[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126165-en":3,"doc-seo-126165-105":30,"detail-sidebar-cat-0-en-105":92},{"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":11,"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},126165,3985741905716,"Rowan","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Solving physics-based initial value problems with unsupervised machine learning","Solving physics-based initial value problems is addressed using an unsupervised deep learning framework that represents dynamical evolution with neural networks. The approach targets nonlinear, coupled, and chaotic systems by learning trajectories that can preserve physical properties such as energy and stationary action. Demonstrations cover a free particle, motion under gravity, a classical pendulum, and the Hénon–Heiles system. The method further clarifies that probabilistic activation functions and coupled networks are required to learn solutions in the strict sense for single and coupled systems.","Solving 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 (Received 10 July 2024; revised 15 March 2025; accepted 25 April 2025; published 15 May 2025)  \nInitial value problems—a system of ordinary differential equations and corresponding initial conditions—can be used to describe many physical phenomena including those arise in classical mechanics. We have developed an approach to solve physics-based initial value problems using unsupervised machine learning. We 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 nonlinear, 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 nonlinear 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.  \nDOI: 10.1103/PhysRevE.111.055302  \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 generalization, and offer deeper insights into the underlying mechanisms of the systems they are designed to analyze.  \nMachine learning is an optimization procedure. Machine learning is physics-informed if physical data or principles are present in (1) the training data, (2) the underlying neural network architecture, (3) the cost function, and (4) the optimization algorithms. Training data may include data from physical experiments or simulations [14, 15] . Training data may be 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 enforce obedience to physical laws. Ling, Kurzawski, and Templeton [17] predict turbulent fluid flows using a deep neural  \n*[Contact author: me@jackg.co](Contact author: me@jackg.co)  \n†Contact author: [s.a.wrathmall@durham.ac.uk](s.a.wrathmall@durham.ac.uk)  \n‡Contact author: [s.a.gardiner@durham.ac.uk](s.a.gardiner@durham.ac.uk)  \nPublished by the American Physical Society under the terms of the Creative Commons Attribution 4 .0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.  \nnetwork with an auxiliary network which ensures invariance to Galilean transformations (which one would expect from such classical fluid problems) .  \nRegularization 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 (1) the differential equation (or othe","cbCaidQLT1kLE4By","https://ap.wps.com/l/cbCaidQLT1kLE4By","pdf",4397245,1,19,"English","en",105,"# Introduction\n## Physics-informed machine learning framework\n## Role of training data and network architecture\n## Cost function design and constraints\n## Regularization and limited-data extremum principle","[{\"question\":\"What type of problems does the paper target?\",\"answer\":\"It targets physics-based initial value problems formulated as ordinary differential equations together with their initial conditions.\"},{\"question\":\"How does the proposed method learn solutions?\",\"answer\":\"It uses a deep learning framework that models system dynamics with neural networks, aiming to produce physically consistent trajectories.\"},{\"question\":\"Which physical systems are used to demonstrate the approach?\",\"answer\":\"The paper demonstrates on a free particle, a particle in a gravitational field, a classical pendulum, and the Hénon–Heiles system.\"}]","Solving physics-based initial value problems with unsupervised machine learning | 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type of problems does the paper target?","Question",{"text":76,"@type":77},"It targets physics-based initial value problems formulated as ordinary differential equations together with their initial conditions.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed method learn solutions?",{"text":81,"@type":77},"It uses a deep learning framework that models system dynamics with neural networks, aiming to produce physically consistent trajectories.",{"name":83,"@type":74,"acceptedAnswer":84},"Which physical systems are used to demonstrate the approach?",{"text":85,"@type":77},"The paper demonstrates on a free particle, a particle in a gravitational field, a classical pendulum, and the Hénon–Heiles 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