新论文 | 参数化多重 Poisson 白噪声激励下高维非线性动力系统的随机响应分析

Probabilistic response determination of high-dimensional nonlinear dynamical systems enforced by parametric multiple Poisson white noises

研究背景

然而，对于路径非连续的高维问题，例如 Poisson 激励下的高维系统，其瞬时概率密度函数仍然难以确定。现有的降维概率密度演化方程理论框架无法给出其控制偏微分方程的有限阶截断，而高阶偏微分项的影响难以作为小量被忽略。在此背景下，本文针对 Poisson 白噪声激励下的高维系统，推导了其任意感兴趣量瞬时概率密度函数的一维偏微分-积分控制方程，将原本难以有效截断无穷阶偏微分项转化为积分项，从而巧妙避免了概率密度控制方程难以精确截断的问题，并证明了这一控制方程 (文中仍称为降维概率密度演化方程) 是精确成立的，保证了有效数值求解的精度与效率。

方法概述

不失一般性，Poisson 白噪声激励下的高维随机动力系统的运动方程可以写为

其中系统遭受的多重参数化 Poisson 激励可以表达为

经推导可以给出，系统中任意感兴趣响应过程的瞬时概率密度函数满足的控制方程为

这是一个一维偏微分-积分方程，文中称为降维概率密度演化方程。一旦确定了方程中的本征漂移函数与本征发生率函数，方程就很容易数值求解。具体推导过程可详见论文。

算例 1: Poisson 白噪声下的多维 Ornstein-Uhlenbeck 过程

Poisson 白噪声驱动下 Ornstein-Uhlenbeck 系统的瞬时概率分布

Poisson 白噪声驱动下 Ornstein-Uhlenbeck 系统的标准差与峰度

Poisson 白噪声驱动下 Ornstein-Uhlenbeck 系统的典型样本路径

算例 2: 带 SOS 通路的基因开关

蛋白质 B 的瞬时概率分布

蛋白质 B 的前四阶矩

本征发生率函数与蛋白质 A 与 B 的典型样本路径

算例 3: 陆地水平衡系统

土壤相对湿度的瞬时概率分布

土壤相对湿度的前四阶矩

土壤相对湿度的本征漂移函数与本征发生率函数

求解降维概率密度演化方程的挑战之一在于获得本征漂移函数和本征发生率函数的表达。对于 Poisson 白噪声驱动下的一般高维非线性系统，其通常难以解析获得。然而可证明，本征漂移函数和本征发生率函数可以通过条件期望函数表达，从而通过确定性分析数据进行识别。结合根据数据数值识别出的本征漂移函数和本征发生率函数表达，数值求解降维概率密度演化方程，可以较高精度获得感兴趣量瞬时概率密度函数的数值解。

总之，本研究推动了参数化 Poisson 白噪声激励下高维非线性随机动力系统的进展。所推导的降维概率密度演化方程及其相关数值算法为精确高效地预测某一感兴趣量的瞬时概率密度函数提供了统一框架。通过克服以往方法的局限性，所提方法为研究复杂随机系统开辟了新途径，并有潜力推动多个科学和工程学科的进一步发展。

DOI:
10.1007/s11071-024-09592-x

陈建兵 Jian-Bing Chen
同济大学土木工程学院教授, 国家杰出青年科学基金获得者

邮箱：chenjb@tongji.edu.cn

https://jcersm.tongji.edu.cn/1e/f9/c13092a139001/page.htm

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