[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116835-en":3,"doc-seo-116835-105":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},116835,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Block Sparsity and Gauge Mediated Weight Sharing for Learning Dynamical Laws from Data - Block Sparsity and Gauge Mediated Weight Sharing","Recent years have increased interest in recovering dynamical laws of complex systems in a data-driven way under meaningful assumptions. This work introduces a scalable method to learn dynamical laws of classical dynamical systems from data by using block sparse tensor trains, a tensor-network-based function dictionary, and exploiting problem self-similarity. A gauge mediated weight sharing scheme, inspired by machine learning, significantly improves performance over prior approaches. Numerical demonstrations use three 1D systems, and sample-efficiency and robustness to additive Gaussian noise are reported.","arXiv :2208 .01591v2 [math .DS] 4 Aug 2022  \nBLOCK SPARSITY AND GAUGE MEDIATED WEIGHT SHARING  \nFOR LEARNING DYNAMICAL LAWS FROM DATA  \nM. Gtte 1 ;y, J. Fuksa2 , I. Roth3 , and J. Eisert2 ;4  \n1 Institute of Mathematics, Technische Universitt Berlin, Germany  \n2 Dahlem Center for Complex Quantum Systems, Freie Universitt Berlin, Germany  \n3 Quantum Research Centre, Technology Innovation Institute, Abu Dhabi  \n4 Fraunhofer Heinrich Hertz Institute, Germany [y](y jonas.fuksa@gmail.com)[ jonas.fuksa@gmail.com](y jonas.fuksa@gmail.com)  \nABSTRACT  \nRecent years have witnessed an increased interest in recovering dynamical laws of complex systems in a largely data-driven fashion under meaningful hypotheses. In this work, we propose a method for scalably learning dynamical laws of classical dynamical systems from data. As a novel ingredient, to achieve an efficient scaling with the system size, block sparse tensor trains – instances of tensor networks applied to function dictionaries – are used and the self similarity of the problem is exploited.  \nFor the latter, we propose an approach of gauge mediated weight sharing, inspired by notions of machine learning, which significantly improves performance over previous approaches. The practical performance of the method is demonstrated numerically on three one-dimensional systems – the Fermi-Pasta-Ulam-Tsingou system, rotating magnetic dipoles and classical particles interacting via modified Lennard-Jones potentials. We highlight the ability of the method to recover these systems, requiring 1400 samples to recover the 50 particle Fermi-Pasta-Ulam-Tsingou system to residuum of  \n5 􀀂 10 􀀀7 , 900 samples to recover the 50 particle magnetic dipole chain to residuum of 1:5 􀀂 10 􀀀4 and 7000 samples to recover the Lennard-Jones system of 10 particles to residuum 1:5 􀀂 10 􀀀2 . The robustness against additive Gaussian noise is demonstrated for the magnetic dipole system.  \nKeywords Dynamical laws recovery 􀀁 machine learning 􀀁 tensor trains 􀀁 block sparse tensor trains 􀀁 tensor networks 􀀁 gauge mediated weight sharing  \nA PREPRINT-AUGUST 5, 2022  \n1 Introduction  \nComing-up with dynamical laws that govern the behaviour of a complex physical many-body systems has been a daunting and challenging task of great practical importance for centuries. The availability of large amounts of data and the increase in today's computing power has drastically changed our perspective on these types of problems, shifting the focus to automated or largely `data-driven' approaches [SL09, BPK16, GKES19, GGR+ 20, IMW+ 20, CPSW21, KBK22, CDA+ 21], augmented by meaningful hypotheses about the general form of the underlying dynamical laws. In this way, one can think of `discovering physical laws' from data.  \nOne prominent recent approach is the sparse identification of non-linear dynamics (SINDy) algorithm [BPK16, SBK21, dSCQ+ 20] . The state of a dynamical system is described by a set of d real variables (x(1) ; : : : ; x (d)) =: x, which evolve along smooth trajectories t !7 x (t) 2 Rd , as given by a set of d ordinary differential equations (ODE), such as  \nx_ (t) = f(x(t); t) or (t) = f(x(t); t): (1)  \nThe first form can be thought of as a Hamiltonian system, where f = fx; Hg with H the Hamiltonian and f􀀁 ; 􀀁g the Poisson bracket; the second form can be interpreted as Newton's equations, in which case f are the forces. In this formulation, learning dynamical laws means obtaining an approximation of f : Rd 􀀂 R ! Rd from data, i.e., given M data pairs (xi ; yi) 2 Rd􀀂d for i = 1; : : : ; M , with the relation yi 􀀙 f (xi) . From now on we restrict ourselves to the case where f does not explicitly depend on time. In the SINDy approach, the learning task is phrased asa linear inversion problem. Introducing a dictionary of functions, the problem reduces to finding the coefficients that linearly combine functions in the dictionary to f. By promoting sparse solution one favours simple models and yield interpretable and generalizable r","cbCaijLqRXhnvWx3","https://ap.wps.com/l/cbCaijLqRXhnvWx3","pdf",546316,1,13,"English","en",105,"# Introduction\n## Data-driven discovery of dynamical laws\n## Sparse identification of nonlinear dynamics (SINDy)\n## Curse of dimensionality and locality\n## Tensor networks and scalable parametrizations","[{\"question\":\"What problem does the method address?\",\"answer\":\"It focuses on learning dynamical laws of classical dynamical systems from data efficiently under meaningful hypotheses.\"},{\"question\":\"What is the key new ingredient for scalability?\",\"answer\":\"Block sparse tensor trains are used as function dictionaries, leveraging self-similarity to improve scaling with system size.\"},{\"question\":\"How does gauge mediated weight sharing help?\",\"answer\":\"It provides a weight-sharing mechanism that is inspired by machine learning and yields significantly better performance than previous approaches.\"}]","Block Sparsity and Gauge Mediated Weight Sharing for Learning Dynamical Laws from Data - 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