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However, neural networks persistently face challenges related to theoretical understanding, interpretability, and generalization. Thus, researchers are increasingly adopting a differential equations perspective to propose a unified theoretical framework and systematic design methodologies. We establish a comprehensive taxonomy of the literature from two distinct perspectives. First, we classify the models based on their mathematical formulations, covering ordinary, partial, and stochastic differential equations (ODEs, PDEs, and SDEs). Second, we categorize them by their methodological roles, distinguishing between equation-guided networks and networks designed for solving equations. Moreover, we conduct a qualitative meta-analysis supported by consolidated evidence summaries from existing literature to illustrate their characteristics and performance. Finally, we explore promising research directions in integrating differential equations with deep learning to offer new insights for developing computational methods with improved interpretability and generalization.<\/jats:p>","DOI":"10.1145\/3828753","type":"journal-article","created":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T21:03:46Z","timestamp":1782939826000},"page":"1-37","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Deep Neural Networks Inspired by Differential Equations"],"prefix":"10.1145","volume":"58","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-8388-5374","authenticated-orcid":false,"given":"Yongshuai","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-1062-2108","authenticated-orcid":false,"given":"Lianfang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-8205-9187","authenticated-orcid":false,"given":"Kuilin","family":"Qin","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-9647-2300","authenticated-orcid":false,"given":"Qinghua","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5317-562X","authenticated-orcid":false,"given":"Faqiang","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3311-9054","authenticated-orcid":false,"given":"Li","family":"Cui","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8697-3089","authenticated-orcid":false,"given":"Jun","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2433-3056","authenticated-orcid":false,"given":"Yuping","family":"Duan","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beijing Normal University","place":["Beijing, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-0626-8782","authenticated-orcid":false,"given":"Tieyong","family":"Zeng","sequence":"additional","affiliation":[{"name":"The Chinese University of Hong Kong","place":["Hong Kong, China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,7,28]]},"reference":[{"issue":"1","key":"e_1_3_2_2_2","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1038\/s41746-021-00438-z","article-title":"Diagnostic accuracy of deep learning in medical imaging: A systematic review and meta-analysis","volume":"4","author":"Aggarwal Ravi","year":"2021","unstructured":"Ravi Aggarwal, Viknesh Sounderajah, Guy Martin, Daniel S. 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Retrieved from https:\/\/arxiv.org\/abs\/2307.08698"},{"key":"e_1_3_2_43_2","first-page":"139054","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Davis Oscar","year":"2024","unstructured":"Oscar Davis, Samuel Kessler, Mircea Petrache, Ismail Ceylan, Michael Bronstein, and Joey Bose. 2024. Fisher flow matching for generative modeling over discrete data. In Proceedings of the Advances in Neural Information Processing Systems. 139054\u2013139084."},{"key":"e_1_3_2_44_2","unstructured":"Arturo De Marinis Nicola Guglielmi Stefano Sicilia and Francesco Tudisco. 2025. Stability of neural ODEs by a control over the expansivity of their flows. arXiv:2501.10740. Retrieved from https:\/\/arxiv.org\/abs\/2501.10740"},{"key":"e_1_3_2_45_2","first-page":"2676","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Dhifallah Omar","year":"2021","unstructured":"Omar Dhifallah and Yitong Lu. 2021. On the inherent regularization effects of noise injection during training. In Proceedings of the International Conference on Machine Learning. 2676\u20132686."},{"key":"e_1_3_2_46_2","unstructured":"Finale Doshi-Velez and Been Kim. 2017. Towards a rigorous science of interpretable machine learning. arXiv:1702.08608. Retrieved from https:\/\/arxiv.org\/abs\/1702.08608"},{"key":"e_1_3_2_47_2","first-page":"8678","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Du Weitao","year":"2023","unstructured":"Weitao Du, He Zhang, Tao Yang, and Yuanqi Du. 2023. A flexible diffusion model. In Proceedings of the International Conference on Machine Learning. 8678\u20138696."},{"key":"e_1_3_2_48_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Dupont Emilien","year":"2019","unstructured":"Emilien Dupont, Arnaud Doucet, and Yee Whye Teh. 2019. Augmented neural odes. 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In Proceedings of the Advances in Neural Information Processing Systems. 133345\u2013133385."},{"key":"e_1_3_2_60_2","first-page":"75460","article-title":"Mean flows for one-step generative modeling","volume":"38","author":"Geng Zhengyang","year":"2026","unstructured":"Zhengyang Geng, Mingyang Deng, Xingjian Bai, Zico Kolter, and Kaiming He. 2026. Mean flows for one-step generative modeling. Advances in Neural Information Processing Systems 38 (2026), 75460\u201375482.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_3_2_61_2","first-page":"30467","article-title":"Improved mean flows: On the challenges of fast-forward generative models","author":"Geng Zhengyang","year":"2026","unstructured":"Zhengyang Geng, Yiyang Lu, Zongze Wu, Eli Shechtman, J. Zico Kolter, and Kaiming He. 2026. Improved mean flows: On the challenges of fast-forward generative models. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR). 30467\u201330476.","journal-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR)"},{"key":"e_1_3_2_62_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Ghiasi Golnaz","year":"2018","unstructured":"Golnaz Ghiasi, Tsung-Yi Lin, and Quoc V. Le. 2018. Dropblock: A regularization method for convolutional networks. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_63_2","first-page":"14831","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Ghosh Arnab","year":"2020","unstructured":"Arnab Ghosh, Harkirat Behl, Emilien Dupont, Philip Torr, and Vinay Namboodiri. 2020. Steer: Simple temporal regularization for neural ode. In Proceedings of the Advances in Neural Information Processing Systems. 14831\u201314843."},{"key":"e_1_3_2_64_2","doi-asserted-by":"crossref","unstructured":"Patryk Gierjatowicz Marc Sabate-Vidales David \u0160i\u0161ka Lukasz Szpruch and \u017dan \u017duri\u010d. 2020. Robust pricing and hedging via neural SDEs. arXiv:2007.04154. Retrieved from https:\/\/arxiv.org\/abs\/2007.04154","DOI":"10.2139\/ssrn.3646241"},{"key":"e_1_3_2_65_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Gomez Aidan N.","year":"2017","unstructured":"Aidan N. Gomez, Mengye Ren, Raquel Urtasun, and Roger B. Grosse. 2017. The reversible residual network: Backpropagation without storing activations. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_66_2","unstructured":"Aidan N. Gomez Ivan Zhang Siddhartha Rao Kamalakara Divyam Madaan Kevin Swersky Yarin Gal and Geoffrey E. Hinton. 2019. Learning sparse networks using targeted dropout. arXiv:1905.13678. Retrieved from https:\/\/arxiv.org\/abs\/1905.13678"},{"key":"e_1_3_2_67_2","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/978-3-031-36644-4_6","volume-title":"Proceedings of the Machine Learning in Modeling and Simulation: Methods and Applications","author":"Goswami Somdatta","year":"2023","unstructured":"Somdatta Goswami, Aniruddha Bora, Yue Yu, and George Em Karniadakis. 2023. Physics-informed deep neural operator networks. In Proceedings of the Machine Learning in Modeling and Simulation: Methods and Applications. Springer, 219\u2013254."},{"key":"e_1_3_2_68_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Greydanus Samuel","year":"2019","unstructured":"Samuel Greydanus, Misko Dzamba, and Jason Yosinski. 2019. Hamiltonian neural networks. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_69_2","unstructured":"Albert Gu and Tri Dao. 2023. Mamba: Linear-time sequence modeling with selective state spaces. arXiv:2312.00752. Retrieved from https:\/\/arxiv.org\/abs\/2312.00752"},{"key":"e_1_3_2_70_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Gu Albert","year":"2022","unstructured":"Albert Gu, Karan Goel, and Christopher Re. 2022. Efficiently modeling long sequences with structured state spaces. In Proceedings of the International Conference on Learning Representations. Retrieved from https:\/\/openreview.net\/forum?id=uYLFoz1vlAC"},{"key":"e_1_3_2_71_2","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Guo Pengsheng","year":"2025","unstructured":"Pengsheng Guo and Alex Schwing. 2025. Variational rectified flow matching. In Proceedings of the International Conference on Machine Learning."},{"key":"e_1_3_2_72_2","first-page":"15642","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Guo Yuan","year":"2025","unstructured":"Yuan Guo, Jingyu Kong, Yu Wang, and Yuping Duan. 2025. Take the bull by the horns: Learning to segment hard samples. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 15642\u201315652."},{"issue":"1","key":"e_1_3_2_73_2","doi-asserted-by":"crossref","first-page":"014004","DOI":"10.1088\/1361-6420\/aa9a90","article-title":"Stable architectures for deep neural networks","volume":"34","author":"Haber Eldad","year":"2017","unstructured":"Eldad Haber and Lars Ruthotto. 2017. Stable architectures for deep neural networks. Inverse Problems 34, 1 (2017), 014004.","journal-title":"Inverse Problems"},{"key":"e_1_3_2_74_2","doi-asserted-by":"publisher","DOI":"10.1145\/3594869"},{"key":"e_1_3_2_75_2","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v35i9.16936"},{"key":"e_1_3_2_76_2","first-page":"15464","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Hayou Soufiane","year":"2021","unstructured":"Soufiane Hayou and Fadhel Ayed. 2021. Regularization in resnet with stochastic depth. In Proceedings of the Advances in Neural Information Processing Systems. 15464\u201315474."},{"key":"e_1_3_2_77_2","doi-asserted-by":"crossref","first-page":"117130","DOI":"10.1016\/j.cma.2024.117130","article-title":"Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries","volume":"429","author":"He Junyan","year":"2024","unstructured":"Junyan He, Seid Koric, Diab Abueidda, Ali Najafi, and Iwona Jasiuk. 2024. Geom-deeponet: A point-cloud-based deep operator network for field predictions on 3d parameterized geometries. Computer Methods in Applied Mechanics and Engineering 429 (2024), 117130.