Uniqueness in Deep Neural Networks: The Inevitable Singularity of Learning Trajectories

Uniqueness in Deep Neural Networks: The Inevitable Singularity of Learning Trajectories | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 4 February 2026 V1 Latest version Share on Uniqueness in Deep Neural Networks: The Inevitable Singularity of Learning Trajectories Author : Haamed Ghiassian 0009-0003-3279-5709 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177023111.18939115/v1 99 views 52 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We establish fundamental physical and mathematical constraints in learning trajectory on parameter-level reproducibility in deep neural networks. Using dynamical systems theory, information thermodynamics, and high-dimensional geometry, we prove three impossibility theorems regarding exact structural replication of trained networks. Specifically, we demonstrate that: (i) Gradient-based training dynamics inherently exhibit positive Lyapunov exponents, leading to exponential sensitivity to initial conditions; (ii) The learning process constitutes a thermodynamically irreversible non-equilibrium process with strictly positive entropy production; (iii) In the learning process, different training trajectories occupy distinct topological equivalence classes in parameter space, as measured by persistent homology invariants. These complementary constraints collectively imply that each trained neural network traverses a structurally unique path and states with probability one, establishing inherent limits to exact reproducibility in deep learning and necessitating a paradigm shift from exact replication to distributional reproducibility and state uniqueness in machine learning science. Supplementary Material File (uniqueness_in_deep_neural_networks__the_inevitable_singularity_of_learning_trajectories.pdf) Download 524.42 KB Information & Authors Information Version history V1 Version 1 04 February 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords chaotic dynamics deep neural networks information theory learning trajectories physics of learning topology uniqueness Authors Affiliations Haamed Ghiassian 0009-0003-3279-5709 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 99 views 52 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Haamed Ghiassian. Uniqueness in Deep Neural Networks: The Inevitable Singularity of Learning Trajectories. Authorea . 04 February 2026. DOI: https://doi.org/10.22541/au.177023111.18939115/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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