[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126111-en":3,"doc-seo-126111-105":31,"detail-sidebar-cat-0-en-105":93},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},126111,5909887254083,"Miles","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Machine learning for recognition of simple physical systems","Machine learning methods, especially neural networks, are increasingly used to solve problems in physics, including the recognition and simulation of simple physical systems from data. This bachelor thesis investigates the Direct Poisson Neural Network (DPNN), which incorporates the structure of Hamilton’s equations of motion to learn the Hamiltonian and the Poisson bivector, enabling identification of the system type and properties. The work analyzes DPNN performance under noisy measurements and limited datasets, evaluates extrapolation ability, and extends the model with Energy Ehrenfest regularisation to better capture and simulate dissipative dynamics.","BACHELOR THESIS  \nJan Benda  \nMachine learning for recognition of simple  \nphysical systems  \nMathematical Institute of Charles University  \nSupervisor of the bachelor thesis: doc. RNDr. Michal Pavelka, Ph.D.  \nStudy programme: Mathematical Modelling  \nStudy branch: MMOP  \nI declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. It has not been used to obtain another or the same degree.  \nI understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.  \n[In . . . . . . . . . . . . . date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .](In . . . . . . . . . . . . . date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .) .  \nAuthor’s signature  \nI would like to thank my supervisor, doc. Michal Pavelka, for his patient guidance and advice.  \nTitle: Machine learning for recognition of simple physical systems  \nAuthor: Jan Benda  \nInstitute: Mathematical Institute of Charles University  \nSupervisor: doc. RNDr. Michal Pavelka, Ph.D., Mathematical Institute of Charles University  \nAbstract: The rise of machine learning, particularly through the use of neural networks, has begun to change how we solve problems, including understanding simple physical systems. This thesis focuses on the Direct Poisson Neural Network (DPNN), a network that uses the structure of Hamilton’s equations of motion to learn from data. This method allows us to extract the Hamiltonian and Poisson bivector from the data, helping to identify the type of physical systems. We explore how DPNN works with noisy data and when data is limited, checking its ability to make predictions in challenging conditions. Moreover, we have implemented Energy Ehrenfest regularisationto the model, which helps it recognise and simulate dissipative systems better.  \nKeywords: Machine Learning Hamiltonian Systems System Recognition Neural Networks  \nContents  \nList of Abbreviations 2  \nIntroduction 3  \n1 Introduction to Deep Learning 4  \n1.1 Neural networks basics ......................... 4  \n1.2 Training the model ........................... 5  \n1.2.1 Training and Test set ...................... 5  \n1.2.2 Gradient Descent ........................ 6  \n1.3 Model capacity and regularisation ................... 7  \n1.4 Coping with noise ............................ 8  \n2 Hamiltonian Systems 10  \n2.1 Hamilton’s equations .......................... 10  \n2.2 Poisson Brackets ............................ 11  \n2.2.1 Properties of Poisson brackets ................. 11  \n2.3 Non-canonical Poisson brackets .................... 12  \n2.3.1 Jacobiator ............................ 13  \n2.4 3D Hamiltonian systems ........................ 13  \n3 Direct Poisson Neural Network 15  \n3.1 The Architecture and workflow ..................... 15  \n3.1.1 The workflow .......................... 16  \n3.2 Demonstration .............................. 17  \n4 Results and improvements 21  \n4.1 Robustness against noise ........................ 21  \n4.2 Extrapolation .............................. 23  \n4.3 RK4 movement loss ........................... 24  \n4.4 Ehrenfest dissipation .......................... 27  \nConclusion 30  \nBibliography 31  \nList of Figures 32  \nList of Tables 33  \nList of Abbreviations  \nDPNN Direct Poisson Neural Network  \nGD Gradient Descent  \nSGD Stochastic Gradient Descent MSE Means Square Error  \nMLE Maximum Likelihood Estimator  \nWJ DPNN without Jacobi’s identity regularisation SJ DPNN with soft Jacobi’s identity regularisation IJ DPNN with implict Jacobi’s identity  \nFE Forward Euler Method  \nCN Cranck-Nicolson scheme IMR Implicit Midpoint Rule RK4 Runge-Kutta o","cbCaidDxCqeZXZsn","https://ap.wps.com/l/cbCaidDxCqeZXZsn","pdf",1909522,5,1,37,"English","en",105,"# List of Abbreviations\n# Introduction\n## Introduction to Deep Learning\n## Hamiltonian Systems\n## Direct Poisson Neural Network\n## Results and improvements\n## Conclusion\n# Bibliography\n# List of Figures\n# List of Tables","[{\"question\":\"What is the Direct Poisson Neural Network (DPNN) used for in this thesis?\",\"answer\":\"DPNN is used to learn the Hamiltonian and Poisson bivector from data so the underlying physical system type can be recognized.\"},{\"question\":\"How does the thesis evaluate DPNN under real-world data conditions?\",\"answer\":\"It investigates DPNN behavior with noisy data and with limited datasets, including its ability to extrapolate and make predictions in challenging regimes.\"},{\"question\":\"What improvement is introduced to help recognize dissipative systems?\",\"answer\":\"Energy Ehrenfest regularisation is implemented, improving DPNN’s ability to recognize and simulate dissipative dynamics.\"}]","Machine learning for recognition of simple physical systems | 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