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"e_1_3_2_78_2","doi-asserted-by":"crossref","first-page":"107258","DOI":"10.1016\/j.engappai.2023.107258","article-title":"Sequential deep operator networks (s-deeponet) for predicting full-field solutions under time-dependent loads","volume":"127","author":"He Junyan","year":"2024","unstructured":"Junyan He, Shashank Kushwaha, Jaewan Park, Seid Koric, Diab Abueidda, and Iwona Jasiuk. 2024. Sequential deep operator networks (s-deeponet) for predicting full-field solutions under time-dependent loads. Engineering Applications of Artificial Intelligence 127 (2024), 107258.","journal-title":"Engineering Applications of Artificial Intelligence"},{"key":"e_1_3_2_79_2","first-page":"770","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"He Kaiming","year":"2016","unstructured":"Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. Deep residual learning for image recognition. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 770\u2013778."},{"key":"e_1_3_2_80_2","first-page":"1732","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"He Xiangyu","year":"2019","unstructured":"Xiangyu He, Zitao Mo, Peisong Wang, Yang Liu, Mingyuan Yang, and Jian Cheng. 2019. Ode-inspired network design for single image super-resolution. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 1732\u20131741."},{"key":"e_1_3_2_81_2","first-page":"588","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"He Zhezhi","year":"2019","unstructured":"Zhezhi He, Adnan Siraj Rakin, and Deliang Fan. 2019. Parametric noise injection: Trainable randomness to improve deep neural network robustness against adversarial attack. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 588\u2013597."},{"key":"e_1_3_2_82_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Herrera Calypso","year":"2021","unstructured":"Calypso Herrera, Florian Krach, and Josef Teichmann. 2021. Neural jump ordinary differential equations: Consistent continuous-time prediction and filtering. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_83_2","unstructured":"Geoffrey E. Hinton Nitish Srivastava Alex Krizhevsky Ilya Sutskever and Ruslan R. Salakhutdinov. 2012. Improving neural networks by preventing co-adaptation of feature detectors. arXiv:1207.0580. Retrieved from https:\/\/arxiv.org\/abs\/1207.0580"},{"key":"e_1_3_2_84_2","first-page":"6840","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Ho Jonathan","year":"2020","unstructured":"Jonathan Ho, Ajay Jain, and Pieter Abbeel. 2020. Denoising diffusion probabilistic models. In Proceedings of the Advances in Neural Information Processing Systems. 6840\u20136851."},{"key":"e_1_3_2_85_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Holderrieth Peter","year":"2025","unstructured":"Peter Holderrieth, Marton Havasi, Jason Yim, Neta Shaul, Itai Gat, Tommi Jaakkola, Brian Karrer, Ricky T. Q. Chen, and Yaron Lipman. 2025. Generator matching: Generative modeling with arbitrary Markov processes. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_86_2","doi-asserted-by":"crossref","first-page":"109246","DOI":"10.1016\/j.compeleceng.2024.109246","article-title":"Explainable AI approaches in deep learning: Advancements, applications and challenges","volume":"117","author":"Hosain Md Tanzib","year":"2024","unstructured":"Md Tanzib Hosain, Jamin Rahman Jim, M. F. Mridha, and Md Mohsin Kabir. 2024. Explainable AI approaches in deep learning: Advancements, applications and challenges. Computers and Electrical Engineering 117 (2024), 109246.","journal-title":"Computers and Electrical Engineering"},{"key":"e_1_3_2_87_2","volume-title":"Proceedings of the NeurIPS 2024 Workshop on AI for New Drug Modalities","author":"Hou Xiaoyang","year":"2024","unstructured":"Xiaoyang Hou, Tian Zhu, Milong Ren, Dongbo Bu, Xin Gao, Chunming Zhang, and Shiwei Sun. 2024. Improving molecular graph generation with flow matching and optimal transport. In Proceedings of the NeurIPS 2024 Workshop on AI for New Drug Modalities."},{"key":"e_1_3_2_88_2","unstructured":"Xing Hua Haodong Chen Qianqian Duan Danfeng Hong Ruijiao Li Huiliang Shang Linghua Jiang Haima Yang and Dawei Zhang. 2025. A comprehensive review of diffusion models in smart agriculture: Progress applications and challenges. arXiv:2507.18376. Retrieved from https:\/\/arxiv.org\/abs\/2507.18376"},{"key":"e_1_3_2_89_2","first-page":"4700","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Huang Gao","year":"2017","unstructured":"Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q. Weinberger. 2017. Densely connected convolutional networks. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 4700\u20134708."},{"key":"e_1_3_2_90_2","first-page":"646","volume-title":"Proceedings of the European Conference on Computer Vision","author":"Huang Gao","year":"2016","unstructured":"Gao Huang, Yu Sun, Zhuang Liu, Daniel Sedra, and Kilian Q. Weinberger. 2016. Deep networks with stochastic depth. In Proceedings of the European Conference on Computer Vision. 646\u2013661."},{"issue":"8","key":"e_1_3_2_91_2","doi-asserted-by":"crossref","first-page":"13649","DOI":"10.1109\/TNNLS.2025.3545967","article-title":"Partial differential equations meet deep neural networks: A survey","volume":"36","author":"Huang Shudong","year":"2025","unstructured":"Shudong Huang, Wentao Feng, Chenwei Tang, Zhenan He, Caiyang Yu, and Jiancheng Lv. 2025. Partial differential equations meet deep neural networks: A survey. IEEE Transactions on Neural Networks and Learning Systems 36, 8 (2025), 13649\u201313669.","journal-title":"IEEE Transactions on Neural Networks and Learning Systems"},{"key":"e_1_3_2_92_2","first-page":"1","volume-title":"Proceedings of the ACM SIGGRAPH 2024 Conference Papers","author":"Huang Xingchang","year":"2024","unstructured":"Xingchang Huang, Corentin Salaun, Cristina Vasconcelos, Christian Theobalt, Cengiz Oztireli, and Gurprit Singh. 2024. Blue noise for diffusion models. In Proceedings of the ACM SIGGRAPH 2024 Conference Papers. 1\u201311."},{"issue":"6","key":"e_1_3_2_93_2","doi-asserted-by":"crossref","first-page":"4409","DOI":"10.1109\/TPAMI.2025.3541625","article-title":"Diffusion model-based image editing: A survey","volume":"47","author":"Huang Yi","year":"2025","unstructured":"Yi Huang, Jiancheng Huang, Yifan Liu, Mingfu Yan, Jiaxi Lv, Jianzhuang Liu, Wei Xiong, He Zhang, Liangliang Cao, and Shifeng Chen. 2025. Diffusion model-based image editing: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence 47, 6 (2025), 4409\u20134437.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"e_1_3_2_94_2","first-page":"11102","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Issa Zacharia","year":"2023","unstructured":"Zacharia Issa, Blanka Horvath, Maud Lemercier, and Cristopher Salvi. 2023. Non-adversarial training of Neural SDEs with signature kernel scores. In Proceedings of the Advances in Neural Information Processing Systems. 11102\u201311126."},{"key":"e_1_3_2_95_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Jacobsen J\u00f6rn-Henrik","year":"2018","unstructured":"J\u00f6rn-Henrik Jacobsen, Arnold Smeulders, and Edouard Oyallon. 2018. i-RevNet: Deep invertible networks. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_96_2","article-title":"Neural tangent kernel: Convergence and generalization in neural networks","volume":"31","author":"Jacot Arthur","year":"2018","unstructured":"Arthur Jacot, Franck Gabriel, and Cl\u00e9ment Hongler. 2018. Neural tangent kernel: Convergence and generalization in neural networks. Advances in Neural Information Processing Systems 31 (2018).","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_3_2_97_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Jia Junteng","year":"2019","unstructured":"Junteng Jia and Austin R. Benson. 2019. Neural jump stochastic differential equations. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_98_2","doi-asserted-by":"crossref","first-page":"109567","DOI":"10.1016\/j.knosys.2022.109567","article-title":"An adaptive Drop method for deep neural networks regularization: Estimation of DropConnect hyperparameter using generalization gap","volume":"253","author":"Hssayni El Houssaine","year":"2022","unstructured":"El Houssaine Hssayni,Nour-Eddine Joudar,and Mohamed Ettaouil. 2022. An adaptive Drop method for deep neural networks regularization: Estimation of DropConnect hyperparameter using generalization gap. Knowledge-Based Systems 253 (2022), 109567.","journal-title":"Knowledge-Based Systems"},{"key":"e_1_3_2_99_2","first-page":"610","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Kag Anil","year":"2022","unstructured":"Anil Kag and Venkatesh Saligrama. 2022. Condensing cnns with partial differential equations. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 610\u2013619."},{"issue":"5","key":"e_1_3_2_100_2","doi-asserted-by":"crossref","first-page":"1245","DOI":"10.1109\/TPAMI.2017.2701831","article-title":"Shakeout: A new approach to regularized deep neural network training","volume":"40","author":"Kang Guoliang","year":"2017","unstructured":"Guoliang Kang, Jun Li, and Dacheng Tao. 2017. Shakeout: A new approach to regularized deep neural network training. IEEE Transactions on Pattern Analysis and Machine Intelligence 40, 5 (2017), 1245\u20131258.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"e_1_3_2_101_2","first-page":"14925","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kang Qiyu","year":"2021","unstructured":"Qiyu Kang, Yang Song, Qinxu Ding, and Wee Peng Tay. 2021. Stable neural ode with lyapunov-stable equilibrium points for defending against adversarial attacks. In Proceedings of the Advances in Neural Information Processing Systems. 14925\u201314937."},{"key":"e_1_3_2_102_2","first-page":"32","volume-title":"Proceedings of the Neural Information Processing Systems.","author":"Kapu\u0144niak Kacper","year":"2025","unstructured":"Kacper Kapu\u0144niak, Peter Potaptchik, Teodora Reu, Leo Zhang, Alexander Tong, Michael Bronstein, Avishek Joey Bose, and Francesco Di Giovanni. 2025. Metric flow matching for smooth interpolations on the data manifold. In Proceedings of the Neural Information Processing Systems.32 pages."},{"key":"e_1_3_2_103_2","first-page":"26565","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Karras Tero","year":"2022","unstructured":"Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. 2022. Elucidating the design space of diffusion-based generative models. In Proceedings of the Advances in Neural Information Processing Systems. 26565\u201326577."},{"key":"e_1_3_2_104_2","first-page":"4370","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kelly Jacob","year":"2020","unstructured":"Jacob Kelly, Jesse Bettencourt, Matthew J. Johnson, and David K. Duvenaud. 2020. Learning differential equations that are easy to solve. In Proceedings of the Advances in Neural Information Processing Systems. 4370\u20134380."},{"key":"e_1_3_2_105_2","unstructured":"Ehsan Kharazmi Zhongqiang Zhang and George Em Karniadakis. 