{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:30:36Z","timestamp":1760146236928,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T00:00:00Z","timestamp":1729209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"EPSRC grant","doi-asserted-by":"publisher","award":["EP\/X019918\/1"],"award-info":[{"award-number":["EP\/X019918\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The moving sofa problem, introduced by Leo Moser in 1966, seeks to determine the maximal area of a 2D shape that can navigate an L-shaped corridor of unit width. Joseph Gerver\u2019s 1992 solution, providing a lower bound of approximately 2.2195, is the best known, though its global optimality remains unproven. This paper leverages neural networks\u2019 approximation power and recent advances in invexity optimization to explore global optimality. We propose two approaches supporting Gerver\u2019s conjecture that his sofa is the unique global maximum. The first approach uses continuous function learning, discarding assumptions about the monotonicity, symmetry, and differentiability of sofa movements. The sofa area is computed as a differentiable function using our \u201cwaterfall\u201d algorithm, with the loss function incorporating both differential terms and initial conditions based on physics-informed machine learning. Extensive training with diverse network initialization consistently converges to Gerver\u2019s solution. The second approach applies discrete optimization to the Kallus\u2013Romik upper bound, improving it from 2.37 to 2.3337 for five rotation angles. As the number of angles increases, our model asymptotically converges to Gerver\u2019s area from above, indicating that no larger sofa exists.<\/jats:p>","DOI":"10.3390\/sym16101388","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T06:46:52Z","timestamp":1729234012000},"page":"1388","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Deep Learning Evidence for Global Optimality of Gerver\u2019s Sofa"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3007-1825","authenticated-orcid":false,"given":"Kuangdai","family":"Leng","sequence":"first","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}]},{"given":"Jia","family":"Bi","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}]},{"given":"Jaehoon","family":"Cha","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}]},{"given":"Samuel","family":"Pinilla","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2167-1343","authenticated-orcid":false,"given":"Jeyan","family":"Thiyagalingam","sequence":"additional","affiliation":[{"name":"Scientific Computing Department, Science and Technology Facilities Council\u2014Rutherford Appleton Laboratory, Didcot OX11 0QX, UK"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/1008074","article-title":"Problem 66-11, Moving furniture through a hallway","volume":"8","author":"Moser","year":"1966","journal-title":"SIAM Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF00426226","article-title":"On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities","volume":"1","author":"Hammersley","year":"1968","journal-title":"Educ. Stud. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/BF02414066","article-title":"On moving a sofa around a corner","volume":"42","author":"Gerver","year":"1992","journal-title":"Geom. Dedicata"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1080\/10586458.2016.1270858","article-title":"Differential equations and exact solutions in the moving sofa problem","volume":"27","author":"Romik","year":"2018","journal-title":"Exp. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Batsch, M. (2022). A numerical approach for analysing the moving sofa problem. Symmetry, 14.","DOI":"10.3390\/sym14071409"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1016\/j.aim.2018.10.022","article-title":"Improved upper bounds in the moving sofa problem","volume":"340","author":"Kallus","year":"2018","journal-title":"Adv. Math."},{"key":"ref_7","first-page":"1097","article-title":"ImageNet classification with deep convolutional neural networks","volume":"Volume 25","author":"Krizhevsky","year":"2012","journal-title":"Proceedings of the Advances in Neural Information Processing Systems"},{"key":"ref_8","first-page":"5998","article-title":"Attention is all you need","volume":"Volume 30","author":"Vaswani","year":"2017","journal-title":"Proceedings of the Advances in Neural Information Processing Systems"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1038\/s41591-018-0316-z","article-title":"A guide to deep learning in healthcare","volume":"25","author":"Esteva","year":"2019","journal-title":"Nat. Med."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.isprsjprs.2023.03.005","article-title":"Developing an intelligent cloud attention network to support global urban green spaces mapping","volume":"198","author":"Chen","year":"2023","journal-title":"ISPRS J. Photogramm. Remote Sens."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1038\/s41586-022-05172-4","article-title":"Discovering faster matrix multiplication algorithms with reinforcement learning","volume":"610","author":"Fawzi","year":"2022","journal-title":"Nature"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1038\/s41586-023-06924-6","article-title":"Mathematical discoveries from program search with large language models","volume":"625","author":"Barekatain","year":"2024","journal-title":"Nature"},{"key":"ref_13","unstructured":"Gukov, S., Halverson, J., Manolescu, C., and Ruehle, F. (2023). Searching for ribbons with machine learning. arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1002\/cnm.1640100303","article-title":"Neural-network-based approximations for solving partial differential equations","volume":"10","author":"Dissanayake","year":"1994","journal-title":"Commun. Numer. Methods Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/72.712178","article-title":"Artificial neural networks for solving ordinary and partial differential equations","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. Comput. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_18","unstructured":"Pinilla, S., and Thiyagalingam, J. (2024, January 7\u201311). Global Optimality for Non-linear Constrained Restoration Problems via Invexity. Proceedings of the Twelfth International Conference on Learning Representations, Vienna, Austria."},{"key":"ref_19","first-page":"10780","article-title":"Improved imaging by invex regularizers with global optima guarantees","volume":"35","author":"Pinilla","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_20","unstructured":"Esporesto (2016, May 15). Moving Sofa. Online Math Tools, GeoGebra. Available online: https:\/\/www.geogebra.org\/m\/vemEtGyj."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1109\/34.87344","article-title":"Watersheds in digital spaces: An efficient algorithm based on immersion simulations","volume":"13","author":"Vincent","year":"1991","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_22","unstructured":"De Bock, J., De Smet, P., and Philips, W. (2005, January 20\u201323). A fast sequential rainfalling watershed segmentation algorithm. Proceedings of the Advanced Concepts for Intelligent Vision Systems: 7th International Conference, ACIVS 2005, Antwerp, Belgium. Proceedings 7."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1162\/neco.1991.3.2.246","article-title":"Universal approximation using radial-basis-function networks","volume":"3","author":"Park","year":"1991","journal-title":"Neural Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1038\/s42256-021-00302-5","article-title":"Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators","volume":"3","author":"Lu","year":"2021","journal-title":"Nat. Mach. Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1038\/s43586-022-00125-7","article-title":"Piecewise linear neural networks and deep learning","volume":"2","author":"Tao","year":"2022","journal-title":"Nat. Rev. Methods Prim."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Leng, K., and Thiyagalingam, J. (2022). On the compatibility between neural networks and partial differential equations for physics-informed learning. arXiv.","DOI":"10.2139\/ssrn.4392241"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mishra, S.K., and Giorgi, G. (2008). Invexity and Optimization, Springer Science & Business Media.","DOI":"10.1007\/978-3-540-78562-0"},{"key":"ref_28","unstructured":"Kingma, D.P., and Ba, J. (2015, January 7\u20139). Adam: A Method for Stochastic Optimization. Proceedings of the ICLR, San Diego, CA, USA."},{"key":"ref_29","unstructured":"Loshchilov, I., and Hutter, F. (2017). Decoupled weight decay regularization. arXiv."},{"key":"ref_30","unstructured":"Post, J.V. (2007, May 05). Sequence A128463 in the On-Line Encyclopedia of Integer Sequences. Available online: https:\/\/oeis.org\/A128463."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/BF01589116","article-title":"On the limited memory BFGS method for large scale optimization","volume":"45","author":"Liu","year":"1989","journal-title":"Math. Program."},{"key":"ref_32","first-page":"6391","article-title":"Visualizing the loss landscape of neural nets","volume":"31","author":"Li","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"023401","DOI":"10.1088\/1742-5468\/ad13fc","article-title":"Visualizing high-dimensional loss landscapes with Hessian directions","volume":"2024","author":"Wheeler","year":"2024","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Math. Control Signals Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","article-title":"Approximation capabilities of multilayer feedforward networks","volume":"4","author":"Hornik","year":"1991","journal-title":"Neural Netw."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0925-2312(98)00111-8","article-title":"Lower bounds for approximation by MLP neural networks","volume":"25","author":"Maiorov","year":"1999","journal-title":"Neurocomputing"},{"key":"ref_37","first-page":"6232","article-title":"The expressive power of neural networks: A view from the width","volume":"30","author":"Lu","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/0022-247X(81)90123-2","article-title":"On sufficiency of the kuhn-tucker conditions","volume":"80","author":"Hanson","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S1446788700022126","article-title":"Invex functions and duality","volume":"39","author":"Craven","year":"1985","journal-title":"J. Aust. Math. Soc."},{"key":"ref_40","first-page":"23245","article-title":"Fair sparse regression with clustering: An invex relaxation for a combinatorial problem","volume":"34","author":"Barik","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1109\/LSP.2013.2283425","article-title":"Invexity of the minimum error entropy criterion","volume":"20","author":"Syed","year":"2013","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"3376","DOI":"10.1109\/TSP.2016.2539127","article-title":"Generalized correntropy for robust adaptive filtering","volume":"64","author":"Chen","year":"2016","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1388\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:16:02Z","timestamp":1760112962000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1388"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,18]]},"references-count":42,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101388"],"URL":"https:\/\/doi.org\/10.3390\/sym16101388","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,10,18]]}}}