{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/hamiltonian-deep-neural-networks-guaranteeing","title":"Hamiltonian Deep Neural Networks Guaranteeing Non-vanishing Gradients by Design","arxiv_id":"2105.13205","date":"2021-05-27","proceeding":null,"authors":["Clara Lucía Galimberti","Luca Furieri","Liang Xu","Giancarlo Ferrari-Trecate"],"abstract":"Deep Neural Networks (DNNs) training can be difficult due to vanishing and exploding gradients during weight optimization through backpropagation. To address this problem, we propose a general class of Hamiltonian DNNs (H-DNNs) that stem from the discretization of continuous-time Hamiltonian systems and include several existing DNN architectures based on ordinary differential equations. Our main result is that a broad set of H-DNNs ensures non-vanishing gradients by design for an arbitrary network depth. This is obtained by proving that, using a semi-implicit Euler discretization scheme, the backward sensitivity matrices involved in gradient computations are symplectic. We also provide an upper-bound to the magnitude of sensitivity matrices and show that exploding gradients can be controlled through regularization. Finally, we enable distributed implementations of backward and forward propagation algorithms in H-DNNs by characterizing appropriate sparsity constraints on the weight matrices. The good performance of H-DNNs is demonstrated on benchmark classification problems, including image classification with the MNIST dataset.","url_abs":"https://arxiv.org/abs/2105.13205v2","url_pdf":"https://arxiv.org/pdf/2105.13205v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"hamiltonian-deep-neural-networks-guaranteeing","repo_url":"https://github.com/DecodEPFL/HamiltonianNet","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"hamiltonian-deep-neural-networks-guaranteeing","repo_url":"https://github.com/ClaraGalimberti/HamiltonianNet","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}},{"paper_slug":"hamiltonian-deep-neural-networks-guaranteeing","repo_url":"https://github.com/decodepfl/deepdiscoph","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[{"task_slug":"image-classification","task_name":"Image Classification"},{"task_slug":"sensitivity","task_name":"Sensitivity"},{"task_slug":"image-classification","task_name":"image-classification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2105.13205","atlas_url":"https://app.syntology.ai/?focus=2105.13205","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2105.13205"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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