2019. Variational physics-informed neural networks for solving partial differential equations. arXiv:1912.00873. Retrieved from https:\/\/arxiv.org\/abs\/1912.00873"},{"key":"e_1_3_2_106_2","doi-asserted-by":"crossref","first-page":"113547","DOI":"10.1016\/j.cma.2020.113547","article-title":"hp-VPINNs: Variational physics-informed neural networks with domain decomposition","volume":"374","author":"Kharazmi Ehsan","year":"2021","unstructured":"Ehsan Kharazmi, Zhongqiang Zhang, and George Em Karniadakis. 2021. hp-VPINNs: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering 374 (2021), 113547.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"e_1_3_2_107_2","volume-title":"On Neural Differential Equations","author":"Kidger P.","year":"2021","unstructured":"P. Kidger. 2021. On Neural Differential Equations. Ph. D. Dissertation. University of Oxford."},{"key":"e_1_3_2_108_2","first-page":"5453","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Kidger Patrick","year":"2021","unstructured":"Patrick Kidger, James Foster, Xuechen Li, and Terry J. Lyons. 2021. Neural sdes as infinite-dimensional gans. In Proceedings of the International Conference on Machine Learning. 5453\u20135463."},{"key":"e_1_3_2_109_2","first-page":"18747","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kidger Patrick","year":"2021","unstructured":"Patrick Kidger, James Foster, Xuechen Chen Li, and Terry Lyons. 2021. Efficient and accurate gradients for neural sdes. In Proceedings of the Advances in Neural Information Processing Systems. 18747\u201318761."},{"key":"e_1_3_2_110_2","first-page":"6696","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kidger Patrick","year":"2020","unstructured":"Patrick Kidger, James Morrill, James Foster, and Terry Lyons. 2020. Neural controlled differential equations for irregular time series. In Proceedings of the Advances in Neural Information Processing Systems. 6696\u20136707."},{"issue":"1","key":"e_1_3_2_111_2","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1080\/00036810108840931","article-title":"Differential equations of fractional order: methods results and problem\u2013I","volume":"78","author":"Kilbas A. A.","year":"2001","unstructured":"A. A. Kilbas and J. J. Trujillo. 2001. Differential equations of fractional order: methods results and problem\u2013I. Applicable Analysis 78, 1\u20132 (2001), 153\u2013192.","journal-title":"Applicable Analysis"},{"key":"e_1_3_2_112_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kingma Durk P.","year":"2015","unstructured":"Durk P. Kingma, Tim Salimans, and Max Welling. 2015. Variational dropout and the local reparameterization trick. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_113_2","first-page":"7220","volume-title":"Proceedings of the AAAI Conference on Artificial Intelligence","author":"Kong Lin","year":"2022","unstructured":"Lin Kong, Wei Sun, Fanhua Shang, Yuanyuan Liu, and Hongying Liu. 2022. HNO: High-order numerical architecture for ODE-inspired deep unfolding networks. In Proceedings of the AAAI Conference on Artificial Intelligence. 7220\u20137228."},{"key":"e_1_3_2_114_2","volume-title":"Proceedings of the ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models","author":"Kong Zhifeng","year":"2021","unstructured":"Zhifeng Kong and Wei Ping. 2021. On fast sampling of diffusion probabilistic models. In Proceedings of the ICML Workshop on Invertible Neural Networks, Normalizing Flows, and Explicit Likelihood Models."},{"key":"e_1_3_2_115_2","first-page":"104180","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Kornilov Nikita","year":"2024","unstructured":"Nikita Kornilov, Petr Mokrov, Alexander Gasnikov, and Aleksandr Korotin. 2024. Optimal flow matching: Learning straight trajectories in just one step. In Proceedings of the Advances in Neural Information Processing Systems. 104180\u2013104204."},{"issue":"89","key":"e_1_3_2_116_2","first-page":"1","article-title":"Neural operator: Learning maps between function spaces with applications to pdes","volume":"24","author":"Kovachki Nikola","year":"2023","unstructured":"Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. 2023. Neural operator: Learning maps between function spaces with applications to pdes. Journal of Machine Learning Research 24, 89 (2023), 1\u201397.","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_3_2_117_2","unstructured":"Siddharth Krishna Kumar. 2017. On weight initialization in deep neural networks. arXiv:1704.08863. Retrieved from https:\/\/arxiv.org\/abs\/1704.08863"},{"key":"e_1_3_2_118_2","unstructured":"Alex Labach Hojjat Salehinejad and Shahrokh Valaee. 2019. Survey of dropout methods for deep neural networks. arXiv:1904.13310. Retrieved from https:\/\/arxiv.org\/abs\/1904.13310"},{"key":"e_1_3_2_119_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Larsson Gustav","year":"2017","unstructured":"Gustav Larsson, Michael Maire, and Gregory Shakhnarovich. 2017. FractalNet: Ultra-deep neural networks without residuals. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_120_2","first-page":"63082","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Lee Sangyun","year":"2024","unstructured":"Sangyun Lee, Zinan Lin, and Giulia Fanti. 2024. Improving the training of rectified flows. In Proceedings of the Advances in Neural Information Processing Systems. 63082\u201363109."},{"issue":"1","key":"e_1_3_2_121_2","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1109\/TNNLS.2022.3175757","article-title":"Accelerating neural odes using model order reduction","volume":"35","author":"Lehtim\u00e4ki Mikko","year":"2022","unstructured":"Mikko Lehtim\u00e4ki, Lassi Paunonen, and Marja-Leena Linne. 2022. Accelerating neural odes using model order reduction. IEEE Transactions on Neural Networks and Learning Systems 35, 1 (2022), 519\u2013531.","journal-title":"IEEE Transactions on Neural Networks and Learning Systems"},{"key":"e_1_3_2_122_2","first-page":"10795","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Li Mufan","year":"2022","unstructured":"Mufan Li, Mihai Nica, and Dan Roy. 2022. The neural covariance SDE: Shaped infinite depth-and-width networks at initialization. In Proceedings of the Advances in Neural Information Processing Systems. 10795\u201310808."},{"key":"e_1_3_2_123_2","first-page":"13178","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Li Shufan","year":"2025","unstructured":"Shufan Li, Konstantinos Kallidromitis, Akash Gokul, Zichun Liao, Yusuke Kato, Kazuki Kozuka, and Aditya Grover. 2025. Omniflow: Any-to-any generation with multi-modal rectified flows. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 13178\u201313188."},{"key":"e_1_3_2_124_2","unstructured":"Shan Li and Yiu-Ming Liu. 2016. Whiteout: Gaussian adaptive noise regularization in deep neural networks. arXiv:1612.01490. Retrieved from https:\/\/arxiv.org\/abs\/1612.01490"},{"key":"e_1_3_2_125_2","first-page":"3870","volume-title":"Proceedings of the International Conference on Artificial Intelligence and Statistics","author":"Li Xuechen","year":"2020","unstructured":"Xuechen Li, Ting-Kam Leonard Wong, Ricky T. Q. Chen, and David Duvenaud. 2020. Scalable gradients for stochastic differential equations. In Proceedings of the International Conference on Artificial Intelligence and Statistics. 3870\u20133882."},{"key":"e_1_3_2_126_2","unstructured":"Zongyi Li Nikola Kovachki Kamyar Azizzadenesheli Burigede Liu Kaushik Bhattacharya Andrew Stuart and Anima Anandkumar. 2020. Fourier neural operator for parametric partial differential equations. arXiv:2010.08895. Retrieved from https:\/\/arxiv.org\/abs\/2010.08895"},{"issue":"1","key":"e_1_3_2_127_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/s44387-025-00066-y","article-title":"Flow matching meets biology and life science: A survey","volume":"2","author":"Li Zihao","year":"2026","unstructured":"Zihao Li,Zhichen Zeng,Xiao Lin,Feihao Fang,Yanru Qu,Zhe Xu,Zhining Liu,Xuying Ning,Tianxin Wei,Ge Liu,Hanghang Tong,andJingrui He. 2026. Flow matching meets biology and life science: A survey. npj Artificial Intelligence 2, 1 (2026), 17.","journal-title":"npj Artificial Intelligence"},{"issue":"3","key":"e_1_3_2_128_2","first-page":"1","article-title":"Physics-informed neural operator for learning partial differential equations","volume":"1","author":"Li Zongyi","year":"2024","unstructured":"Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. 2024. Physics-informed neural operator for learning partial differential equations. ACM\/IMS Journal of Data Science 1, 3 (2024), 1\u201327.","journal-title":"ACM\/IMS Journal of Data Science"},{"issue":"1","key":"e_1_3_2_129_2","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1049\/cje.2016.01.023","article-title":"DropConnect regularization method with sparsity constraint for neural networks","volume":"25","author":"Lian Zifeng","year":"2016","unstructured":"Zifeng Lian, Xiaojun Jing, Xiaohan Wang, Hai Huang, Youheng Tan, and Yuanhao Cui. 2016. DropConnect regularization method with sparsity constraint for neural networks. Chinese Journal of Electronics 25, 1 (2016), 152\u2013158.","journal-title":"Chinese Journal of Electronics"},{"key":"e_1_3_2_130_2","first-page":"2330","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Liao Kang","year":"2025","unstructured":"Kang Liao, Zongsheng Yue, Zhouxia Wang, and Chen Change Loy. 2025. Denoising as adaptation: Noise-space domain adaptation for image restoration. In Proceedings of the International Conference on Learning Representations. 2330\u20132357."},{"key":"e_1_3_2_131_2","doi-asserted-by":"crossref","first-page":"111838","DOI":"10.1016\/j.jcp.2022.111838","article-title":"Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems","volume":"474","author":"Linot Alec J.","year":"2023","unstructured":"Alec J. Linot, Joshua W. Burby, Qi Tang, Prasanna Balaprakash, Michael D. Graham, and Romit Maulik. 2023. Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems. Journal of Computational Physics 474 (2023), 111838.","journal-title":"Journal of Computational Physics"},{"key":"e_1_3_2_132_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Lipman Yaron","year":"2023","unstructured":"Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. 2023. Flow matching for generative modeling. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_133_2","unstructured":"Yaron Lipman Marton Havasi Peter Holderrieth Neta Shaul Matt Le Brian Karrer Ricky T. Q. Chen David Lopez-Paz Heli Ben-Hamu and Itai Gat. 2024. Flow matching guide and code. arXiv:2412.06264. Retrieved from https:\/\/arxiv.org\/abs\/2412.06264"},{"key":"e_1_3_2_134_2","first-page":"25267","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Liu Guan-Horng","year":"2021","unstructured":"Guan-Horng Liu, Tianrong Chen, and Evangelos Theodorou. 2021. Second-order neural ode optimizer. In Proceedings of the Advances in Neural Information Processing Systems. 25267\u201325279."},{"key":"e_1_3_2_135_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Liu Luping","year":"2022","unstructured":"Luping Liu, Yi Ren, Zhijie Lin, and Zhou Zhao. 2022. Pseudo numerical methods for diffusion models on manifolds. