[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126681-en":3,"doc-seo-126681-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},126681,962084925502,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8",8,"Research & Report","Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning","A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. The method trains stencil weights of a discretization using the truncation error as the objective for training. Centered finite differences initialize optimization, and a second-order implicit-explicit time solver provides the framework. Symmetry constraints enforce stability of the learned operator. Applied to the Korteweg–de Vries equation, it achieves higher accuracy than finite difference and spectral methods on coarse grids when initial data lies near the training set.","Air Force Institute of Technology  \nAFIT Scholar  \nFaculty Publications  \n6-21-2023  \nNumerical Simulation of the Korteweg–de Vries Equation with Machine Learning  \nKristina O. F. Williams *  \nAir Force Institute of Technology  \nBenjamin F. Akers  \nAir Force Institute of Technology  \nFollow this and additional works at: [https://scholar.afit.edu/facpub](https://scholar.afit.edu/facpub)  \n Part of the Computer Sciences Commons, and the Mathematics Commons  \nRecommended Citation  \nWilliams, K. O. F., & Akers, B. F. (2023) . Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning. Mathematics, 11(13), 2791. [https://doi.org/10.3390/math11132791](https://doi.org/10.3390/math11132791)  \n[This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in](This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in)[ ](This Article is brought to you for free and open access by AFIT Scholar. It has been accepted for inclusion in)[Faculty Publications by an authorized administrator of AFIT Scholar. For more information](Faculty Publications by an authorized administrator of AFIT Scholar. For more information), please contact [AFIT.ENWL.Repository@us.af.mil](AFIT.ENWL.Repository@us.af.mil).  \n mathematics  \nArticle  \nNumerical Simulation of the Korteweg–de Vries Equation with Machine Learning  \nKristina O. F. Williams * and Benjamin F. Akers  \nCitation: Williams, K.O.F.; Akers, B.F. Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning. Mathematics 2023, 11, 2791. [https://doi.org/10.3390/](https://doi.org/10.3390/)[ ](https://doi.org/10.3390/)math11132791  \nAcademic Editor: Xiangmin Jiao  \nReceived: 17 May 2023  \nRevised: 13 June 2023  \nAccepted: 16 June 2023  \nPublished: 21 June 2023  \nCopyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://](https://)[ ](https://)[creativecommons.org/licenses/by/](creativecommons.org/licenses/by/)[ ](creativecommons.org/licenses/by/)[4.0/](4.0/)) .  \nDepartment of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433, USA; benjamin.akers@afi[t.edu](t.edu)  \n* Correspondence: kristina.williams@afi[t.edu](t.edu)  \nAbstract: A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg–de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse grids when the initial data is near enough to the training set.  \nKeywords: machine learning; Korteweg–de Vries equation; coarse grid  \nMSC: 65M25  \n1. Introduction  \nNumerical methods for nonlinear wave equations have a long history, from the seminal works of Courant, Friedrichs, and Lewy [1] almost a century ago, to more recent contributions of Fornberg, Trefethen, LeVeque, and many others [2–10] . By and large, these methods are successful when a sufficiently fine discretization is used. Many classical numerical methods for partial differential equations (PDE) perform poorly on coarse grids, i.e., with few data points [11–13] . Recently, a number of authors have used machine learning to augment numerical solvers in the coarse discretization regime [14–16] . In this work, a procedure for numerically solving a nonlinear dispersive wave equation is proposed using a machine learning model to optimize stencil weights. A simple ","cbCaiq1iWvOlfXbt","https://ap.wps.com/l/cbCaiq1iWvOlfXbt","pdf",1491288,1,15,"English","en",105,"# Introduction\n## Nonlinear wave equations and coarse-grid challenges\n## Machine learning for augmenting numerical solvers\n## Proposed learning-based stencil optimization\n## Application to the Korteweg–de Vries equation\n## Traveling-wave solutions (solitons) and stability considerations","[{\"question\":\"What does the proposed machine learning method optimize for the numerical scheme?\",\"answer\":\"It trains stencil weights by using the truncation error of the discretization as the objective function during training.\"},{\"question\":\"How is the numerical solver structured in the proposed framework?\",\"answer\":\"The approach initializes the optimization with centered finite differences and uses a second-order implicit-explicit time solver as the solution framework.\"},{\"question\":\"How is stability ensured for the learned discretization operator?\",\"answer\":\"Symmetry conditions are enforced on the learned operator to ensure a stable numerical method.\"}]","Numerical Simulation of the Korteweg–de Vries Equation with Machine Learning | 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does the proposed machine learning method optimize for the numerical scheme?","Question",{"text":75,"@type":76},"It trains stencil weights by using the truncation error of the discretization as the objective function during training.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the numerical solver structured in the proposed framework?",{"text":80,"@type":76},"The approach initializes the optimization with centered finite differences and uses a second-order implicit-explicit time solver as the solution framework.",{"name":82,"@type":73,"acceptedAnswer":83},"How is stability ensured for the learned discretization operator?",{"text":84,"@type":76},"Symmetry conditions are enforced on the learned operator to ensure a stable numerical 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