Mengge Du, Yuntian Chen, Zhongzheng Wang, Longfeng Nie, Dongxiao Zhang
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The best equations are preserved and further refined iteratively using the reasoning capacity of LLMs. We propose two alternately iterated strategies to collaboratively optimize the generated equations. The first strategy uses LLMs as a black-box optimizer to achieve equation self-improvement based on historical samples and their performance. The second strategy instructs LLMs to perform evolutionary operations for a global search. Experiments are conducted on various nonlinear systems described by partial differential equations, including the Burgers equation, the Chafee–Infante equation, and the Navier–Stokes equation. The results demonstrate that our framework can discover correct equations that reveal the underlying physical laws. Further comparisons with state-of-the-art models on extensive ordinary differential equations showcase that the equations discovered by our framework possess physical meaning and better generalization capability on unseen data.","PeriodicalId":20066,"journal":{"name":"Physics of Fluids","volume":null,"pages":null},"PeriodicalIF":4.1000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Large language models for automatic equation discovery of nonlinear dynamics\",\"authors\":\"Mengge Du, Yuntian Chen, Zhongzheng Wang, Longfeng Nie, Dongxiao Zhang\",\"doi\":\"10.1063/5.0224297\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Equation discovery aims to directly extract physical laws from data and has emerged as a pivotal research domain in nonlinear systems. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require handcrafted representation rules and complex optimization algorithms. In this paper, we introduce a novel framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically extracting governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse candidate equations in string form and then evaluate the generated equations based on observations. The best equations are preserved and further refined iteratively using the reasoning capacity of LLMs. We propose two alternately iterated strategies to collaboratively optimize the generated equations. The first strategy uses LLMs as a black-box optimizer to achieve equation self-improvement based on historical samples and their performance. The second strategy instructs LLMs to perform evolutionary operations for a global search. Experiments are conducted on various nonlinear systems described by partial differential equations, including the Burgers equation, the Chafee–Infante equation, and the Navier–Stokes equation. The results demonstrate that our framework can discover correct equations that reveal the underlying physical laws. Further comparisons with state-of-the-art models on extensive ordinary differential equations showcase that the equations discovered by our framework possess physical meaning and better generalization capability on unseen data.\",\"PeriodicalId\":20066,\"journal\":{\"name\":\"Physics of Fluids\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.1000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physics of Fluids\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0224297\",\"RegionNum\":2,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physics of Fluids","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1063/5.0224297","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MECHANICS","Score":null,"Total":0}
Large language models for automatic equation discovery of nonlinear dynamics
Equation discovery aims to directly extract physical laws from data and has emerged as a pivotal research domain in nonlinear systems. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require handcrafted representation rules and complex optimization algorithms. In this paper, we introduce a novel framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically extracting governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse candidate equations in string form and then evaluate the generated equations based on observations. The best equations are preserved and further refined iteratively using the reasoning capacity of LLMs. We propose two alternately iterated strategies to collaboratively optimize the generated equations. The first strategy uses LLMs as a black-box optimizer to achieve equation self-improvement based on historical samples and their performance. The second strategy instructs LLMs to perform evolutionary operations for a global search. Experiments are conducted on various nonlinear systems described by partial differential equations, including the Burgers equation, the Chafee–Infante equation, and the Navier–Stokes equation. The results demonstrate that our framework can discover correct equations that reveal the underlying physical laws. Further comparisons with state-of-the-art models on extensive ordinary differential equations showcase that the equations discovered by our framework possess physical meaning and better generalization capability on unseen data.
期刊介绍:
Physics of Fluids (PoF) is a preeminent journal devoted to publishing original theoretical, computational, and experimental contributions to the understanding of the dynamics of gases, liquids, and complex or multiphase fluids. Topics published in PoF are diverse and reflect the most important subjects in fluid dynamics, including, but not limited to:
-Acoustics
-Aerospace and aeronautical flow
-Astrophysical flow
-Biofluid mechanics
-Cavitation and cavitating flows
-Combustion flows
-Complex fluids
-Compressible flow
-Computational fluid dynamics
-Contact lines
-Continuum mechanics
-Convection
-Cryogenic flow
-Droplets
-Electrical and magnetic effects in fluid flow
-Foam, bubble, and film mechanics
-Flow control
-Flow instability and transition
-Flow orientation and anisotropy
-Flows with other transport phenomena
-Flows with complex boundary conditions
-Flow visualization
-Fluid mechanics
-Fluid physical properties
-Fluid–structure interactions
-Free surface flows
-Geophysical flow
-Interfacial flow
-Knudsen flow
-Laminar flow
-Liquid crystals
-Mathematics of fluids
-Micro- and nanofluid mechanics
-Mixing
-Molecular theory
-Nanofluidics
-Particulate, multiphase, and granular flow
-Processing flows
-Relativistic fluid mechanics
-Rotating flows
-Shock wave phenomena
-Soft matter
-Stratified flows
-Supercritical fluids
-Superfluidity
-Thermodynamics of flow systems
-Transonic flow
-Turbulent flow
-Viscous and non-Newtonian flow
-Viscoelasticity
-Vortex dynamics
-Waves