{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T12:29:02Z","timestamp":1786364942332,"version":"build-2736575974"},"reference-count":24,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T00:00:00Z","timestamp":1786320000000},"content-version":"vor","delay-in-days":221,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:p>Currently, ordinary differential equations play an important role in various fields, which is of great significance for scientific research. However, traditional inversion methods for ordinary differential equations often suffer from poor inversion bias and poor ability to optimize equation parameters. To improve the inversion performance of ordinary differential equations, a linear and nonlinear equation inversion method based on an improved multistep neural network (MSNN) is proposed. A new method for ordinary differential equation inversion based on improved MSNNs and a two\u2010stage Monte\u2013Carlo optimization is proposed. An MSNN architecture with a time\u2010step adaptive mechanism is designed to uniformly handle linear and nonlinear problems, and a two\u2010stage optimization strategy combining coarse sampling and covariance matrix fine\u2010tuning significantly improves parameter estimation efficiency. L2 regularization and momentum sampling are introduced to enhance noise resistance. Experiments showed that this method reduced the maximum inversion deviation by 3.2 and shortened the convergence time by 24\u2009s, providing an efficient and stable solution for complex system parameter inversion.<\/jats:p>","DOI":"10.1155\/jama\/9972766","type":"journal-article","created":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T11:39:06Z","timestamp":1786361946000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linear and Nonlinear Ordinary Differential Equation Inversion Method Based on Improved Multistep Neural Network"],"prefix":"10.1155","volume":"2026","author":[{"given":"Yingping","family":"Song","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-5103-785X","authenticated-orcid":false,"given":"Qingqing","family":"Cao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Feng","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tian","family":"Gan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,10]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.nucengdes.2024.113094"},{"key":"e_1_2_9_2_2","unstructured":"JiangE. PengJ. MaZ. andYanX. B. ODE-DPS: ODE-Based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation 21 no. 6 13496\u201313497 arXiv: 2404.2024."},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.59849\/2218-6816.2024.1.3"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2023.10.007"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-023-09415-5"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.geomphys.2024.105168"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0219887824501317"},{"key":"e_1_2_9_8_2","first-page":"395","volume-title":"European Conference on Computer Vision","author":"Garibi D.","year":"2025"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.28924\/2291-8639-22-2024-65"},{"key":"e_1_2_9_10_2","unstructured":"ChoH. LeeJ. KimS. B. OhT. H. andJeongY. 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