[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122090-en":3,"doc-seo-122090-105":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},122090,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","NUMERICAL SIMULATION OF NONLINEAR WAVE EQUATIONS WITH MACHINE LEARNING - Dissertation","A machine learning procedure is proposed to construct numerical schemes for solving selected nonlinear wave equations on coarse grids. The method trains stencil weights for a discretization using the scheme’s truncation error as the training objective. A neural network models the stencil weights, initialized via centered finite differences and embedded in a second-order implicit-explicit time-integration framework. Symmetry constraints are applied to the learned operator to promote stability.","Air Force Institute of Technology  \nAFIT Scholar  \n\n| Theses and Dissertations | Student Graduate Works |\n| --- | --- |\n| 9-2023\u003Cbr>Numerical Simulation of Nonlinear Wave Learning\u003Cbr>Kristina O. F. Williams\u003Cbr>Follow this and additional works at: [https://scholar.afit.edu/etd](https://scholar.afit.edu/etd)[ ](https://scholar.afit.edu/etd) Part of the Applied Mathematics Commons | Equations with Machine |\n\nRecommended Citation  \nWilliams, Kristina O. F., \"Numerical Simulation of Nonlinear Wave Equations with Machine Learning\"(2023) . Theses and Dissertations. 7661.  \n[https://scholar.afit.edu/etd/7661](https://scholar.afit.edu/etd/7661)  \nThis Dissertation is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please [contact AFIT.ENWL.Repository@us.af.mil](contact AFIT.ENWL.Repository@us.af.mil).  \nNUMERICAL SIMULATION OF NONLINEAR WAVE EQUATIONS WITH MACHINE LEARNING  \nDISSERTATION  \nKristina O.F. Williams, Captain, USAFAFIT-ENC-DS-23-S-004  \nDEPARTMENT OF THE AIR FORCE  \nAIR UNIVERSITY  \nAIR FORCE INSTITUTE OF TECHNOLOGY  \nWright-Patterson Air Force Base, Ohio  \nDISTRIBUTION STATEMENT A  \nAPPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.  \nThe views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.  \nAFIT-ENC-DS-23-S-004  \nNUMERICAL SIMULATION OF NONLINEAR WAVE EQUATIONS WITH  \nMACHINE LEARNING  \nDISSERTATION  \nPresented to the Faculty  \nGraduate School of Engineering and Management Air Force Institute of Technology  \nAir University  \nAir Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics  \nKristina O.F. Williams, M.S.  \nCaptain, USAF  \n14 September 2023  \nDISTRIBUTION STATEMENT A  \nAPPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.  \nAFIT-ENC-DS-23-S-004  \nNUMERICAL SIMULATION OF NONLINEAR WAVE EQUATIONS WITH  \nMACHINE LEARNING  \nDISSERTATION  \nKristina O.F. Williams, M.S.  \nCaptain, USAF  \nCommittee Membership:  \nBenjamin F. Akers, Ph.D.  \nChair  \nAihua W. Wood, Ph.D.  \nMember  \nTony Liu, Ph.D.  \nMember  \nSteven T. Fiorino, Ph.D.  \nMember  \nAFIT-ENC-DS-23-S-004  \nAbstract  \nA machine learning procedure is proposed to create numerical schemes for solutions of certain types 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. A neural network is used as a model for the stencil weights. 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.  \nThe procedure is applied to the Korteweg–de Vries and nonlinear Schrödinger equations. It is observed to be more accurate than finite difference or spectral methodson coarse grids when the initial data is near enough to the training set.  \nAFIT-ENC-DS-23-S-004  \nTo my husband, Kris, who was there for me through this entire endeavor.  \nAcknowledgements  \nI would like to express my deepest appreciation to my advisor, Dr. Ben Akers, for his guidance and feedback during my doctoral studies. His knowledge and expertise in the field of mathematics were invaluable throughout this project.  \nI would also like to thank the members of my committee, Dr. Aihua Wood, Dr. Steven Fiorino, and Maj Tony Liu, for their helpful discussions and guidance.  \nKristina O.F. Williams  \nContents  \nPage  \nAbstract ............................................................","cbCaim8vzV7vbvGs","https://ap.wps.com/l/cbCaim8vzV7vbvGs","pdf",4830341,1,122,"English","en",105,"# Abstract\n## Acknowledgements\n## List of Figures\n## I. Introduction\n### 1.1 Motivation and Background\n### 1.2 Research Accomplishments\n### 1.3 Document Overview\n## II. Methodology\n### 2.1 Approach\n### 2.2 Korteweg-de Vries and Nonlinear Schrödinger Equations\n## III. Numerical Simulation of the Korteweg-de Vries Equation with Machine Learning\n### 3.1 Introduction","[{\"question\":\"What is the main goal of the proposed method?\",\"answer\":\"To create numerical schemes for certain nonlinear wave equations on coarse grids using machine learning.\"},{\"question\":\"How are the machine learning parameters trained?\",\"answer\":\"Stencil weights are trained with the scheme’s truncation error as the objective function, using a neural network model for the weights.\"},{\"question\":\"Which equations are used to demonstrate the method and how does it perform?\",\"answer\":\"The Korteweg–de Vries and nonlinear Schrödinger equations are used, and the method is observed to be more accurate than finite difference or spectral methods on coarse grids when initial data is close to the training set.\"}]","NUMERICAL SIMULATION OF NONLINEAR WAVE EQUATIONS WITH MACHINE LEARNING - Dissertation | PDF",1785808769,307,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"numerical-simulation-of-nonlinear-wave-equations-with-machine-learning-dissertation","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/numerical-simulation-of-nonlinear-wave-equations-with-machine-learning-dissertation/122090/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of the proposed method?","Question",{"text":75,"@type":76},"To create numerical schemes for certain nonlinear wave equations on coarse grids using machine learning.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the machine learning parameters trained?",{"text":80,"@type":76},"Stencil weights are trained with the scheme’s truncation error as the objective function, using a neural network model for the weights.",{"name":82,"@type":73,"acceptedAnswer":83},"Which equations are used to demonstrate the method and how does it perform?",{"text":84,"@type":76},"The Korteweg–de Vries and nonlinear Schrödinger equations are used, and the method is observed to be more accurate than finite difference or spectral methods on coarse grids when initial data is close to the training set.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]