在线弱形稀疏的部分微分方程的识别.
Daniel A Messenger1, Emiliano Dall'anese2, David M Bortz1
1Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526.
概括
本研究介绍了一种在线算法,用于识别部分微分方程 (PDEs),使用弱形式稀疏识别非线性动力学 (WSINDy) 方法. 该方法有效地处理顺序数据,以实时识别系统,即使具有时间变化的系数.
科学领域:
- 计算数学 计算数学 计算数学
- 应用数学 应用数学 应用数学
- 动态系统 动态系统
背景情况:
- 从数据中识别部分微分方程 (PDEs) 对于建模复杂现象至关重要.
- 传统方法通常需要完整的数据集,并与顺序或流数据作斗争.
- 非线性动力学稀疏识别 (SINDy) 算法为发现治理方程提供了一个强大的框架.
研究的目的:
- 开发一个在线算法,从顺序到达的数据快照中识别PDEs.
- 适应非线性动态的弱形式稀疏识别 (WSINDy) 算法用于流式识别任务.
- 为了能够实时跟踪具有时间变化的系数的系统.
主要方法:
- 拟议的方法结合了候选PDE的弱形式离散和在线近接梯度下降方法.
- 它使用了硬值策略 (L0伪规范的近接运算符) 以对杂的顺序数据进行高效的稀疏回归.
- 算法在到达时处理解决方案快照,从而实现连续的系统识别.
主要成果:
- 成功确定了Kuramoto-Sivashinsky方程的PDEs,与时变波速的非线性波方程以及线性波方程.
- 在一个,两个和三个空间维度中表现出能力.
- 展示了系统的有效识别和跟踪,其系数突然变化.
结论:
- 在线WSINDy算法为PDE识别提供了一个高效和强大的流媒体替代方案.
- 该方法适用于高维问题和具有动态参数变化的系统.
- 这种方法推进了基于数据的微分方程发现领域.
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