{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T20:34:39Z","timestamp":1773002079980,"version":"3.50.1"},"reference-count":37,"publisher":"Institution of Engineering and Technology (IET)","issue":"1","license":[{"start":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T00:00:00Z","timestamp":1759881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"},{"start":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T00:00:00Z","timestamp":1759881600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/http\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["ietresearch.onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["CAAI Trans on Intel Tech"],"published-print":{"date-parts":[[2026,2]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>Monge\u2013Amp\u00e8re equations (MAEs) are fully nonlinear second\u2010order partial differential equations (PDEs), which are closely related to various fields including optimal transport (OT) theory, geometrical optics and affine geometry. Despite their significance, MAEs are extremely challenging to solve. Although some classical numerical approaches can solve MAEs, their computational efficiency deteriorates significantly on fine grids, with convergence often heavily dependent on the quality of initial estimate. Research on deep learning methods for solving MAEs is still in its early stages, which predominantly addresses simple formulations with basic Dirichlet boundary conditions. Here, we propose a deep learning method based on physics\u2010driven deep neural networks, enabling the solution of both simple and generalised MAEs with transport boundary conditions. In this method, we deal with two first\u2010order sub\u2010equations separated from MAE instead of solving the single MAE directly, which facilitates the imposition of transport boundary conditions and simplifies the training of neural networks. Moreover, we constrain the convexity of solution using the Lagrange multiplier method and maintain the optimisation process differentiable with bilinear interpolation. We provide three progressively complex examples ranging from a simple MAE with an analytical solution to a highly nonlinear variant arising in phase retrieval to validate the effectiveness of our method. For comparison, we benchmark against state\u2010of\u2010the\u2010art deep learning approaches that have been systematically adapted to accommodate the specific requirements of each example.<\/jats:p>","DOI":"10.1049\/cit2.70067","type":"journal-article","created":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T21:24:37Z","timestamp":1759958677000},"page":"15-25","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Physics\u2010Driven Deep Neural Networks for Solving the Optimal Transport Problem Associated With the Monge\u2013Amp\u00e8re Equation"],"prefix":"10.1049","volume":"11","author":[{"given":"Xinghua","family":"Pan","sequence":"first","affiliation":[{"name":"Beijing Engineering Research Center of Mixed Reality and Advanced Display School of Optics and Photonics Beijing Institute of Technology  Beijing China"},{"name":"MOE Key Laboratory of Optoelectronic Imaging Technology and Systems Beijing Institute of Technology  Beijing China"}]},{"given":"Zexin","family":"Feng","sequence":"additional","affiliation":[{"name":"Beijing Engineering Research Center of Mixed Reality and Advanced Display School of Optics and Photonics Beijing Institute of Technology  Beijing China"},{"name":"MOE Key Laboratory of Optoelectronic Imaging Technology and Systems Beijing Institute of Technology  Beijing China"}]},{"given":"Kang","family":"Yang","sequence":"additional","affiliation":[{"name":"Huawei Technologies Co. Ltd  Hangzhou China"}]}],"member":"265","published-online":{"date-parts":[[2025,10,8]]},"reference":[{"key":"e_1_2_9_2_1","first-page":"666","article-title":"M\u00e9moire Sur la Th\u00e9orie Des D\u00e9blais Et Des Remblais","author":"Monge G.","year":"1781","journal-title":"Histoire de l\u2019Acadmie Royale des Sciences"},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICIP46576.2022.9897650"},{"key":"e_1_2_9_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/tpami.2015.2408346"},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.1364\/ol.38.000229"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.eng.2019.09.010"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2010017"},{"key":"e_1_2_9_8_1","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2008.10.221"},{"key":"e_1_2_9_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10915\u2010018\u20100698\u20106"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2016.1247443"},{"key":"e_1_2_9_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2004.12.018"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.1364\/ol.44.005501"},{"key":"e_1_2_9_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0095270"},{"key":"e_1_2_9_15_1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0071616"},{"key":"e_1_2_9_16_1","doi-asserted-by":"publisher","DOI":"10.3390\/axioms12100982"},{"key":"e_1_2_9_17_1","doi-asserted-by":"publisher","DOI":"10.1142\/s0218348x23401035"},{"key":"e_1_2_9_18_1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0116038"},{"key":"e_1_2_9_19_1","doi-asserted-by":"publisher","DOI":"10.1063\/5.0153705"},{"key":"e_1_2_9_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcmds.2023.100080"},{"key":"e_1_2_9_21_1","unstructured":"A.Singh M.Bauer andS.Joshi \u201cPhysics Informed Convex Artificial Neural Networks (Picanns) for Optimal Transport Based Density Estimation \u201d arXiv preprint arXiv:2104.01194 (2021)."},{"key":"e_1_2_9_22_1","first-page":"146","volume-title":"Proceedings of Machine Learning Research (PMLR)","author":"Amos B.","year":"2017"},{"key":"e_1_2_9_23_1","doi-asserted-by":"publisher","DOI":"10.1364\/josaa.34.001490"},{"key":"e_1_2_9_24_1","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.90"},{"key":"e_1_2_9_25_1","unstructured":"P.Ramachandran B.Zoph andQ. 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