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_136_2","unstructured":"Qiang Liu. 2022. Rectified flow: A marginal preserving approach to optimal transport. arXiv:2209.14577. Retrieved from https:\/\/arxiv.org\/abs\/2209.14577"},{"key":"e_1_3_2_137_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2016.12.038"},{"key":"e_1_3_2_138_2","first-page":"133825","article-title":"Learning to integrate diffusion ODEs by averaging the derivatives","volume":"38","author":"Liu Wenze","year":"2025","unstructured":"Wenze Liu and Xiangyu Yue. 2025. Learning to integrate diffusion ODEs by averaging the derivatives. In Advances in Neural Information Processing Systems 38 (2025), 133825\u2013133859.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"e_1_3_2_139_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Liu Xingchao","year":"2023","unstructured":"Xingchao Liu, Chengyue Gong, and qiang liu. 2023. Flow straight and fast: Learning to generate and transfer data with rectified flow. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_140_2","unstructured":"Xuanqing Liu Tesi Xiao Si Si Qin Cao Sanjiv Kumar and Cho-Jui Hsieh. 2019. Neural sde: Stabilizing neural ode networks with stochastic noise. arXiv:1906.02355. Retrieved from https:\/\/arxiv.org\/abs\/1906.02355"},{"key":"e_1_3_2_141_2","first-page":"282","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Liu Xuanqing","year":"2020","unstructured":"Xuanqing Liu, Tesi Xiao, Si Si, Qin Cao, Sanjiv Kumar, and Cho-Jui Hsieh. 2020. How does noise help robustness? explanation and exploration under the neural sde framework. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 282\u2013290."},{"key":"e_1_3_2_142_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Liu Xingchao","year":"2023","unstructured":"Xingchao Liu, Xiwen Zhang, Jianzhu Ma, Jian Peng, et\u00a0al. 2023. Instaflow: One step is enough for high-quality diffusion-based text-to-image generation. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_143_2","doi-asserted-by":"crossref","first-page":"108925","DOI":"10.1016\/j.jcp.2019.108925","article-title":"PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network","volume":"399","author":"Long Zichao","year":"2019","unstructured":"Zichao Long, Yiping Lu, and Bin Dong. 2019. PDE-Net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network. Journal of Computational Physics 399 (2019), 108925.","journal-title":"Journal of Computational Physics"},{"key":"e_1_3_2_144_2","first-page":"3208","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Long Zichao","year":"2018","unstructured":"Zichao Long, Yiping Lu, Xianzhong Ma, and Bin Dong. 2018. Pde-net: Learning pdes from data. In Proceedings of the International Conference on Machine Learning. 3208\u20133216."},{"issue":"4","key":"e_1_3_2_145_2","first-page":"4023","article-title":"A deterministic approximation to neural SDEs","volume":"45","author":"Look Andreas","year":"2022","unstructured":"Andreas Look, Melih Kandemir, Barbara Rakitsch, and Jan Peters. 2022. A deterministic approximation to neural SDEs. IEEE Transactions on Pattern Analysis and Machine Intelligence 45, 4 (2022), 4023\u20134037.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"e_1_3_2_146_2","first-page":"22675","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Lou Aaron","year":"2023","unstructured":"Aaron Lou and Stefano Ermon. 2023. Reflected diffusion models. In Proceedings of the International Conference on Machine Learning. 22675\u201322701."},{"key":"e_1_3_2_147_2","first-page":"5775","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Lu Cheng","year":"2022","unstructured":"Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. 2022. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. In Proceedings of the Advances in Neural Information Processing Systems. 5775\u20135787."},{"issue":"4","key":"e_1_3_2_148_2","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1007\/s11633-025-1562-4","article-title":"DPM-Solver++: Fast solver for guided sampling of diffusion probabilistic models","volume":"22","author":"Lu Cheng","year":"2025","unstructured":"Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. 2025. DPM-Solver++: Fast solver for guided sampling of diffusion probabilistic models. Machine Intelligence Research 22, 4 (2025), 730\u2013751.","journal-title":"Machine Intelligence Research"},{"issue":"3","key":"e_1_3_2_149_2","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1038\/s42256-021-00302-5","article-title":"Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators","volume":"3","author":"Lu Lu","year":"2021","unstructured":"Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. 2021. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence 3, 3 (2021), 218\u2013229.","journal-title":"Nature Machine Intelligence"},{"key":"e_1_3_2_150_2","doi-asserted-by":"crossref","first-page":"114778","DOI":"10.1016\/j.cma.2022.114778","article-title":"A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data","volume":"393","author":"Lu Lu","year":"2022","unstructured":"Lu Lu, Xuhui Meng, Shengze Cai, Zhiping Mao, Somdatta Goswami, Zhongqiang Zhang, and George Em Karniadakis. 2022. A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Computer Methods in Applied Mechanics and Engineering 393 (2022), 114778.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"5","key":"e_1_3_2_151_2","first-page":"3576","article-title":"Speed-up ddpm for real-time underwater image enhancement","volume":"34","author":"Lu Siqi","year":"2023","unstructured":"Siqi Lu, Fengxu Guan, Hanyu Zhang, and Haitao Lai. 2023. Speed-up ddpm for real-time underwater image enhancement. IEEE Transactions on Circuits and Systems for Video Technology 34, 5 (2023), 3576\u20133588.","journal-title":"IEEE Transactions on Circuits and Systems for Video Technology"},{"key":"e_1_3_2_152_2","first-page":"3276","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Lu Yiping","year":"2018","unstructured":"Yiping Lu, Aoxiao Zhong, Quanzheng Li, and Bin Dong. 2018. Beyond finite layer neural networks: Bridging deep architectures and numerical differential equations. In Proceedings of the International Conference on Machine Learning. 3276\u20133285."},{"key":"e_1_3_2_153_2","first-page":"23045","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Luo Ziwei","year":"2023","unstructured":"Ziwei Luo, Fredrik K. Gustafsson, Zheng Zhao, Jens Sj\u00f6lund, and Thomas B. Sch\u00f6n. 2023. Image restoration with mean-reverting stochastic differential equations. In Proceedings of the International Conference on Machine Learning. 23045\u201323066."},{"issue":"4","key":"e_1_3_2_154_2","doi-asserted-by":"crossref","first-page":"3395","DOI":"10.1007\/s40747-022-00671-3","article-title":"Rethinking ResNets: Improved stacking strategies with high-order schemes for image classification","volume":"8","author":"Luo Zhengbo","year":"2022","unstructured":"Zhengbo Luo, Zitang Sun, Weilian Zhou, Zizhang Wu, and Sei-ichiro Kamata. 2022. Rethinking ResNets: Improved stacking strategies with high-order schemes for image classification. Complex and Intelligent Systems 8, 4 (2022), 3395\u20133407.","journal-title":"Complex and Intelligent Systems"},{"key":"e_1_3_2_155_2","first-page":"38154","article-title":"Learning straight flows: Variational flow matching for efficient generation","author":"Ma Chenrui","year":"2026","unstructured":"Chenrui Ma, Xi Xiao, Tianyang Wang, Xiao Wang, and Yanning Shen. 2026. Learning straight flows: Variational flow matching for efficient generation. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR). 38154\u201338164.","journal-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR)"},{"key":"e_1_3_2_156_2","unstructured":"Zhiyuan Ma Ruixun Liu Sixian Liu Jianjun Li and Bowen Zhou. 2025. Flow diverse and efficient: Learning momentum flow matching via stochastic velocity field sampling. arXiv:2506.08796. Retrieved from https:\/\/arxiv.org\/abs\/2506.08796"},{"issue":"1","key":"e_1_3_2_157_2","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/BF02846028","article-title":"Continuous markov processes and stochastic equations","volume":"4","author":"Maruyama Gisiro","year":"1955","unstructured":"Gisiro Maruyama. 1955. Continuous markov processes and stochastic equations. Rendiconti del Circolo Matematico di Palermo 4, 1 (1955), 48\u201390.","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"e_1_3_2_158_2","unstructured":"Stefano Massaroli Michael Poli Michelangelo Bin Jinkyoo Park Atsushi Yamashita and Hajime Asama. 2020. Stable neural flows. arXiv:2003.08063. Retrieved from https:\/\/arxiv.org\/abs\/2003.08063"},{"key":"e_1_3_2_159_2","first-page":"3952","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Massaroli Stefano","year":"2020","unstructured":"Stefano Massaroli, Michael Poli, Jinkyoo Park, Atsushi Yamashita, and Hajime Asama. 2020. Dissecting neural odes. In Proceedings of the Advances in Neural Information Processing Systems. 3952\u20133963."},{"key":"e_1_3_2_160_2","unstructured":"Sam McCallum and James Foster. 2024. Efficient accurate and stable gradients for neural odes. arXiv:2410.11648. Retrieved from https:\/\/arxiv.org\/abs\/2410.11648"},{"key":"e_1_3_2_161_2","doi-asserted-by":"crossref","first-page":"111722","DOI":"10.1016\/j.jcp.2022.111722","article-title":"Self-adaptive physics-informed neural networks","volume":"474","author":"McClenny Levi D.","year":"2023","unstructured":"Levi D. McClenny and Ulisses M. Braga-Neto. 2023. Self-adaptive physics-informed neural networks. Journal of Computational Physics 474 (2023), 111722.","journal-title":"Journal of Computational Physics"},{"issue":"6","key":"e_1_3_2_162_2","doi-asserted-by":"crossref","first-page":"3947","DOI":"10.1007\/s10462-019-09784-7","article-title":"A survey of regularization strategies for deep models","volume":"53","author":"Moradi Reza","year":"2020","unstructured":"Reza Moradi, Reza Berangi, and Behrouz Minaei. 2020. A survey of regularization strategies for deep models. Artificial Intelligence Review 53, 6 (2020), 3947\u20133986.","journal-title":"Artificial Intelligence Review"},{"key":"e_1_3_2_163_2","first-page":"3544","volume-title":"Proceedings of the IEEE\/CVF International Conference on Computer Vision","author":"Morerio Pietro","year":"2017","unstructured":"Pietro Morerio, Jacopo Cavazza, Riccardo Volpi, Ren\u00e9 Vidal, and Vittorio Murino. 2017. Curriculum dropout. In Proceedings of the IEEE\/CVF International Conference on Computer Vision. 3544\u20133552."},{"key":"e_1_3_2_164_2","first-page":"7829","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Morrill James","year":"2021","unstructured":"James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster. 2021. Neural rough differential equations for long time series. In Proceedings of the International Conference on Machine Learning. 7829\u20137838."},{"key":"e_1_3_2_165_2","doi-asserted-by":"publisher","DOI":"10.1109\/TITS.2020.3032227"},{"issue":"1","key":"e_1_3_2_166_2","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/s10462-021-10033-z","article-title":"A review on weight initialization strategies for neural networks","volume":"55","author":"Narkhede Meenal V.","year":"2022","unstructured":"Meenal V. Narkhede, Prashant P. Bartakke, and Mukul S. Sutaone. 