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To address the challenge of discovering Lax pairs from observational data, we propose D-Lax-Pair-FIND, a data-driven methodology that synergistically combines deep learning with sparse regression to automatically discover the evolution operators in Lax pairs of discrete integrable equations with known spectral operators. The approach employs fully connected feedforward neural networks to approximate solutions of semi-discrete systems, treating the discrete spatial variable as continuous to enable interpolation at non-integer lattice points. By constructing a predefined library of candidate operators and leveraging sparse regularization techniques, the method automatically identifies the linear evolution operators. The loss function comprises three key components: data-fitting loss, discrete Lax compatibility loss, and sparse regularization loss. Through joint optimization of the neural network parameters and the sparse coefficient vector, the method achieves robust discovery of Lax pairs. Numerical experiments demonstrate that the proposed framework successfully recovers the evolution operators in Lax pairs for several classical discrete integrable systems, including the self-dual network equation, the Ablowitz\u2013Ladik equation, and the Toda lattice\u2014covering both the form with explicit spectral parameters and the operator form. This research provides an effective and general computational framework for automatically uncovering the intrinsic structure of discrete integrable systems from limited data, with potential implications for advancing data-driven studies in integrable systems theory.<\/jats:p>","DOI":"10.1063\/5.0336825","type":"journal-article","created":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T12:50:36Z","timestamp":1791204636000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["D-Lax-Pair-FIND: Data-driven reconstruction of the evolution operators in Lax pairs for discrete integrable equations"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3438-0860","authenticated-orcid":false,"given":"Shuning","family":"Lin","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science, Fudan University 1 , Shanghai 200433,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6008-6542","authenticated-orcid":false,"given":"Yong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) and Shanghai Key Laboratory of PMMP, East China Normal University 2 , Shanghai 200241,","place":["China"]},{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology 3 , Qingdao 266590,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,5]]},"reference":[{"issue":"45","key":"2026100508503409700_c1","doi-asserted-by":"publisher","first-page":"22445","DOI":"10.1073\/pnas.1906995116","article-title":"Data-driven discovery of coordinates and governing equations","volume":"116","year":"2019","journal-title":"Proc. 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