对于依赖时间的随机局部微分方程的非侵入性减少顺序模型,使用动态模式分解和多项式混乱扩展
Shuman Wang1, Afshan Batool2, Xiang Sun3
1Department of Mathematics, Shanghai University, Shanghai 200444, China.
Chaos (Woodbury, N.Y.)
|July 1, 2024
概括
我们介绍了一种结合正确直角分解,动态模式分解和多项式混乱扩展 (POD-DMD-PCE) 的新方法,以有效地解决复杂的随机局部微分方程 (SPDEs). 这种方法准确地预测了超出培训数据的解决方案.
科学领域:
- 计算数学 计算数学 计算数学
- 数字分析 数字分析
- 科学计算科学计算
背景情况:
- 随机局部微分方程 (SPDEs) 对于模拟具有固有随机性的复杂系统至关重要.
- 解决时间依赖的SPDEs通常需要计算密集的高保真模拟.
- 减少顺序建模 (ROM) 提供了一个高效的替代方案,通过用低维表示方法近似解决方案.
研究的目的:
- 为时间依赖的随机局部微分方程 (SPDEs) 开发一种新的,非侵入性的减少顺序模型 (ROM).
- 通过在培训时间间隔之外提供准确的解决方案来增强预测能力.
- 在各种非线性SPDEs上验证拟议方法的效率和推断能力.
主要方法:
- 适当的直角分解 (POD) 来从高保真度解决方案中提取主导空间模式.
- 动态模式分解 (DMD) 应用于垂直堆叠的投影系数,用于预测未来的系数场.
- 多项式混沌扩展 (PCE) 建立随机参数和DMD预测系数场之间的映射,形成POD-DMD-PCE方法.
主要成果:
- POD-DMD-PCE方法显示了高效的模型缩小和强大的时间推算能力.
- 该技术成功地在训练时间间隔内和超越训练时间间隔内恢复现场解决方案.
- 对各种非线性SPDEs进行了验证,包括具有多个参数的反应-扩散,热和汉堡方程.
结论:
- 拟议的POD-DMD-PCE方法为解决时间依赖的SPDEs提供了一种计算效率高,准确的方法.
- POD,DMD和PCE的新集成,特别是系数的垂直堆叠,提高了预测准确性和推断.
- 该方法为分析由随机局部微分方程控制的复杂系统提供了有价值的工具.
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