2022. A review on weight initialization strategies for neural networks. Artificial Intelligence Review 55, 1 (2022), 291\u2013322.","journal-title":"Artificial Intelligence Review"},{"key":"e_1_3_2_167_2","doi-asserted-by":"crossref","first-page":"116546","DOI":"10.1016\/j.cma.2023.116546","article-title":"Physics informed WNO","volume":"418","author":"Navaneeth N.","year":"2024","unstructured":"N. Navaneeth, Tapas Tripura, and Souvik Chakraborty. 2024. Physics informed WNO. Computer Methods in Applied Mechanics and Engineering 418 (2024), 116546.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"e_1_3_2_168_2","first-page":"7712","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Nguyen Ho Huu Nghia","year":"2022","unstructured":"Ho Huu Nghia Nguyen, Tan Nguyen, Huyen Vo, Stanley Osher, and Thieu Vo. 2022. Improving neural ordinary differential equations with nesterov\u2019s accelerated gradient method. In Proceedings of the Advances in Neural Information Processing Systems. 7712\u20137726."},{"key":"e_1_3_2_169_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Nie Shen","year":"2024","unstructured":"Shen Nie, Hanzhong Allan Guo, Cheng Lu, Yuhao Zhou, Chenyu Zheng, and Chongxuan Li. 2024. The blessing of randomness: Sde beats ode in general diffusion-based image editing. In Proceedings of the International Conference on Learning Representations."},{"issue":"9","key":"e_1_3_2_170_2","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s10462-024-10894-0","article-title":"On the applications of neural ordinary differential equations in medical image analysis","volume":"57","author":"Niu Hao","year":"2024","unstructured":"Hao Niu, Yuxiang Zhou, Xiaohao Yan, Jun Wu, Yuncheng Shen, Zhang Yi, and Junjie Hu. 2024. On the applications of neural ordinary differential equations in medical image analysis. Artificial Intelligence Review 57, 9 (2024), 236.","journal-title":"Artificial Intelligence Review"},{"key":"e_1_3_2_171_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Noh Hyeonwoo","year":"2017","unstructured":"Hyeonwoo Noh, Tackgeun You, Jonghwan Mun, and Bohyung Han. 2017. Regularizing deep neural networks by noise: Its interpretation and optimization. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_172_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Norcliffe Alexander","year":"2021","unstructured":"Alexander Norcliffe, Cristian Bodnar, Ben Day, Jacob Moss, and Pietro Li\u00f2. 2021. Neural ODE processes. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_173_2","first-page":"5911","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Norcliffe Alexander","year":"2020","unstructured":"Alexander Norcliffe, Cristian Bodnar, Ben Day, Nikola Simidjievski, and Pietro Li\u00f2. 2020. On second order behaviour in augmented neural odes. In Proceedings of the Advances in Neural Information Processing Systems. 5911\u20135921."},{"key":"e_1_3_2_174_2","first-page":"10610","article-title":"Comprehensive review of neural differential equations for time series analysis","author":"Oh YongKyung","year":"2025","unstructured":"YongKyung Oh, Seungsu Kam, Jonghun Lee, Dong-Young Lim, Sungil Kim, and Alex A. T. Bui. 2025. Comprehensive review of neural differential equations for time series analysis. In Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence (IJCAI). 10610\u201310620.","journal-title":"Proceedings of the Thirty-Fourth International Joint Conference on Artificial Intelligence (IJCAI)"},{"key":"e_1_3_2_175_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Oh YongKyung","year":"2024","unstructured":"YongKyung Oh, Dongyoung Lim, and Sungil Kim. 2024. Stable neural stochastic differential equations in analyzing irregular time series data. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_176_2","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1007\/978-3-030-17795-9_10","volume-title":"Proceedings of the Advances in Computer Vision","author":"O\u2019Mahony Niall","year":"2020","unstructured":"Niall O\u2019Mahony, Sean Campbell, Anderson Carvalho, Suman Harapanahalli, Gustavo Velasco Hernandez, Lenka Krpalkova, Daniel Riordan, and Joseph Walsh. 2020. Deep learning vs. traditional computer vision. In Proceedings of the Advances in Computer Vision, Kohei Arai and Supriya Kapoor (Eds.). Springer International Publishing, Cham, 128\u2013144."},{"key":"e_1_3_2_177_2","first-page":"7265","volume-title":"Proceedings of the International Conference on Artificial Intelligence and Statistics","author":"Orvieto Antonio","year":"2023","unstructured":"Antonio Orvieto, Anant Raj, Hans Kersting, and Francis Bach. 2023. Explicit regularization in overparametrized models via noise injection. In Proceedings of the International Conference on Artificial Intelligence and Statistics. 7265\u20137287."},{"key":"e_1_3_2_178_2","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2020.2979670"},{"key":"e_1_3_2_179_2","first-page":"7671","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Pal Ambar","year":"2020","unstructured":"Ambar Pal, Connor Lane, Ren\u00e9 Vidal, and Benjamin D. Haeffele. 2020. On the regularization properties of structured dropout. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 7671\u20137679."},{"key":"e_1_3_2_180_2","first-page":"90030","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Park Dogyun","year":"2024","unstructured":"Dogyun Park, Sojin Lee, Sihyeon Kim, Taehoon Lee, Youngjoon Hong, and Hyunwoo J. Kim. 2024. Constant acceleration flow. In Proceedings of the Advances in Neural Information Processing Systems. 90030\u201390060."},{"key":"e_1_3_2_181_2","first-page":"1434","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Park Sung Woo","year":"2022","unstructured":"Sung Woo Park, Hyomin Kim, Kyungjae Lee, and Junseok Kwon. 2022. Riemannian neural SDE: Learning stochastic representations on manifolds. In Proceedings of the Advances in Neural Information Processing Systems. 1434\u20131444."},{"key":"e_1_3_2_182_2","first-page":"17246","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Perugachi-Diaz Yura","year":"2021","unstructured":"Yura Perugachi-Diaz, Jakub Tomczak, and Sandjai Bhulai. 2021. Invertible densenets with concatenated lipswish. In Proceedings of the Advances in Neural Information Processing Systems. 17246\u201317257."},{"key":"e_1_3_2_183_2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511721373","volume-title":"Stochastic Partial Differential Equations with L\u00e9vy Noise: An Evolution Equation Approach","author":"Peszat Szymon","year":"2007","unstructured":"Szymon Peszat and Jerzy Zabczyk. 2007. Stochastic Partial Differential Equations with L\u00e9vy Noise: An Evolution Equation Approach. Cambridge University Press."},{"key":"e_1_3_2_184_2","doi-asserted-by":"crossref","DOI":"10.1201\/9780203749319","volume-title":"Mathematical Theory of Optimal Processes","author":"Pontryagin Lev Semenovich","year":"2018","unstructured":"Lev Semenovich Pontryagin. 2018. Mathematical Theory of Optimal Processes. Routledge."},{"key":"e_1_3_2_185_2","unstructured":"Kuilin Qin Lianfang Wang Xu Sun Jiwei Jia Yu Wang Yong Wang and Yuping Duan. 2026. Starter-iterator neural operator: A unified architecture for high-fidelity forward and inverse PDE problems. arXiv:2606.18305. Retrieved from https:\/\/arxiv.org\/abs\/2606.18305"},{"key":"e_1_3_2_186_2","doi-asserted-by":"crossref","unstructured":"Christopher Rackauckas Yingbo Ma Julius Martensen Collin Warner Kirill Zubov Rohit Supekar Dominic Skinner Ali Ramadhan and Alan Edelman. 2020. Universal differential equations for scientific machine learning. arXiv:2001.04385. Retrieved from https:\/\/arxiv.org\/abs\/2001.04385","DOI":"10.21203\/rs.3.rs-55125\/v1"},{"key":"e_1_3_2_187_2","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi Maziar","year":"2019","unstructured":"Maziar Raissi, Paris Perdikaris, and George E Karniadakis. 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics 378 (2019), 686\u2013707.","journal-title":"Journal of Computational physics"},{"key":"e_1_3_2_188_2","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1109\/SLT.2014.7078567","volume-title":"Proceedings of the 2014 IEEE Spoken Language Technology Workshop","author":"Rennie Steven J.","year":"2014","unstructured":"Steven J. Rennie, Vaibhava Goel, and Samuel Thomas. 2014. Annealed dropout training of deep networks. In Proceedings of the 2014 IEEE Spoken Language Technology Workshop. IEEE, 159\u2013164."},{"key":"e_1_3_2_189_2","volume-title":"Proceedings of the ICML 2023 Workshop: Sampling and Optimization in Discrete Space","author":"Richemond Pierre Harvey","year":"2023","unstructured":"Pierre Harvey Richemond, Sander Dieleman, and Arnaud Doucet. 2023. Categorical SDEs with simplex diffusion. In Proceedings of the ICML 2023 Workshop: Sampling and Optimization in Discrete Space."},{"key":"e_1_3_2_190_2","first-page":"18687","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Rodriguez Ivan Dario Jimenez","year":"2022","unstructured":"Ivan Dario Jimenez Rodriguez, Aaron Ames, and Yisong Yue. 2022. Lyanet: A lyapunov framework for training neural odes. In Proceedings of the International Conference on Machine Learning. 18687\u201318703."},{"key":"e_1_3_2_191_2","unstructured":"Sumanta Roy Bahador Bahmani Ioannis G. Kevrekidis and Michael D. Shields. 2025. A physics-informed multi-resolution neural operator. arXiv:2510.23810. Retrieved from https:\/\/arxiv.org\/abs\/2510.23810"},{"key":"e_1_3_2_192_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Rubanova Yulia","year":"2019","unstructured":"Yulia Rubanova, Ricky T. Q. Chen, and David K Duvenaud. 2019. Latent ordinary differential equations for irregularly-sampled time series. In Proceedings of the Advances in Neural Information Processing Systems."},{"issue":"5","key":"e_1_3_2_193_2","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1038\/s42256-019-0048-x","article-title":"Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead","volume":"1","author":"Rudin Cynthia","year":"2019","unstructured":"Cynthia Rudin. 2019. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence 1, 5 (2019), 206\u2013215.","journal-title":"Nature Machine Intelligence"},{"issue":"3","key":"e_1_3_2_194_2","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1007\/s10851-019-00903-1","article-title":"Deep neural networks motivated by partial differential equations","volume":"62","author":"Ruthotto Lars","year":"2020","unstructured":"Lars Ruthotto and Eldad Haber. 2020. Deep neural networks motivated by partial differential equations. Journal of Mathematical Imaging and Vision 62, 3 (2020), 352\u2013364.","journal-title":"Journal of Mathematical Imaging and Vision"},{"issue":"3","key":"e_1_3_2_195_2","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1134\/S0361768823030088","article-title":"Enhancing image recognition with pre-defined convolutional layers based on PDEs","volume":"49","author":"Sahakyan Vardan R.","year":"2023","unstructured":"Vardan R. Sahakyan, Vahagn G. Melkonyan, Gor A. Gharagyozyan, and Arman S. Avetisyan. 