知的光谱运算子 (L) 识别线性演算子 (A),而无需事先了解方程的形式.
主要方法:
- 构建一个候选操作员术语库,用于识别操作员A.
- 使用稀疏优化和其他技术来选择构成运算符A的关键词.
- 通过操作员组合和验证操作员方程与设计的测试函数来计算Lax兼容性残留值.
主要成果:
- 成功发现了 Lax 对向导,KdV,mKdV,Boussinesq-Burgers 和非线性施罗丁格方程的对.
- 通过数值模拟,在处理不同稀疏度和噪声水平的数据方面表现出卓越的有效性和稳定性.
- 验证了算法的识别新型可集成系统和现有系统的新的Lax对表示的能力.
结论:
- Lax-Pair-FIND算法提供了一个强大的计算框架,将深度学习与物理信息集成为Lax对发现.
- 这种数据驱动的方法显示出在发现可集成系统及其相关的Lax对领域的进步方面有很大的潜力.
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