{"title":"基于分数的同质朗道方程粒子法","authors":"Yan Huang, Li Wang","doi":"arxiv-2405.05187","DOIUrl":null,"url":null,"abstract":"We propose a novel score-based particle method for solving the Landau\nequation in plasmas, that seamlessly integrates learning with\nstructure-preserving particle methods [arXiv:1910.03080]. Building upon the\nLagrangian viewpoint of the Landau equation, a central challenge stems from the\nnonlinear dependence of the velocity field on the density. Our primary\ninnovation lies in recognizing that this nonlinearity is in the form of the\nscore function, which can be approximated dynamically via techniques from\nscore-matching. The resulting method inherits the conservation properties of\nthe deterministic particle method while sidestepping the necessity for kernel\ndensity estimation in [arXiv:1910.03080]. This streamlines computation and\nenhances scalability with dimensionality. Furthermore, we provide a theoretical\nestimate by demonstrating that the KL divergence between our approximation and\nthe true solution can be effectively controlled by the score-matching loss.\nAdditionally, by adopting the flow map viewpoint, we derive an update formula\nfor exact density computation. Extensive examples have been provided to show\nthe efficiency of the method, including a physically relevant case of Coulomb\ninteraction.","PeriodicalId":501061,"journal":{"name":"arXiv - CS - Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A score-based particle method for homogeneous Landau equation\",\"authors\":\"Yan Huang, Li Wang\",\"doi\":\"arxiv-2405.05187\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We propose a novel score-based particle method for solving the Landau\\nequation in plasmas, that seamlessly integrates learning with\\nstructure-preserving particle methods [arXiv:1910.03080]. Building upon the\\nLagrangian viewpoint of the Landau equation, a central challenge stems from the\\nnonlinear dependence of the velocity field on the density. Our primary\\ninnovation lies in recognizing that this nonlinearity is in the form of the\\nscore function, which can be approximated dynamically via techniques from\\nscore-matching. The resulting method inherits the conservation properties of\\nthe deterministic particle method while sidestepping the necessity for kernel\\ndensity estimation in [arXiv:1910.03080]. This streamlines computation and\\nenhances scalability with dimensionality. Furthermore, we provide a theoretical\\nestimate by demonstrating that the KL divergence between our approximation and\\nthe true solution can be effectively controlled by the score-matching loss.\\nAdditionally, by adopting the flow map viewpoint, we derive an update formula\\nfor exact density computation. Extensive examples have been provided to show\\nthe efficiency of the method, including a physically relevant case of Coulomb\\ninteraction.\",\"PeriodicalId\":501061,\"journal\":{\"name\":\"arXiv - CS - Numerical Analysis\",\"volume\":\"37 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2405.05187\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.05187","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A score-based particle method for homogeneous Landau equation
We propose a novel score-based particle method for solving the Landau
equation in plasmas, that seamlessly integrates learning with
structure-preserving particle methods [arXiv:1910.03080]. Building upon the
Lagrangian viewpoint of the Landau equation, a central challenge stems from the
nonlinear dependence of the velocity field on the density. Our primary
innovation lies in recognizing that this nonlinearity is in the form of the
score function, which can be approximated dynamically via techniques from
score-matching. The resulting method inherits the conservation properties of
the deterministic particle method while sidestepping the necessity for kernel
density estimation in [arXiv:1910.03080]. This streamlines computation and
enhances scalability with dimensionality. Furthermore, we provide a theoretical
estimate by demonstrating that the KL divergence between our approximation and
the true solution can be effectively controlled by the score-matching loss.
Additionally, by adopting the flow map viewpoint, we derive an update formula
for exact density computation. Extensive examples have been provided to show
the efficiency of the method, including a physically relevant case of Coulomb
interaction.