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We provide rigorous error analyses of continuous and discrete MGDL models, showing that the discrete model retains the convergence and stability of its continuous counterpart under sufficiently small quadrature error. We identify the DNN training error as the primary source of approximation error, motivating a novel adaptive MGDL algorithm that selects the network grade based on training performance. Numerical experiments with highly oscillatory (including wavenumber 500) and singular solutions confirm the accuracy, effectiveness and robustness of the proposed approach.<\/jats:p>","DOI":"10.1007\/s10915-026-03189-9","type":"journal-article","created":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T02:02:18Z","timestamp":1769824938000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adaptive Multi-Grade Deep Learning for Highly Oscillatory Fredholm Integral Equations of the Second Kind"],"prefix":"10.1007","volume":"106","author":[{"given":"Jie","family":"Jiang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuesheng","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,31]]},"reference":[{"key":"3189_CR1","volume-title":"Collectively Compact Operator Approximation Theory and Applications to Integral Equations","author":"PM Anselone","year":"1971","unstructured":"Anselone, P.M.: Collectively Compact Operator Approximation Theory and Applications to Integral Equations. 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