Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems

IF 3 3区 工程技术 Q2 ENGINEERING, MECHANICAL Probabilistic Engineering Mechanics Pub Date : 2024-03-11 DOI:10.1016/j.probengmech.2024.103594
Xi Wang , Siyuan Xing , Jun Jiang , Ling Hong , Jian-Qiao Sun
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Abstract

This paper extends the recently developed method of separable Gaussian neural networks (SGNN) to obtain solutions of the Fokker–Planck–Kolmogorov (FPK) equation in high-dimensional state space. Several challenges when extending SGNN to high-dimensional state space are addressed including proper definition of domain for placing Gaussian neurons and region for data sampling, and numerical integration issue of evaluating marginal probability density functions. Three benchmark nonlinear dynamic systems with increasing complexity and dimension are examined with the SGNN method. In particular, the steady-state probability density of the response is obtained with the SGNN method and compared with the results of extensive Monte Carlo simulations. It should be pointed out that some solutions of high-dimensional FPK equations for nonlinear dynamic systems would be very difficult to obtain without SGNN.

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用于高维非线性随机系统的可分离高斯神经网络
本文扩展了最近开发的可分离高斯神经网络(SGNN)方法,以获得高维状态空间中福克-普朗克-科尔莫戈罗夫(FPK)方程的解。本文探讨了将 SGNN 扩展到高维状态空间时面临的几个挑战,包括高斯神经元放置域和数据采样区域的正确定义,以及评估边际概率密度函数的数值积分问题。利用 SGNN 方法研究了复杂度和维度不断增加的三个基准非线性动态系统。特别是,利用 SGNN 方法获得了响应的稳态概率密度,并与大量蒙特卡罗模拟的结果进行了比较。需要指出的是,如果没有 SGNN,某些非线性动态系统的高维 FPK 方程解将很难获得。
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来源期刊
Probabilistic Engineering Mechanics
Probabilistic Engineering Mechanics 工程技术-工程:机械
CiteScore
3.80
自引率
15.40%
发文量
98
审稿时长
13.5 months
期刊介绍: This journal provides a forum for scholarly work dealing primarily with probabilistic and statistical approaches to contemporary solid/structural and fluid mechanics problems encountered in diverse technical disciplines such as aerospace, civil, marine, mechanical, and nuclear engineering. The journal aims to maintain a healthy balance between general solution techniques and problem-specific results, encouraging a fruitful exchange of ideas among disparate engineering specialities.
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