{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T07:50:20Z","timestamp":1750751420513,"version":"3.37.3"},"reference-count":41,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["Mach. Learn.: Sci. Technol."],"published-print":{"date-parts":[[2024,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The time evolution of physical systems is described by differential equations, which depend on abstract quantities like energy and force. Traditionally, these quantities are derived as functionals based on observables such as positions and velocities. Discovering these governing symbolic laws is the key to comprehending the interactions in nature. Here, we present a Hamiltonian graph neural network (<jats:sc>Hgnn<\/jats:sc>), a physics-enforced <jats:sc>Gnn<\/jats:sc> that learns the dynamics of systems directly from their trajectory. We demonstrate the performance of <jats:sc>Hgnn<\/jats:sc> on <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>n<\/mml:mi>\n                           <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula>springs, <jats:inline-formula>\n                     <jats:tex-math\/>\n                     <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                           <mml:mi>n<\/mml:mi>\n                           <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:mrow>\n                     <\/mml:math>\n                  <\/jats:inline-formula>pendulums, gravitational systems, and binary Lennard Jones systems; <jats:sc>Hgnn<\/jats:sc> learns the dynamics in excellent agreement with the ground truth from small amounts of data. We also evaluate the ability of <jats:sc>Hgnn<\/jats:sc> to generalize to larger system sizes, and to a hybrid spring-pendulum system that is a combination of two original systems (spring and pendulum) on which the models are trained independently. Finally, employing symbolic regression on the learned <jats:sc>Hgnn<\/jats:sc>, we infer the underlying equations relating to the energy functionals, even for complex systems such as the binary Lennard-Jones liquid. Our framework facilitates the interpretable discovery of interaction laws directly from physical system trajectories. Furthermore, this approach can be extended to other systems with topology-dependent dynamics, such as cells, polydisperse gels, or deformable bodies.<\/jats:p>","DOI":"10.1088\/2632-2153\/ad6be6","type":"journal-article","created":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T22:57:25Z","timestamp":1722985045000},"page":"035049","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Discovering symbolic laws directly from trajectories with hamiltonian graph neural networks"],"prefix":"10.1088","volume":"5","author":[{"given":"Suresh","family":"Bishnoi","sequence":"first","affiliation":[]},{"given":"Ravinder","family":"Bhattoo","sequence":"additional","affiliation":[]},{"family":"Jayadeva","sequence":"additional","affiliation":[]},{"given":"Sayan","family":"Ranu","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1500-4947","authenticated-orcid":true,"given":"N M Anoop","family":"Krishnan","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2024,8,20]]},"reference":[{"year":"2004","author":"Rapaport","key":"mlstad6be6bib1"},{"key":"mlstad6be6bib2","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1126\/science.1165893","article-title":"Distilling free-form natural laws from experimental data","volume":"324","author":"Schmidt","year":"2009","journal-title":"Science"},{"key":"mlstad6be6bib3","doi-asserted-by":"publisher","first-page":"eaay2631","DOI":"10.1126\/sciadv.aay2631","article-title":"Ai feynman: a physics-inspired method for symbolic regression","volume":"6","author":"Udrescu","year":"2020","journal-title":"Sci. Adv."},{"key":"mlstad6be6bib4","doi-asserted-by":"publisher","first-page":"1777","DOI":"10.1038\/s41467-023-37236-y","article-title":"Combining data and theory for derivable scientific discovery with ai-descartes","volume":"14","author":"Cornelio","year":"2023","journal-title":"Nat. Commun."},{"key":"mlstad6be6bib5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.126.180604","article-title":"Machine learning conservation laws from trajectories","volume":"126","author":"Liu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"mlstad6be6bib6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.055302","article-title":"Machine-learning nonconservative dynamics for new-physics detection","volume":"104","author":"Liu","year":"2021","journal-title":"Phys. Rev. E"},{"article-title":"Interpretable machine learning for science with pysr and symbolicregression.jl","year":"2023","author":"Cranmer","key":"mlstad6be6bib7"},{"key":"mlstad6be6bib8","first-page":"p 33","article-title":"Discovering symbolic models from deep learning with inductive biases","author":"Cranmer","year":"2020"},{"key":"mlstad6be6bib9","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"Em Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"mlstad6be6bib10","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1038\/s41567-020-0842-8","article-title":"Unveiling the predictive power of static structure in glassy systems","volume":"16","author":"Bapst","year":"2020","journal-title":"Nat. Phys."},{"key":"mlstad6be6bib11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-021-00543-3","article-title":"Accurate and scalable graph neural network force field and molecular dynamics with direct force architecture","volume":"7","author":"Woo Park","year":"2021","journal-title":"npj Comput. Mater."