2023. Enhancing image recognition with pre-defined convolutional layers based on PDEs. Programming and Computer Software 49, 3 (2023), 192\u2013197.","journal-title":"Programming and Computer Software"},{"issue":"14","key":"e_1_3_2_196_2","doi-asserted-by":"crossref","first-page":"3106","DOI":"10.3390\/electronics12143106","article-title":"A review on dropout regularization approaches for deep neural networks within the scholarly domain","volume":"12","author":"Salehin Imrus","year":"2023","unstructured":"Imrus Salehin and Dae-Ki Kang. 2023. A review on dropout regularization approaches for deep neural networks within the scholarly domain. Electronics 12, 14 (2023), 3106. Retrieved from https:\/\/www.mdpi.com\/2079-9292\/12\/14\/3106","journal-title":"Electronics"},{"issue":"10","key":"e_1_3_2_197_2","doi-asserted-by":"crossref","first-page":"5279","DOI":"10.1109\/TNNLS.2021.3069970","article-title":"Edropout: Energy-based dropout and pruning of deep neural networks","volume":"33","author":"Salehinejad Hojjat","year":"2021","unstructured":"Hojjat Salehinejad and Shahrokh Valaee. 2021. Edropout: Energy-based dropout and pruning of deep neural networks. IEEE Transactions on Neural Networks and Learning Systems 33, 10 (2021), 5279\u20135292.","journal-title":"IEEE Transactions on Neural Networks and Learning Systems"},{"key":"e_1_3_2_198_2","first-page":"1333","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Salvi Cristopher","year":"2022","unstructured":"Cristopher Salvi, Maud Lemercier, and Andris Gerasimovics. 2022. Neural stochastic pdes: Resolution-invariant learning of continuous spatiotemporal dynamics. In Proceedings of the Advances in Neural Information Processing Systems. 1333\u20131344."},{"key":"e_1_3_2_199_2","doi-asserted-by":"crossref","first-page":"112790","DOI":"10.1016\/j.cma.2019.112790","article-title":"An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications","volume":"362","author":"Samaniego Esteban","year":"2020","unstructured":"Esteban Samaniego, Cosmin Anitescu, Somdatta Goswami, Vien Minh Nguyen-Thanh, Hongwei Guo, Khader Hamdia, Xiaoying Zhuang, and Timon Rabczuk. 2020. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering 362 (2020), 112790.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"issue":"10","key":"e_1_3_2_200_2","first-page":"1","article-title":"Avoiding overfitting: A survey on regularization methods for convolutional neural networks","volume":"54","author":"Santos Claudio Filipi Gon\u00e7alves Dos","year":"2022","unstructured":"Claudio Filipi Gon\u00e7alves Dos Santos and Jo\u00e3o Paulo Papa. 2022. Avoiding overfitting: A survey on regularization methods for convolutional neural networks. Computing Surveys 54, 10s (2022), 1\u201325.","journal-title":"Computing Surveys"},{"key":"e_1_3_2_201_2","first-page":"28347","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Schusterbauer Johannes","year":"2025","unstructured":"Johannes Schusterbauer, Ming Gui, Frank Fundel, and Bj\u00f6rn Ommer. 2025. Diff2Flow: Training flow matching models via diffusion model alignment. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 28347\u201328357."},{"key":"e_1_3_2_202_2","doi-asserted-by":"crossref","DOI":"10.1002\/0471742015","volume-title":"The Numerical Solution of Ordinary and Partial Differential Equations","author":"Sewell Granville","year":"2005","unstructured":"Granville Sewell. 2005. The Numerical Solution of Ordinary and Partial Differential Equations. John Wiley & Sons."},{"key":"e_1_3_2_203_2","first-page":"96991","article-title":"Denoising L\u00e9vy probabilistic models","volume":"2025","author":"Shariatian Dario","year":"2025","unstructured":"Dario Shariatian, Umut Simsekli, and Alain Durmus. 2025. Denoising L\u00e9vy probabilistic models. In International Conference on Learning Representations (ICLR) 2025 (2025), 96991\u201397024.","journal-title":"International Conference on Learning Representations (ICLR)"},{"key":"e_1_3_2_204_2","unstructured":"Macheng Shen and Chen Cheng. 2025. Neural SDEs as a unified approach to continuous-domain sequence modeling. arXiv:2501.18871. Retrieved from https:\/\/arxiv.org\/abs\/2501.18871"},{"key":"e_1_3_2_205_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","volume":"29","author":"Singh Saurabh","year":"2016","unstructured":"Saurabh Singh, Derek Hoiem, and David Forsyth. 2016. Swapout: Learning an ensemble of deep architectures. In Proceedings of the Advances in Neural Information Processing Systems 29 (2016)."},{"key":"e_1_3_2_206_2","doi-asserted-by":"publisher","DOI":"10.3390\/s20185097"},{"key":"e_1_3_2_207_2","unstructured":"Luke Snow and Vikram Krishnamurthy. 2025. Efficient neural SDE training using wiener-space cubature. arXiv:2502.12395. Retrieved from https:\/\/arxiv.org\/abs\/2502.12395"},{"key":"e_1_3_2_208_2","first-page":"2256","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Sohl-Dickstein Jascha","year":"2015","unstructured":"Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. 2015. Deep unsupervised learning using nonequilibrium thermodynamics. In Proceedings of the International Conference on Machine Learning. 2256\u20132265."},{"key":"e_1_3_2_209_2","unstructured":"Hwijae Son Jin Woo Jang Woo Jin Han and Hyung Ju Hwang. 2021. Sobolev training for physics informed neural networks. arXiv:2101.08932. Retrieved from https:\/\/arxiv.org\/abs\/2101.08932"},{"key":"e_1_3_2_210_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Song Jiaming","year":"2021","unstructured":"Jiaming Song, Chenlin Meng, and Stefano Ermon. 2021. Denoising diffusion implicit models. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_211_2","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Song Yang","year":"2019","unstructured":"Yang Song and Stefano Ermon. 2019. Generative modeling by estimating gradients of the data distribution. In Proceedings of the Advances in Neural Information Processing Systems."},{"key":"e_1_3_2_212_2","first-page":"12438","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","volume":"33","author":"Song Yang","year":"2020","unstructured":"Yang Song and Stefano Ermon. 2020. Improved techniques for training score-based generative models. In Proceedings of the Advances in Neural Information Processing Systems, H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (Eds.). Vol. 33, Curran Associates, Inc., 12438\u201312448."},{"key":"e_1_3_2_213_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Song Yang","year":"2021","unstructured":"Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. 2021. Score-based generative modeling through stochastic differential equations. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_214_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Song Yang","year":"2021","unstructured":"Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. 2021. Score-based generative modeling through stochastic differential equations. In Proceedings of the International Conference on Learning Representations."},{"issue":"1","key":"e_1_3_2_215_2","first-page":"1929","article-title":"Dropout: A simple way to prevent neural networks from overfitting","volume":"15","author":"Srivastava Nitish","year":"2014","unstructured":"Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. 2014. Dropout: A simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research 15, 1 (2014), 1929\u20131958.","journal-title":"The Journal of Machine Learning Research"},{"key":"e_1_3_2_216_2","unstructured":"Qi Sun Yunzhe Tao and Qiang Du. 2018. Stochastic training of residual networks: A differential equation viewpoint. arXiv:1812.00174. Retrieved from https:\/\/arxiv.org\/abs\/1812.00174"},{"key":"e_1_3_2_217_2","first-page":"352","volume-title":"Proceedings of the Mathematical and Scientific Machine Learning","author":"Sun Yifan","year":"2020","unstructured":"Yifan Sun, Linan Zhang, and Hayden Schaeffer. 2020. NeuPDE: Neural network based ordinary and partial differential equations for modeling time-dependent data. In Proceedings of the Mathematical and Scientific Machine Learning. 352\u2013372."},{"key":"e_1_3_2_218_2","first-page":"1","volume-title":"Proceedings of the 2017 IEEE Custom Integrated Circuits Conference","author":"Sze Vivienne","year":"2017","unstructured":"Vivienne Sze, Yu-Hsin Chen, Joel Emer, Amr Suleiman, and Zhengdong Zhang. 2017. Hardware for machine learning: Challenges and opportunities. In Proceedings of the 2017 IEEE Custom Integrated Circuits Conference. 1\u20138. DOI:10.1109\/CICC.2017.7993626"},{"key":"e_1_3_2_219_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-34372-9","volume-title":"Computer Vision: Algorithms and Applications","author":"Szeliski Richard","year":"2022","unstructured":"Richard Szeliski. 2022. Computer Vision: Algorithms and Applications. Springer Nature."},{"issue":"2","key":"e_1_3_2_220_2","doi-asserted-by":"crossref","first-page":"1364","DOI":"10.1137\/25M1777992","article-title":"Contractive diffusion probabilistic models","volume":"19","author":"Tang Wenpin","year":"2026","unstructured":"Wenpin Tang and Hanyang Zhao. 2026. Contractive diffusion probabilistic models. SIAM Journal on Imaging Sciences 19, 2 (2026), 1364\u20131389.","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"e_1_3_2_221_2","unstructured":"Sasha Targ Diogo Almeida and Kevin Lyman. 2016. Resnet in resnet: Generalizing residual architectures. arXiv:1603.08029. Retrieved from https:\/\/arxiv.org\/abs\/1603.08029"},{"key":"e_1_3_2_222_2","first-page":"648","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Tompson Jonathan","year":"2015","unstructured":"Jonathan Tompson, Ross Goroshin, Arjun Jain, Yann LeCun, and Christoph Bregler. 2015. Efficient object localization using convolutional networks. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 648\u2013656."},{"key":"e_1_3_2_223_2","first-page":"1","article-title":"Improving and generalizing flow-based generative models with minibatch optimal transport","author":"Tong Alexander","year":"2024","unstructured":"Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. 2024. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research (2024), 1\u201334.","journal-title":"Transactions on Machine Learning Research"},{"issue":"2","key":"e_1_3_2_224_2","doi-asserted-by":"crossref","first-page":"531","DOI":"10.3390\/s25020531","article-title":"A comprehensive survey of deep learning approaches in image processing","volume":"25","author":"Trigka Maria","year":"2025","unstructured":"Maria Trigka and Elias Dritsas. 2025. A comprehensive survey of deep learning approaches in image processing. Sensors 25, 2 (2025), 531.","journal-title":"Sensors"},{"key":"e_1_3_2_225_2","doi-asserted-by":"crossref","first-page":"115783","DOI":"10.1016\/j.cma.2022.115783","article-title":"Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems","volume":"404","author":"Tripura Tapas","year":"2023","unstructured":"Tapas Tripura and Souvik Chakraborty. 