基于深度学习的参数化Fokker-Planck方程的伪解析概率解

IF 8 2区 计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS
Xiaolong Wang , Jing Feng , Gege Wang , Tong Li , Yong Xu
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引用次数: 0

摘要

有效求解福克-普朗克方程(FPEs)对于理解动力系统中随机粒子的概率演化至关重要,然而,解析解只能在特定情况下实现。为了加快具有多个系统参数的参数化fpe的求解过程,我们引入了一种基于深度学习的伪解析概率解(PAPS)的求解方法。与以往需要为每组系统参数单独求解fpe的数值方法不同,PAPS在单个训练阶段同时处理预定义的连续系统参数范围内的所有fpe。该方法利用高斯混合分布(GMD)来表示平稳概率密度函数,即fpe的解。通过利用深度残差网络,将系统的每个参数配置映射到GMD的参数,确保高斯分量的权重、均值和方差自适应地与相应的真密度函数对齐。为了有效地训练残差网络,进一步提出了一种无网格算法,得到了符合非负性、归一化和边界条件的可行PAPS。大量的数值研究验证了该方法的准确性和有效性。该方法为FPEs的伪解析解提供了新的见解,并有望显著加快多参数、多维随机非线性系统的响应分析。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The pseudo-analytical probability solution to parameterized Fokker–Planck equations via deep learning

The pseudo-analytical probability solution to parameterized Fokker–Planck equations via deep learning
Efficiently solving Fokker–Planck equations (FPEs) is crucial for understanding the probabilistic evolution of stochastic particles in dynamical systems, however, analytical solutions are only attainable in specific cases. To speed up the solving process of parameterized FPEs with several system parameters, we introduce a deep learning-based method to obtain the pseudo-analytical probability solution (PAPS). Unlike previous numerical methodologies that necessitate solving the FPEs separately for each set of system parameters, the PAPS simultaneously addresses all FPEs within a predefined continuous range of system parameters during a single training phase. The approach utilizes a Gaussian mixture distribution (GMD) to represent the stationary probability density functions, namely, the solutions to FPEs. By leveraging a deep residual network, each parameter configuration of the system is mapped to the parameters of the GMD, ensuring that the weights, means, and variances of Gaussian components adaptively align with the corresponding true density functions. A grid-free algorithm is further developed to effectively train the residual network, resulting in a feasible PAPS obeying nonnegativity, normalization and boundary conditions. Extensive numerical studies validate the accuracy and efficiency of our method. This approach presents new insight to the pseudo-analytical solutions to FPEs, and promises significant acceleration in the response analysis of multi-parameter, multi-dimensional stochastic nonlinear systems.
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来源期刊
Engineering Applications of Artificial Intelligence
Engineering Applications of Artificial Intelligence 工程技术-工程:电子与电气
CiteScore
9.60
自引率
10.00%
发文量
505
审稿时长
68 days
期刊介绍: Artificial Intelligence (AI) is pivotal in driving the fourth industrial revolution, witnessing remarkable advancements across various machine learning methodologies. AI techniques have become indispensable tools for practicing engineers, enabling them to tackle previously insurmountable challenges. Engineering Applications of Artificial Intelligence serves as a global platform for the swift dissemination of research elucidating the practical application of AI methods across all engineering disciplines. Submitted papers are expected to present novel aspects of AI utilized in real-world engineering applications, validated using publicly available datasets to ensure the replicability of research outcomes. Join us in exploring the transformative potential of AI in engineering.
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