Xiaolong Wang , Jing Feng , Gege Wang , Tong Li , Yong Xu
{"title":"基于深度学习的参数化Fokker-Planck方程的伪解析概率解","authors":"Xiaolong Wang , Jing Feng , Gege Wang , Tong Li , Yong Xu","doi":"10.1016/j.engappai.2025.111344","DOIUrl":null,"url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":50523,"journal":{"name":"Engineering Applications of Artificial Intelligence","volume":"157 ","pages":"Article 111344"},"PeriodicalIF":8.0000,"publicationDate":"2025-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The pseudo-analytical probability solution to parameterized Fokker–Planck equations via deep learning\",\"authors\":\"Xiaolong Wang , Jing Feng , Gege Wang , Tong Li , Yong Xu\",\"doi\":\"10.1016/j.engappai.2025.111344\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>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.</div></div>\",\"PeriodicalId\":50523,\"journal\":{\"name\":\"Engineering Applications of Artificial Intelligence\",\"volume\":\"157 \",\"pages\":\"Article 111344\"},\"PeriodicalIF\":8.0000,\"publicationDate\":\"2025-06-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Engineering Applications of Artificial Intelligence\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0952197625013466\",\"RegionNum\":2,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"AUTOMATION & CONTROL SYSTEMS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering Applications of Artificial Intelligence","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0952197625013466","RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
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.
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.