2023. Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems. Computer Methods in Applied Mechanics and Engineering 404 (2023), 115783.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"e_1_3_2_226_2","volume-title":"Proceedings of the ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations","author":"Tuor Aaron","year":"2019","unstructured":"Aaron Tuor, Jan Drgona, and Draguna Vrabie. 2019. Constrained neural ordinary differential equations with stability guarantees. In Proceedings of the ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations."},{"key":"e_1_3_2_227_2","unstructured":"Belinda Tzen and Maxim Raginsky. 2019. Neural stochastic differential equations: Deep latent gaussian models in the diffusion limit. arXiv:1905.09883. Retrieved from https:\/\/arxiv.org\/abs\/1905.09883"},{"issue":"3","key":"e_1_3_2_228_2","doi-asserted-by":"crossref","first-page":"1723","DOI":"10.1007\/s10462-021-10049-5","article-title":"Evolutionary design of neural network architectures: A review of three decades of research","volume":"55","author":"\u00dcnal Hamit Taner","year":"2022","unstructured":"Hamit Taner \u00dcnal and Fatih Ba\u015f\u00e7ift\u00e7i. 2022. Evolutionary design of neural network architectures: A review of three decades of research. Artificial Intelligence Review 55, 3 (2022), 1723\u20131802.","journal-title":"Artificial Intelligence Review"},{"key":"e_1_3_2_229_2","volume-title":"Proceedings of the ICLR 2025 Workshop on Deep Generative Model in Machine Learning: Theory, Principle and Efficacy","author":"Vandersanden Jente","year":"2025","unstructured":"Jente Vandersanden, Sascha Holl, Xingchang Huang, and Gurprit Singh. 2025. Edge-preserving noise for diffusion models. In Proceedings of the ICLR 2025 Workshop on Deep Generative Model in Machine Learning: Theory, Principle and Efficacy."},{"key":"e_1_3_2_230_2","first-page":"839","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Veit Andreas","year":"2017","unstructured":"Andreas Veit, Neil Alldrin, Gal Chechik, Ivan Krasin, Abhinav Gupta, and Serge Belongie. 2017. Learning from noisy large-scale datasets with minimal supervision. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 839\u2013847."},{"key":"e_1_3_2_231_2","volume-title":"Proceedings of the NeurIPS 2022 Workshop on Score-Based Methods","author":"Voleti Vikram","year":"2022","unstructured":"Vikram Voleti, Christopher Pal, and Adam M. Oberman. 2022. Score-based denoising diffusion with non-isotropic gaussian noise models. In Proceedings of the NeurIPS 2022 Workshop on Score-Based Methods."},{"key":"e_1_3_2_232_2","doi-asserted-by":"publisher","DOI":"10.1155\/2018\/7068349"},{"key":"e_1_3_2_233_2","first-page":"1058","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Wan Li","year":"2013","unstructured":"Li Wan, Matthew Zeiler, Sixin Zhang, Yann Le Cun, and Rob Fergus. 2013. Regularization of neural networks using dropconnect. In Proceedings of the International Conference on Machine Learning. 1058\u20131066."},{"key":"e_1_3_2_234_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Wang Fu-Yun","year":"2025","unstructured":"Fu-Yun Wang, Ling Yang, Zhaoyang Huang, Mengdi Wang, and Hongsheng Li. 2025. Rectified diffusion: Straightness is not your need in rectified flow. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_235_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Wang Luran","year":"2025","unstructured":"Luran Wang, Chaoran Cheng, Yizhen Liao, Yanru Qu, and Ge Liu. 2025. Training free guided flow-matching with optimal control. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_236_2","first-page":"88045","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Wang Shengbo","year":"2024","unstructured":"Shengbo Wang, Jose Blanchet, and Peter Glynn. 2024. An efficient high-dimensional gradient estimator for stochastic differential equations. In Proceedings of the Advances in Neural Information Processing Systems. 88045\u201388090."},{"key":"e_1_3_2_237_2","first-page":"118","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Wang Sida","year":"2013","unstructured":"Sida Wang and Christopher Manning. 2013. Fast dropout training. In Proceedings of the International Conference on Machine Learning. 118\u2013126."},{"issue":"40","key":"e_1_3_2_238_2","doi-asserted-by":"crossref","first-page":"eabi8605","DOI":"10.1126\/sciadv.abi8605","article-title":"Learning the solution operator of parametric partial differential equations with physics-informed DeepONets","volume":"7","author":"Wang Sifan","year":"2021","unstructured":"Sifan Wang, Hanwen Wang, and Paris Perdikaris. 2021. Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Science Advances 7, 40 (2021), eabi8605.","journal-title":"Science Advances"},{"key":"e_1_3_2_239_2","doi-asserted-by":"crossref","first-page":"110768","DOI":"10.1016\/j.jcp.2021.110768","article-title":"When and why PINNs fail to train: A neural tangent kernel perspective","volume":"449","author":"Wang Sifan","year":"2022","unstructured":"Sifan Wang, Xinling Yu, and Paris Perdikaris. 2022. When and why PINNs fail to train: A neural tangent kernel perspective. Journal of Computational Physics 449 (2022), 110768.","journal-title":"Journal of Computational Physics"},{"key":"e_1_3_2_240_2","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2025.3540310"},{"issue":"1","key":"e_1_3_2_241_2","first-page":"1","article-title":"A proposal on machine learning via dynamical systems","volume":"5","author":"Weinan Ee","year":"2017","unstructured":"Ee Weinan. 2017. A proposal on machine learning via dynamical systems. Communications in Mathematics and Statistics 5, 1 (2017), 1\u201311.","journal-title":"Communications in Mathematics and Statistics"},{"key":"e_1_3_2_242_2","first-page":"46","volume-title":"Neural Information Processing: 22nd International Conference, ICONIP 2015, Istanbul, Turkey, November 9-12, 2015, Proceedings, Part I 22","author":"Wu Haibing","year":"2015","unstructured":"Haibing Wu and Xiaodong Gu. 2015. Max-pooling dropout for regularization of convolutional neural networks. In Neural Information Processing: 22nd International Conference, ICONIP 2015, Istanbul, Turkey, November 9-12, 2015, Proceedings, Part I 22. Springer, 46\u201354."},{"key":"e_1_3_2_243_2","unstructured":"Yuchen Wu Yuxin Chen and Yuting Wei. 2024. Stochastic runge-kutta methods: Provable acceleration of diffusion models. arXiv:2410.04760. Retrieved from https:\/\/arxiv.org\/abs\/2410.04760"},{"key":"e_1_3_2_244_2","first-page":"18646","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Xia Hedi","year":"2021","unstructured":"Hedi Xia, Vai Suliafu, Hangjie Ji, Tan Nguyen, Andrea Bertozzi, Stanley Osher, and Bao Wang. 2021. Heavy ball neural ordinary differential equations. In Proceedings of the Advances in Neural Information Processing Systems. 18646\u201318659."},{"key":"e_1_3_2_245_2","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.neucom.2022.05.015","article-title":"Self-adaptive loss balanced physics-informed neural networks","volume":"496","author":"Xiang Zixue","year":"2022","unstructured":"Zixue Xiang, Wei Peng, Xu Liu, and Wen Yao. 2022. Self-adaptive loss balanced physics-informed neural networks. Neurocomputing 496 (2022), 11\u201334.","journal-title":"Neurocomputing"},{"key":"e_1_3_2_246_2","first-page":"1492","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Xie Saining","year":"2017","unstructured":"Saining Xie, Ross Girshick, Piotr Doll\u00e1r, Zhuowen Tu, and Kaiming He. 2017. Aggregated residual transformations for deep neural networks. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 1492\u20131500."},{"key":"e_1_3_2_247_2","volume-title":"Proceedings of the International Conference on Machine Learning","author":"Xue Kaiwen","year":"2024","unstructured":"Kaiwen Xue, Yuhao Zhou, Shen Nie, Xu Min, Xiaolu Zhang, JUN ZHOU, and Chongxuan Li. 2024. Unifying bayesian flow networks and diffusion models through stochastic differential equations. In Proceedings of the International Conference on Machine Learning."},{"key":"e_1_3_2_248_2","first-page":"77632","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Xue Shuchen","year":"2023","unstructured":"Shuchen Xue, Mingyang Yi, Weijian Luo, Shifeng Zhang, Jiacheng Sun, Zhenguo Li, and Zhi-Ming Ma. 2023. Sa-solver: Stochastic adams solver for fast sampling of diffusion models. In Proceedings of the Advances in Neural Information Processing Systems. 77632\u201377674."},{"key":"e_1_3_2_249_2","doi-asserted-by":"crossref","first-page":"186126","DOI":"10.1109\/ACCESS.2019.2960566","article-title":"ShakeDrop regularization for deep residual learning","volume":"7","author":"Yamada Yu","year":"2019","unstructured":"Yu Yamada, Masaki Iwamura, Takuya Akiba, and Koichi Kise. 2019. ShakeDrop regularization for deep residual learning. IEEE Access 7 (2019), 186126\u2013186136.","journal-title":"IEEE Access"},{"key":"e_1_3_2_250_2","unstructured":"Yoshihiro Yamada Masakazu Iwamura and Koichi Kise. 2016. Deep pyramidal residual networks with separated stochastic depth. arXiv:1612.01230. Retrieved from http:\/\/arxiv.org\/abs\/1612.01230"},{"key":"e_1_3_2_251_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"YAN Hanshu","year":"2020","unstructured":"Hanshu YAN, Jiawei DU, Vincent TAN, and Jiashi FENG. 2020. On robustness of neural ordinary differential equations. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_252_2","first-page":"78630","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Yan Hanshu","year":"2024","unstructured":"Hanshu Yan, Xingchao Liu, Jiachun Pan, Jun Hao Liew, Qiang Liu, and Jiashi Feng. 2024. Perflow: Piecewise rectified flow as universal plug-and-play accelerator. In Proceedings of the Advances in Neural Information Processing Systems. 78630\u201378652."},{"issue":"4","key":"e_1_3_2_253_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3626235","article-title":"Diffusion models: A comprehensive survey of methods and applications","volume":"56","author":"Yang Ling","year":"2023","unstructured":"Ling Yang, Zhilong Zhang, Yang Song, Shenda Hong, Runsheng Xu, Yue Zhao, Wentao Zhang, Bin Cui, and Ming-Hsuan Yang. 2023. Diffusion models: A comprehensive survey of methods and applications. Computing Surveys 56, 4 (2023), 1\u201339.","journal-title":"Computing Surveys"},{"key":"e_1_3_2_254_2","unstructured":"Ling Yang Zixiang Zhang Zhilong Zhang Xingchao Liu Minkai Xu Wentao Zhang Chenlin Meng Stefano Ermon and Bin Cui. 2024. Consistency flow matching: Defining straight flows with velocity consistency. arXiv:2407.02398. Retrieved from https:\/\/arxiv.org\/abs\/2407.02398"},{"key":"e_1_3_2_255_2","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1007\/978-981-96-8295-9_7","volume-title":"Proceedings of the Data Science: Foundations and Applications","author":"Yang Yuan-Chih","year":"2025","unstructured":"Yuan-Chih Yang and Hung-Hsuan Chen. 2025. Dynamic dropconnect: Enhancing neural network robustness through adaptive edge dropping strategies. In Proceedings of the Data Science: Foundations and Applications. 