},{"key":"mlstad6be6bib12","doi-asserted-by":"publisher","first-page":"18327","DOI":"10.1073\/pnas.1306572110","article-title":"Simulation as an engine of physical scene understanding","volume":"110","author":"Battaglia","year":"2013"},{"key":"mlstad6be6bib13","first-page":"pp 8459","article-title":"Learning to simulate complex physics with graph networks","author":"Sanchez-Gonzalez","year":"2020"},{"key":"mlstad6be6bib14","first-page":"pp 1218","article-title":"Benchmarking energy-conserving neural networks for learning dynamics from data","author":"Desmond Zhong","year":"2021"},{"article-title":"Lagrangian neural networks","year":"2020","author":"Cranmer","key":"mlstad6be6bib15"},{"key":"mlstad6be6bib16","first-page":"pp 13880","article-title":"Simplifying hamiltonian and lagrangian neural networks via explicit constraints","volume":"vol 33","author":"Finzi","year":"2020"},{"article-title":"Deep lagrangian networks: using physics as model prior for deep learning","year":"2019","author":"Lutter","key":"mlstad6be6bib17"},{"article-title":"Deconstructing the inductive biases of hamiltonian neural networks","year":"2021","author":"Gruver","key":"mlstad6be6bib18"},{"key":"mlstad6be6bib19","doi-asserted-by":"publisher","first-page":"18194","DOI":"10.1073\/pnas.2001258117","article-title":"Predicting the long-term stability of compact multiplanet systems","volume":"117","author":"Tamayo","year":"2020","journal-title":"Proc. Natl Acad. Sci."},{"article-title":"Hamiltonian graph networks with ode integrators","year":"2019","author":"Sanchez-Gonzalez","key":"mlstad6be6bib20"},{"key":"mlstad6be6bib21","first-page":"pp 15379","article-title":"Hamiltonian neural networks","volume":"vol 32","author":"Greydanus","year":"2019"},{"article-title":"Dissipative symoden: Encoding hamiltonian dynamics with dissipation and control into deep learning","year":"2020","author":"Desmond Zhong","key":"mlstad6be6bib22"},{"key":"mlstad6be6bib23","first-page":"pp 6572","article-title":"Neural ordinary differential equations","author":"Chen","year":"2018"},{"article-title":"Enhancing the inductive biases of graph neural ode for modeling physical systems","year":"2023","author":"Bishnoi","key":"mlstad6be6bib24"},{"article-title":"Unravelling the performance of physics-informed graph neural networks for dynamical systems","year":"2022","author":"Thangamuthu","key":"mlstad6be6bib25"},{"key":"mlstad6be6bib26","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1109\/TNN.2008.2005605","article-title":"The graph neural network model","volume":"20","author":"Scarselli","year":"2008","journal-title":"IEEE Trans. Neural Netw."},{"key":"mlstad6be6bib27","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/acb03e","article-title":"Learning the dynamics of particle-based systems with lagrangian graph neural networks","volume":"4","author":"Bhattoo","year":"2023","journal-title":"Mach. Learn.: Sci. Technol."},{"article-title":"Learning articulated rigid body dynamics with lagrangian graph neural network","year":"2022","author":"Bhattoo","key":"mlstad6be6bib28"},{"key":"mlstad6be6bib29","doi-asserted-by":"publisher","first-page":"284","DOI":"10.1038\/s42254-022-00456-0","article-title":"Lessons on interpretable machine learning from particle physics","volume":"4","author":"Grojean","year":"2022","journal-title":"Nat. Rev. Phys."},{"year":"2006","author":"LaValle","key":"mlstad6be6bib30"},{"year":"2011","author":"Goldstein","key":"mlstad6be6bib31"},{"year":"2017","author":"Murray","key":"mlstad6be6bib32"},{"key":"mlstad6be6bib33","doi-asserted-by":"publisher","first-page":"1376","DOI":"10.1103\/PhysRevLett.73.1376","article-title":"Scaling behavior in the \u03b2-relaxation regime of a supercooled lennard-jones mixture","volume":"73","author":"Kob","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"mlstad6be6bib34","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1038\/nphys3224","article-title":"Active gel physics","volume":"11","author":"Prost","year":"2015","journal-title":"Nat. Phys."},{"key":"mlstad6be6bib35","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1038\/nature24062","article-title":"Granular materials flow like complex fluids","volume":"551","author":"Kou","year":"2017","journal-title":"Nature"},{"key":"mlstad6be6bib36","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1038\/s41567-018-0194-9","article-title":"Mesoscale physical principles of collective cell organization","volume":"14","author":"Trepat","year":"2018","journal-title":"Nat. Phys."},{"key":"mlstad6be6bib37","doi-asserted-by":"publisher","first-page":"2453","DOI":"10.1038\/s41467-022-29939-5","article-title":"E (3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials","volume":"13","author":"Batzner","year":"2022","journal-title":"Nat. Commun."},{"key":"mlstad6be6bib38","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1038\/s41586-020-2869-5","article-title":"Anatomy of cage formation in a two-dimensional glass-forming liquid","volume":"587","author":"Li","year":"2020","journal-title":"Nature"},{"key":"mlstad6be6bib39","first-page":"p 33","article-title":"Jax md: a framework for differentiable physics","author":"Schoenholz","year":"2020"},{"article-title":"Jax: composable transformations of python+ numpy programs","year":"2018","author":"Bradbury","key":"mlstad6be6bib40"},{"article-title":"Jraph: a library for graph neural networks in jax","year":"2020","author":"Godwin","key":"mlstad6be6bib41"}],"container-title":["Machine Learning: Science and Technology"],"original-title":[],"link":[{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6","content-type":"text\/html","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T09:27:38Z","timestamp":1724146058000},"score":1,"resource":{"primary":{"URL":"https:\/\/iopscience.iop.org\/article\/10.1088\/2632-2153\/ad6be6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,20]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,8,20]]},"published-print":{"date-parts":[[2024,9,1]]}},"URL":"https:\/\/doi.org\/10.1088\/2632-2153\/ad6be6","relation":{},"ISSN":["2632-2153"],"issn-type":[{"type":"electronic","value":"2632-2153"}],"subject":[],"published":{"date-parts":[[2024,8,20]]},"assertion":[{"value":"Discovering symbolic laws directly from trajectories with hamiltonian graph neural networks","name":"article_title","label":"Article Title"},{"value":"Machine Learning: Science and Technology","name":"journal_title","label":"Journal Title"},{"value":"paper","name":"article_type","label":"Article Type"},{"value":"\u00a9 2024 The Author(s). Published by IOP Publishing Ltd","name":"copyright_information","label":"Copyright Information"},{"value":"2024-01-04","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-08-06","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2024-08-20","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}