[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125391-en":3,"doc-seo-125391-105":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},125391,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Solving physics-based initial value problems with unsupervised machine learning","Initial value problems formulated as ordinary differential equations and initial conditions describe many classical physical phenomena. This work develops a novel unsupervised deep-learning framework that learns dynamical evolution of mechanical systems using neural networks, addressing nonlinear, coupled, and chaotic regimes. The method approximates solution trajectories while conserving key physical invariants such as energy and stationary action. It shows that probabilistic activation functions are required to learn solutions in the strict sense and introduces coupled neural networks to handle coupled systems.","arXiv :2407 . 18320v1 [physics .comp-ph] 25 Jul 2024  \nSolving physics-based initial value problems with unsupervised machine learning  \nJack Griffiths ,∗ Steven A. Wrathmall ,† and Simon A. Gardiner ‡  \nJoint Quantum Centre (JQC) Durham– Newcastle, Department of Physics, Durham University, Durham DH1 3LE, United Kingdom  \n(Dated: July 29, 2024)  \nInitial value problems—a system of ordinary differential equations and corresponding initial conditions  \n—can be used to describe many physical phenomena including those arise in classical mechanics. We have developed a novel approach to solve physics-based initial value problems using unsupervised machine learning.  \nWe propose a deep learning framework that models the dynamics of a variety of mechanical systems through neural networks. Our framework is flexible, allowing us to solve non-linear, coupled, and chaotic dynamical systems. We demonstrate the effectiveness of our approach on systems including a free particle, a particle in a gravitational field, a classical pendulum, and the Hénon–Heiles system (a pair of coupled harmonic oscillators with a non-linear perturbation, used in celestial mechanics) . Our results show that deep neural networks can successfully approximate solutions to these problems, producing trajectories which conserve physical properties such as energy and those with stationary action. We note that probabilistic activation functions, as defined in this paper, are required to learn any solutions of initial value problems in their strictest sense, and we introduce coupled neural networks to learn solutions of coupled systems.  \nI. INTRODUCTION  \nMachine learning models embedded with physical constraints, equations, data, or other physical principles (including variational principles such as the Rayleigh–Ritz method or Hamilton’s principle, which is what we present in the current work) offer a powerful framework for ensuring physically consistent solutions for differential equations [1–7] or discovering new physics [8–13] . Incorporating informed priors in machine learning models allows effective handling of sparse data, improved generalisation, and offer deeper insights into the underlying mechanisms of the systems they are designed to analyse.  \nMachine learning is an optimisation procedure. Machine learning is physics-informed if physical data or principles are present in i) the training data, ii) the underlying neural network architecture, iii) the cost function and iv) the optimisation algorithms. Training data may include data from physical experiments or simulations [14, 15] . Training data maybe augmented [16] by applying transformations (e.g., translations, scalings or rotations) in line with known symmetries of the system, without having to generate new data. The network architecture may include structures which obey physical laws. Ling, Kurzawski and Templeton [17] predict turbulent fluid flows using a deep neural network with an auxiliary network which ensures invariance to Galilean transformations (which one would expect from such classical fluid problems) .  \nRegularisation terms may be added to the cost function to constrain the neural network output based on, for example, the underlying differential equation, whether the solution should be divergence free [18], or any other mathematical description of the data [19] . The cost function therefore may consist of a weighted combination of: i) the differential equation (or other equations which constrain the solution), ii) the initial conditions, iii) the boundary conditions (where appropriate),  \n∗  \n†  \n‡  \n[me@jackg.co](me@jackg.co)[s.a.wrathmall@durham.ac.uk](s.a.wrathmall@durham.ac.uk)[ ](s.a.wrathmall@durham.ac.uk)[s.a.gardiner@durham.ac.uk](s.a.gardiner@durham.ac.uk)  \nand iv) sparse data sampled from throughout the problem domain (the training data) . Other problem-specific conditions may also be included in the cost function. Xu, et al. [6], solve Fokker–Planck equations with machine learni","cbCaidNl09gnOIEn","https://ap.wps.com/l/cbCaidNl09gnOIEn","pdf",3367968,1,19,"English","en",105,"# Introduction\n## Physics-informed machine learning and physical constraints\n## Cost functions and training data (including sparsity)\n## Using Hamilton’s principle with minimal constraints\n## Unsupervised framework and model examples","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses solving physics-based initial value problems, i.e., ordinary differential equations with corresponding initial conditions, for a variety of mechanical systems.\"},{\"question\":\"How does the proposed method learn without labeled training data?\",\"answer\":\"It uses an unsupervised deep learning framework where the only mathematical constraints are the governing differential equation and the associated initial conditions.\"},{\"question\":\"Which physical properties does the method preserve in its learned trajectories?\",\"answer\":\"The learned trajectories conserve physical properties such as energy and exhibit stationary action, matching the principles emphasized in the paper.\"}]","Solving physics-based initial value problems with unsupervised machine learning | 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problem does the paper address?","Question",{"text":75,"@type":76},"It addresses solving physics-based initial value problems, i.e., ordinary differential equations with corresponding initial conditions, for a variety of mechanical systems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method learn without labeled training data?",{"text":80,"@type":76},"It uses an unsupervised deep learning framework where the only mathematical constraints are the governing differential equation and the associated initial conditions.",{"name":82,"@type":73,"acceptedAnswer":83},"Which physical properties does the method preserve in its learned trajectories?",{"text":84,"@type":76},"The learned trajectories conserve physical properties such as energy and exhibit stationary action, matching the principles emphasized in the 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