110\u2013121."},{"issue":"1","key":"e_1_3_2_256_2","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1038\/s44172-023-00074-3","article-title":"Improving the robustness of analog deep neural networks through a Bayes-optimized noise injection approach","volume":"2","author":"Ye Nanyang","year":"2023","unstructured":"Nanyang Ye, Linfeng Cao, Liujia Yang, Ziqing Zhang, Zhicheng Fang, Qinying Gu, and Guang-Zhong Yang. 2023. Improving the robustness of analog deep neural networks through a Bayes-optimized noise injection approach. Communications Engineering 2, 1 (2023), 25.","journal-title":"Communications Engineering"},{"key":"e_1_3_2_257_2","first-page":"40694","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Yoon Eun Bi","year":"2023","unstructured":"Eun Bi Yoon, Keehun Park, Sungwoong Kim, and Sungbin Lim. 2023. Score-based generative models with L\u00e9vy processes. In Proceedings of the Advances in Neural Information Processing Systems. 40694\u201340707."},{"issue":"3","key":"e_1_3_2_258_2","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/MCI.2018.2840738","article-title":"Recent trends in deep learning based natural language processing","volume":"13","author":"Young Tom","year":"2018","unstructured":"Tom Young, Devamanyu Hazarika, Soujanya Poria, and Erik Cambria. 2018. Recent trends in deep learning based natural language processing. IEEE Computational Intelligence Magazine 13, 3 (2018), 55\u201375.","journal-title":"IEEE Computational Intelligence Magazine"},{"issue":"1","key":"e_1_3_2_259_2","first-page":"1","article-title":"The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"Yu Bing","year":"2018","unstructured":"Bing Yu et\u00a0al. 2018. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics 6, 1 (2018), 1\u201312.","journal-title":"Communications in Mathematics and Statistics"},{"key":"e_1_3_2_260_2","unstructured":"Hu Yu Jie Huang Kaiwen Zheng and Feng Zhao. 2023. High-quality image dehazing with diffusion model. arXiv:2308.11949. Retrieved from https:\/\/arxiv.org\/abs\/2308.11949"},{"key":"e_1_3_2_261_2","doi-asserted-by":"crossref","first-page":"114823","DOI":"10.1016\/j.cma.2022.114823","article-title":"Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems","volume":"393","author":"Yu Jeremy","year":"2022","unstructured":"Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis. 2022. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Computer Methods in Applied Mechanics and Engineering 393 (2022), 114823.","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"e_1_3_2_262_2","first-page":"16010","volume-title":"Proceedings of the IEEE\/CVF International Conference on Computer Vision","author":"Yuan Ye","year":"2023","unstructured":"Ye Yuan, Jiaming Song, Umar Iqbal, Arash Vahdat, and Jan Kautz. 2023. Physdiff: Physics-guided human motion diffusion model. In Proceedings of the IEEE\/CVF International Conference on Computer Vision. 16010\u201316021."},{"key":"e_1_3_2_263_2","volume-title":"Proceedings of the British Machine Vision Conference","author":"Zagoruyko Sergey","year":"2016","unstructured":"Sergey Zagoruyko and Nikos Komodakis. 2016. Wide residual networks. In Proceedings of the British Machine Vision Conference."},{"key":"e_1_3_2_264_2","doi-asserted-by":"crossref","first-page":"109409","DOI":"10.1016\/j.jcp.2020.109409","article-title":"Weak adversarial networks for high-dimensional partial differential equations","volume":"411","author":"Zang Yaohua","year":"2020","unstructured":"Yaohua Zang, Gang Bao, Xiaojing Ye, and Haomin Zhou. 2020. Weak adversarial networks for high-dimensional partial differential equations. Journal of Computational Physics 411 (2020), 109409.","journal-title":"Journal of Computational Physics"},{"key":"e_1_3_2_265_2","first-page":"22416","volume-title":"Proceedings of the AAAI Conference on Artificial Intelligence","author":"Zhang Hong","year":"2025","unstructured":"Hong Zhang, Ying Liu, and Romit Maulik. 2025. Semi-implicit neural ordinary differential equations. In Proceedings of the AAAI Conference on Artificial Intelligence. 22416\u201322424."},{"key":"e_1_3_2_266_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Zhang Jianxin","year":"2024","unstructured":"Jianxin Zhang, Josh Viktorov, Doosan Jung, and Emily Pitler. 2024. Efficient training of neural stochastic differential equations by matching finite dimensional distributions. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_267_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Zhang Jianxin","year":"2025","unstructured":"Jianxin Zhang, Josh Viktorov, Doosan Jung, and Emily Pitler. 2025. Efficient training of neural stochastic differential equations by matching finite dimensional distributions. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_268_2","doi-asserted-by":"crossref","first-page":"9872","DOI":"10.1109\/ACCESS.2018.2890127","article-title":"Multiple feature reweight densenet for image classification","volume":"7","author":"Zhang Ke","year":"2019","unstructured":"Ke Zhang, Yurong Guo, Xinsheng Wang, Jinsha Yuan, and Qiaolin Ding. 2019. Multiple feature reweight densenet for image classification. IEEE Access 7 (2019), 9872\u20139880.","journal-title":"IEEE Access"},{"key":"e_1_3_2_269_2","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2017.2662206"},{"issue":"3","key":"e_1_3_2_270_2","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1007\/s10851-019-00922-y","article-title":"Forward stability of ResNet and its variants","volume":"62","author":"Zhang Linan","year":"2020","unstructured":"Linan Zhang and Hayden Schaeffer. 2020. Forward stability of ResNet and its variants. Journal of Mathematical Imaging and Vision 62, 3 (2020), 328\u2013351.","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"e_1_3_2_271_2","first-page":"16280","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Zhang Qinsheng","year":"2021","unstructured":"Qinsheng Zhang and Yongxin Chen. 2021. Diffusion normalizing flow. In Proceedings of the Advances in Neural Information Processing Systems. 16280\u201316291."},{"key":"e_1_3_2_272_2","volume-title":"Proceedings of the NeurIPS 2022 Workshop on Score-Based Methods","author":"Zhang Qinsheng","year":"2022","unstructured":"Qinsheng Zhang and Yongxin Chen. 2022. Fast sampling of diffusion models with exponential integrator. In Proceedings of the NeurIPS 2022 Workshop on Score-Based Methods."},{"key":"e_1_3_2_273_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Zhang Qinsheng","year":"2023","unstructured":"Qinsheng Zhang, Molei Tao, and Yongxin Chen. 2023. gDDIM: Generalized denoising diffusion implicit models. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_274_2","first-page":"718","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Zhang Xingcheng","year":"2017","unstructured":"Xingcheng Zhang, Zhizhong Li, Chen Change Loy, and Dahua Lin. 2017. Polynet: A pursuit of structural diversity in very deep networks. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 718\u2013726."},{"key":"e_1_3_2_275_2","unstructured":"Yichi Zhang Yici Yan Alex Schwing and Zhizhen Zhao. 2025. Hierarchical rectified flow matching with mini-batch couplings. arXiv:2507.13350. Retrieved from https:\/\/arxiv.org\/abs\/2507.13350"},{"key":"e_1_3_2_276_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Zhang Yichi","year":"2025","unstructured":"Yichi Zhang, Yici Yan, Alex Schwing, and Zhizhen Zhao. 2025. Towards hierarchical rectified flow. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_277_2","unstructured":"Hongjue Zhao Yizhuo Chen Yuchen Wang Hairong Qi Lui Sha Tarek Abdelzaher and Huajie Shao. 2026. Understanding the theoretical foundations of deep neural networks through differential equations. arXiv:2603.18331. Retrieved from https:\/\/arxiv.org\/abs\/2603.18331"},{"key":"e_1_3_2_278_2","volume-title":"Proceedings of the International Conference on Learning Representations","author":"Zhao Hongjue","year":"2025","unstructured":"Hongjue Zhao, Yuchen Wang, Hairong Qi, Zijie Huang, Han Zhao, Lui Sha, and Huajie Shao. 2025. Accelerating neural ODEs: A variational formulation-based approach. In Proceedings of the International Conference on Learning Representations."},{"key":"e_1_3_2_279_2","first-page":"55502","volume-title":"Proceedings of the Advances in Neural Information Processing Systems","author":"Zheng Kaiwen","year":"2023","unstructured":"Kaiwen Zheng, Cheng Lu, Jianfei Chen, and Jun Zhu. 2023. Dpm-solver-v3: Improved diffusion ode solver with empirical model statistics. In Proceedings of the Advances in Neural Information Processing Systems. 55502\u201355542."},{"key":"e_1_3_2_280_2","first-page":"366","volume-title":"Proceedings of the 2022 IEEE International Conference on Image Processing","author":"Zheng Wenqing","year":"2022","unstructured":"Wenqing Zheng, Jiyang Xie, Xian Sun, and Zhanyu Ma. 2022. Structured dropconnect for uncertainty inference in image classification. In Proceedings of the 2022 IEEE International Conference on Image Processing. IEEE, 366\u2013370."},{"key":"e_1_3_2_281_2","doi-asserted-by":"crossref","first-page":"1971","DOI":"10.18653\/v1\/2020.findings-emnlp.178","volume-title":"Findings of the Association for Computational Linguistics: EMNLP 2020","author":"Zhou Wangchunshu","year":"2020","unstructured":"Wangchunshu Zhou, Tao Ge, Furu Wei, Ming Zhou, and Ke Xu. 2020. Scheduled DropHead: A regularization method for transformer models. In Findings of the Association for Computational Linguistics: EMNLP 2020. 1971\u20131980."},{"key":"e_1_3_2_282_2","first-page":"7777","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Zhou Zhenyu","year":"2024","unstructured":"Zhenyu Zhou, Defang Chen, Can Wang, and Chun Chen. 2024. Fast ode-based sampling for diffusion models in around 5 steps. In Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition. 7777\u20137786."},{"issue":"2","key":"e_1_3_2_283_2","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.1007\/s00521-022-07785-2","article-title":"Convolutional neural networks combined with runge\u2013Kutta methods","volume":"35","author":"Zhu Mai","year":"2023","unstructured":"Mai Zhu, Bo Chang, and Chong Fu. 2023. Convolutional neural networks combined with runge\u2013Kutta methods. Neural Computing and Applications 35, 2 (2023), 1629\u20131643.","journal-title":"Neural Computing and Applications"},{"key":"e_1_3_2_284_2","first-page":"342","volume-title":"Proceedings of the European Conference on Computer Vision","author":"Zhu Yuanzhi","year":"2024","unstructured":"Yuanzhi Zhu, Xingchao Liu, and Qiang Liu. 2024. SlimFlow: Training smaller one-step diffusion models with rectified flow. In Proceedings of the European Conference on Computer Vision. 342\u2013359."},{"key":"e_1_3_2_285_2","first-page":"13","volume-title":"Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition","author":"Zhu Yixuan","year":"2024","unstructured":"Yixuan Zhu, Wenliang Zhao, Ao Li, Yansong Tang, Jie Zhou, and Jiwen Lu. 2024. FlowIE: Efficient image enhancement via rectified flow. 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