{"title":"多折射 PDE 的保结构学习","authors":"Süleyman Yıldız, Pawan Goyal, Peter Benner","doi":"arxiv-2409.10432","DOIUrl":null,"url":null,"abstract":"This paper presents an energy-preserving machine learning method for\ninferring reduced-order models (ROMs) by exploiting the multi-symplectic form\nof partial differential equations (PDEs). The vast majority of\nenergy-preserving reduced-order methods use symplectic Galerkin projection to\nconstruct reduced-order Hamiltonian models by projecting the full models onto a\nsymplectic subspace. However, symplectic projection requires the existence of\nfully discrete operators, and in many cases, such as black-box PDE solvers,\nthese operators are inaccessible. In this work, we propose an energy-preserving\nmachine learning method that can infer the dynamics of the given PDE using data\nonly, so that the proposed framework does not depend on the fully discrete\noperators. In this context, the proposed method is non-intrusive. The proposed\nmethod is grey box in the sense that it requires only some basic knowledge of\nthe multi-symplectic model at the partial differential equation level. We prove\nthat the proposed method satisfies spatially discrete local energy conservation\nand preserves the multi-symplectic conservation laws. We test our method on the\nlinear wave equation, the Korteweg-de Vries equation, and the\nZakharov-Kuznetsov equation. We test the generalization of our learned models\nby testing them far outside the training time interval.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Structure-preserving learning for multi-symplectic PDEs\",\"authors\":\"Süleyman Yıldız, Pawan Goyal, Peter Benner\",\"doi\":\"arxiv-2409.10432\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper presents an energy-preserving machine learning method for\\ninferring reduced-order models (ROMs) by exploiting the multi-symplectic form\\nof partial differential equations (PDEs). The vast majority of\\nenergy-preserving reduced-order methods use symplectic Galerkin projection to\\nconstruct reduced-order Hamiltonian models by projecting the full models onto a\\nsymplectic subspace. However, symplectic projection requires the existence of\\nfully discrete operators, and in many cases, such as black-box PDE solvers,\\nthese operators are inaccessible. In this work, we propose an energy-preserving\\nmachine learning method that can infer the dynamics of the given PDE using data\\nonly, so that the proposed framework does not depend on the fully discrete\\noperators. In this context, the proposed method is non-intrusive. The proposed\\nmethod is grey box in the sense that it requires only some basic knowledge of\\nthe multi-symplectic model at the partial differential equation level. We prove\\nthat the proposed method satisfies spatially discrete local energy conservation\\nand preserves the multi-symplectic conservation laws. We test our method on the\\nlinear wave equation, the Korteweg-de Vries equation, and the\\nZakharov-Kuznetsov equation. We test the generalization of our learned models\\nby testing them far outside the training time interval.\",\"PeriodicalId\":501162,\"journal\":{\"name\":\"arXiv - MATH - Numerical Analysis\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Numerical Analysis\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10432\",\"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 - MATH - Numerical Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10432","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Structure-preserving learning for multi-symplectic PDEs
This paper presents an energy-preserving machine learning method for
inferring reduced-order models (ROMs) by exploiting the multi-symplectic form
of partial differential equations (PDEs). The vast majority of
energy-preserving reduced-order methods use symplectic Galerkin projection to
construct reduced-order Hamiltonian models by projecting the full models onto a
symplectic subspace. However, symplectic projection requires the existence of
fully discrete operators, and in many cases, such as black-box PDE solvers,
these operators are inaccessible. In this work, we propose an energy-preserving
machine learning method that can infer the dynamics of the given PDE using data
only, so that the proposed framework does not depend on the fully discrete
operators. In this context, the proposed method is non-intrusive. The proposed
method is grey box in the sense that it requires only some basic knowledge of
the multi-symplectic model at the partial differential equation level. We prove
that the proposed method satisfies spatially discrete local energy conservation
and preserves the multi-symplectic conservation laws. We test our method on the
linear wave equation, the Korteweg-de Vries equation, and the
Zakharov-Kuznetsov equation. We test the generalization of our learned models
by testing them far outside the training time interval.