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Syst."],"published-print":{"date-parts":[[2027,3]]},"abstract":"<jats:p>Modeling deep neural networks as continuous-time dynamical systems has emerged as a powerful approach for analyzing their evolution and robustness. By leveraging the superior numerical stability of implicit methods, several studies have proposed the implicit Euler-based architecture, which significantly enhances both robustness and generalization under adversarial attack. However, existing methods require solving implicit equations iteratively, which substantially increases the computational cost. To overcome these limitations, this paper proposes an explicit reformulation of the implicit Euler method, termed Implicit-to-EXplicit ResNet\u00a0(IEX-ResNet). IEX-ResNet preserves the inherent numerical stability of implicit methods while eliminating the need for iterative solvers, thereby enabling the efficient training of robust neural models. Theoretical analysis demonstrates that IEX-ResNet, under local linearization and spectral radius constraint, achieves Lipschitz-bounded perturbation propagation and maintains a stable approximation of the underlying implicit dynamics. Experimental results show that, compared to conventional implicit Euler architectures, IEX-ResNet achieves improved robustness across diverse challenging settings and maintains competitive robustness while reducing training time by nearly 50% under standard evaluation. Moreover, IEX-ResNet provides practical benefits in a real-world document image restoration task. These results suggest that implicit Euler explicitization is a practical route toward efficient robust neural architecture design.<\/jats:p>","DOI":"10.1142\/s0129065727500146","type":"journal-article","created":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T02:19:32Z","timestamp":1784859572000},"source":"Crossref","is-referenced-by-count":0,"title":["Robust and Efficient Neural Architecture Design via Implicit Euler Explicitization"],"prefix":"10.1142","volume":"37","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-6274-4676","authenticated-orcid":false,"given":"Yue","family":"Huang","sequence":"first","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"},{"name":"Key Laboratory of Digital Innovation of Tianfu Culture, Sichuan Provincial Department of Culture and Tourism, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8739-9795","authenticated-orcid":false,"given":"Jianwei","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"},{"name":"Key Laboratory of Digital Innovation of Tianfu Culture, Sichuan Provincial Department of Culture and Tourism, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9702-6738","authenticated-orcid":false,"given":"Lei","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"},{"name":"School of Artificial Intelligence, Sichuan University, Chengdu 610065, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3907-9388","authenticated-orcid":false,"given":"Yan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Sichuan University, Chengdu 610065, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4512-6956","authenticated-orcid":false,"given":"Jie","family":"Wu","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-4574-145X","authenticated-orcid":false,"given":"Zhanji","family":"Shi","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-0287-1795","authenticated-orcid":false,"given":"Rui","family":"Gao","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-8961-084X","authenticated-orcid":false,"given":"Yilin","family":"Guo","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2265-0085","authenticated-orcid":false,"given":"Chang","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-3248-4759","authenticated-orcid":false,"given":"Chaobang","family":"Gao","sequence":"additional","affiliation":[{"name":"College of Computer Science, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"},{"name":"Key Laboratory of Digital Innovation of Tianfu Culture, Sichuan Provincial Department of Culture and Tourism, Chengdu University, Chengdu 610106, Sichuan, P.\u00a0R.\u00a0China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2026,8,19]]},"reference":[{"key":"S0129065727500146BIB001","doi-asserted-by":"publisher","DOI":"10.1038\/nature14539"},{"key":"S0129065727500146BIB002","unstructured":"C. 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